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2401.09350 | 351 | R. Guo, P. Sun, E. Lindgren, Q. Geng, D. Simcha, F. Chern, and S. Kumar. Accelerating large-scale inference with anisotropic vector quantization. In Proceedings of the 37th International Conference on Machine Learning, 2020.
Y. K. Jang and N. I. Cho. Self-supervised product quantization for deep un- supervised image retrieval. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 12085â12094, October 2021.
H. J´egou, M. Douze, and C. Schmid. Product quantization for nearest neigh- bor search. IEEE Transactions on Pattern Analysis and Machine Intelli- gence, 33(1):117â128, 2011.
J. Johnson, M. Douze, and H. J´egou. Billion-scale similarity search with gpus. IEEE Transactions on Big Data, 7(3):535â547, 2021.
Y. Kalantidis and Y. Avrithis. Locally optimized product quantization for approximate nearest neighbor search. In 2014 IEEE Conference on Com- puter Vision and Pattern Recognition, pages 2329â2336, 2014. | 2401.09350#351 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 351,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "R. Guo, P. Sun, E. Lindgren, Q. Geng, D. Simcha, F. Chern, and S. Kumar. Accelerating large-scale inference with anisotropic vector quantization. In Proceedings of the 37th International Conference on Machine Learning, 2020.\nY. K. Jang and N. I. Cho. Self-supervised product quantization for deep un- supervised image retrieval. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 12085â12094, October 2021.\nH. J´egou, M. Douze, and C. Schmid. Product quantization for nearest neigh- bor search. IEEE Transactions on Pattern Analysis and Machine Intelli- gence, 33(1):117â128, 2011.\nJ. Johnson, M. Douze, and H. J´egou. Billion-scale similarity search with gpus. IEEE Transactions on Big Data, 7(3):535â547, 2021.\nY. Kalantidis and Y. Avrithis. Locally optimized product quantization for approximate nearest neighbor search. In 2014 IEEE Conference on Com- puter Vision and Pattern Recognition, pages 2329â2336, 2014.",
"title": "Foundations of Vector Retrieval",
"year": 2024
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] |
2401.09350 | 352 | B. Klein and L. Wolf. End-to-end supervised product quantization for im- age search and retrieval. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 6 2019.
A. Krishnan and E. Liberty. Projective clustering product quantization, 2021. M. Liu, Y. Dai, Y. Bai, and L.-Y. Duan. Deep product quantization module for efficient image retrieval. In IEEE International Conference on Acous- tics, Speech and Signal Processing, pages 4382â4386, 2020.
S. Liu, H. Lu, and J. Shao. Improved residual vector quantization for high- dimensional approximate nearest neighbor search, 2015.
Z. Lu, D. Lian, J. Zhang, Z. Zhang, C. Feng, H. Wang, and E. Chen. Dif- ferentiable optimized product quantization and beyond. In Proceedings of the ACM Web Conference 2023, pages 3353â3363, 2023.
Y. Matsui, Y. Uchida, H. J´egou, and S. Satoh. A survey of product quan- tization. ITE Transactions on Media Technology and Applications, 6(1): 2â10, 2018.
141
142
9 Quantization | 2401.09350#352 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 352,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "B. Klein and L. Wolf. End-to-end supervised product quantization for im- age search and retrieval. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 6 2019.\nA. Krishnan and E. Liberty. Projective clustering product quantization, 2021. M. Liu, Y. Dai, Y. Bai, and L.-Y. Duan. Deep product quantization module for efficient image retrieval. In IEEE International Conference on Acous- tics, Speech and Signal Processing, pages 4382â4386, 2020.\nS. Liu, H. Lu, and J. Shao. Improved residual vector quantization for high- dimensional approximate nearest neighbor search, 2015.\nZ. Lu, D. Lian, J. Zhang, Z. Zhang, C. Feng, H. Wang, and E. Chen. Dif- ferentiable optimized product quantization and beyond. In Proceedings of the ACM Web Conference 2023, pages 3353â3363, 2023.\nY. Matsui, Y. Uchida, H. J´egou, and S. Satoh. A survey of product quan- tization. ITE Transactions on Media Technology and Applications, 6(1): 2â10, 2018.\n141\n142\n9 Quantization",
"title": "Foundations of Vector Retrieval",
"year": 2024
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2401.09350 | 353 | 141
142
9 Quantization
L. Niu, Z. Xu, L. Zhao, D. He, J. Ji, X. Yuan, and M. Xue. Residual vec- tor product quantization for approximate nearest neighbor search. Expert Systems with Applications, 232(C), 12 2023.
M. Norouzi and D. J. Fleet. Cartesian k-means. In Proceedings of the 2013 IEEE Conference on Computer Vision and Pattern Recognition, pages 3017â3024, 2013.
E. C. Ozan, S. Kiranyaz, and M. Gabbouj. Competitive quantization for approximate nearest neighbor search. IEEE Transactions on Knowledge and Data Engineering, 28(11):2884â2894, 2016.
P. Sun, R. Guo, and S. Kumar. Automating nearest neighbor search config- uration with constrained optimization. In Proceedings of the 11th Interna- tional Conference on Learning Representations, 2023.
X. Wu, R. Guo, A. T. Suresh, S. Kumar, D. N. Holtmann-Rice, D. Simcha, and F. Yu. Multiscale quantization for fast similarity search. In Advances in Neural Information Processing Systems, volume 30, 2017. | 2401.09350#353 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 353,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "141\n142\n9 Quantization\nL. Niu, Z. Xu, L. Zhao, D. He, J. Ji, X. Yuan, and M. Xue. Residual vec- tor product quantization for approximate nearest neighbor search. Expert Systems with Applications, 232(C), 12 2023.\nM. Norouzi and D. J. Fleet. Cartesian k-means. In Proceedings of the 2013 IEEE Conference on Computer Vision and Pattern Recognition, pages 3017â3024, 2013.\nE. C. Ozan, S. Kiranyaz, and M. Gabbouj. Competitive quantization for approximate nearest neighbor search. IEEE Transactions on Knowledge and Data Engineering, 28(11):2884â2894, 2016.\nP. Sun, R. Guo, and S. Kumar. Automating nearest neighbor search config- uration with constrained optimization. In Proceedings of the 11th Interna- tional Conference on Learning Representations, 2023.\nX. Wu, R. Guo, A. T. Suresh, S. Kumar, D. N. Holtmann-Rice, D. Simcha, and F. Yu. Multiscale quantization for fast similarity search. In Advances in Neural Information Processing Systems, volume 30, 2017.",
"title": "Foundations of Vector Retrieval",
"year": 2024
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] |
2401.09350 | 354 | D. Xu, I. W. Tsang, and Y. Zhang. Online product quantization. IEEE Trans- actions on Knowledge and Data Engineering, 30(11):2185â2198, 2018. T. Yu, J. Yuan, C. Fang, and H. Jin. Product quantization network for fast image retrieval. In Proceedings of the European Conference on Computer Vision, September 2018.
J. Zhang, Q. Liu, D. Lian, Z. Liu, L. Wu, and E. Chen. Anisotropic addi- tive quantization for fast inner product search. Proceedings of the AAAI Conference on Artificial Intelligence, 36(4):4354â4362, 6 2022.
J. Zhang, D. Lian, H. Zhang, B. Wang, and E. Chen. Query-aware quantiza- tion for maximum inner product search. In Proceedings of the 37th AAAI Conference on Artificial Intelligence, 2023.
# Chapter 10 Sketching | 2401.09350#354 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 354,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "D. Xu, I. W. Tsang, and Y. Zhang. Online product quantization. IEEE Trans- actions on Knowledge and Data Engineering, 30(11):2185â2198, 2018. T. Yu, J. Yuan, C. Fang, and H. Jin. Product quantization network for fast image retrieval. In Proceedings of the European Conference on Computer Vision, September 2018.\nJ. Zhang, Q. Liu, D. Lian, Z. Liu, L. Wu, and E. Chen. Anisotropic addi- tive quantization for fast inner product search. Proceedings of the AAAI Conference on Artificial Intelligence, 36(4):4354â4362, 6 2022.\nJ. Zhang, D. Lian, H. Zhang, B. Wang, and E. Chen. Query-aware quantiza- tion for maximum inner product search. In Proceedings of the 37th AAAI Conference on Artificial Intelligence, 2023.\n# Chapter 10 Sketching",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 355 | # Chapter 10 Sketching
Abstract Sketching is a probabilistic tool to summarize high-dimensional vectors into low-dimensional vectors, called sketches, while approximately preserving properties of interest. For example, we may sketch vectors in the Euclidean space such that their L2 norm is approximately preserved; or sketch points in an inner product space such that the inner product between any two points is maintained with high probability. This chapter reviews a few data-oblivious algorithms, cherry-picked from the vast literature on sketching, that are tailored to sparse vectors in an inner product space.
# 10.1 Intuition
We learnt about quantization as a form of vector compression in Chapter 9. There, vectors are decomposed into L subspaces, with each subspace mapped to C geometrically-cohesive buckets. By coding each subspace into only C values, we can encode an entire vector in L log C bits, often dramatically re- ducing the size of a vector collection, though at the cost of losing information in the process.
The challenge, we also learnt, is that not enough can be said about the effects of L, C, and other parameters involved in the process of quantization, on the reconstruction error. We can certainly intuit the asymptotic behavior of quantization, but that is neither interesting nor insightful. That leaves us no option other than settling on a configuration empirically. | 2401.09350#355 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 355,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# Chapter 10 Sketching\nAbstract Sketching is a probabilistic tool to summarize high-dimensional vectors into low-dimensional vectors, called sketches, while approximately preserving properties of interest. For example, we may sketch vectors in the Euclidean space such that their L2 norm is approximately preserved; or sketch points in an inner product space such that the inner product between any two points is maintained with high probability. This chapter reviews a few data-oblivious algorithms, cherry-picked from the vast literature on sketching, that are tailored to sparse vectors in an inner product space.\n# 10.1 Intuition\nWe learnt about quantization as a form of vector compression in Chapter 9. There, vectors are decomposed into L subspaces, with each subspace mapped to C geometrically-cohesive buckets. By coding each subspace into only C values, we can encode an entire vector in L log C bits, often dramatically re- ducing the size of a vector collection, though at the cost of losing information in the process.\nThe challenge, we also learnt, is that not enough can be said about the effects of L, C, and other parameters involved in the process of quantization, on the reconstruction error. We can certainly intuit the asymptotic behavior of quantization, but that is neither interesting nor insightful. That leaves us no option other than settling on a configuration empirically.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 356 | Additionally, learning codebooks can become involved and cumbersome. It involves tuning parameters and running clustering algorithms, whose ex- pected behavior is itself ill-understood when handling improper distance func- tions. The resulting codebooks too may become obsolete in the event of a distributional shift.
143
144
10 Sketching
This chapter reviews a different class of compression techniques known as data-oblivious sketching. Let us break down this phrase and understand each part better.
The data-oblivious qualifier is rather self-explanatory: We make no as- sumptions about the input data, and in fact, do not even take advantage of the statistical properties of the data. We are, in other words, completely agnostic and oblivious to our input.
While oblivion may put us at a disadvantage and lead to a larger mag- nitude of error, it creates two opportunities. First, we can often easily quantify the average qualities of the resulting compressed vectors. Sec- ond, by design, the compressed vectors are robust under any data drift. Once a vector collection has been compressed, in other words, we can safely assume that any guarantees we were promised will continue to hold. | 2401.09350#356 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 356,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Additionally, learning codebooks can become involved and cumbersome. It involves tuning parameters and running clustering algorithms, whose ex- pected behavior is itself ill-understood when handling improper distance func- tions. The resulting codebooks too may become obsolete in the event of a distributional shift.\n143\n144\n10 Sketching\nThis chapter reviews a different class of compression techniques known as data-oblivious sketching. Let us break down this phrase and understand each part better.\nThe data-oblivious qualifier is rather self-explanatory: We make no as- sumptions about the input data, and in fact, do not even take advantage of the statistical properties of the data. We are, in other words, completely agnostic and oblivious to our input.\nWhile oblivion may put us at a disadvantage and lead to a larger mag- nitude of error, it creates two opportunities. First, we can often easily quantify the average qualities of the resulting compressed vectors. Sec- ond, by design, the compressed vectors are robust under any data drift. Once a vector collection has been compressed, in other words, we can safely assume that any guarantees we were promised will continue to hold.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 357 | Sketching, to continue our unpacking of the concept, is a probabilistic tool to reduce the dimensionality of a vector space while preserving certain properties of interest with high probability. In its simplest form, sketching is a function Ï : Rd â Rd⦠, where d⦠< d. If the âproperty of interestâ is the Euclidean distance between any pair of points in a collection X , for instance, then Ï(·) must satisfy the following for random points U and V :
p [lise = 6(V)|l2 = |W = Va] < | >1-6,
for δ, ϵ â (0, 1).
The output of Ï(u), which we call the sketch of vector u, is a good sub- stitute for u itself. If all we care about, as we do in top-k retrieval, is the distance between pairs of points, then we retain the ability to deduce that information with high probability just from the sketches of a collection of vectors. Considering that d⦠is smaller than d, we not only compress the col- lection through sketching, but, as with quantization, we are able to perform distance computations directly on the compressed vectors. | 2401.09350#357 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 357,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Sketching, to continue our unpacking of the concept, is a probabilistic tool to reduce the dimensionality of a vector space while preserving certain properties of interest with high probability. In its simplest form, sketching is a function Ï : Rd â Rd⦠, where d⦠< d. If the âproperty of interestâ is the Euclidean distance between any pair of points in a collection X , for instance, then Ï(·) must satisfy the following for random points U and V :\np [lise = 6(V)|l2 = |W = Va] < | >1-6,\nfor δ, ϵ â (0, 1).\nThe output of Ï(u), which we call the sketch of vector u, is a good sub- stitute for u itself. If all we care about, as we do in top-k retrieval, is the distance between pairs of points, then we retain the ability to deduce that information with high probability just from the sketches of a collection of vectors. Considering that d⦠is smaller than d, we not only compress the col- lection through sketching, but, as with quantization, we are able to perform distance computations directly on the compressed vectors.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 358 | The literature on sketching offers numerous algorithms that are designed to approximate a wide array of norms, distances, and other properties of data. We refer the reader to the excellent monograph by Woodruff [2014] for a tour of this rich area of research. But to give the reader a better understanding of the connection between sketching and top-k retrieval, we use the remainder of this chapter to delve into three algorithms. To make things more interesting, we specifically review these algorithms in the context of inner product for sparse vectors.
10.2 Linear Sketching with the JL Transform
The first is the quintessential linear algorithm due to Johnson and Linden- strauss [1984]. It is linear in the sense that Ï is simply a linear transformation, so that Ï(u) = Φu for some (random) matrix Φ â Rdâ¦Ãd. We will learn how to construct the required matrix and discuss what guarantees it has to offer. We then move to two sketching algorithms [Bruch et al., 2023] and [Daliri et al., 2023] whose output space is not Euclidean. Instead, the sketch of a vec- tor is a data structure, equipped with a distance function that approximates the inner product between vectors in the original space.
# 10.2 Linear Sketching with the JL Transform | 2401.09350#358 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 358,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The literature on sketching offers numerous algorithms that are designed to approximate a wide array of norms, distances, and other properties of data. We refer the reader to the excellent monograph by Woodruff [2014] for a tour of this rich area of research. But to give the reader a better understanding of the connection between sketching and top-k retrieval, we use the remainder of this chapter to delve into three algorithms. To make things more interesting, we specifically review these algorithms in the context of inner product for sparse vectors.\n10.2 Linear Sketching with the JL Transform\nThe first is the quintessential linear algorithm due to Johnson and Linden- strauss [1984]. It is linear in the sense that Ï is simply a linear transformation, so that Ï(u) = Φu for some (random) matrix Φ â Rdâ¦Ãd. We will learn how to construct the required matrix and discuss what guarantees it has to offer. We then move to two sketching algorithms [Bruch et al., 2023] and [Daliri et al., 2023] whose output space is not Euclidean. Instead, the sketch of a vec- tor is a data structure, equipped with a distance function that approximates the inner product between vectors in the original space.\n# 10.2 Linear Sketching with the JL Transform",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 359 | # 10.2 Linear Sketching with the JL Transform
Let us begin by repeating the well-known result due to Johnson and Linden- strauss [1984], which we refer to as the JL Lemma:
Lemma 10.1 For ϵ â (0, 1) and any set X of m points in Rd, and an integer d⦠= â¦(ϵâ2 ln m), there exists a Lipschitz mapping Ï : Rd â Rd⦠such that
(1 â ϵ)â¥u â vâ¥2 2 ⤠â¥Ï(u) â Ï(v)â¥2 2 ⤠(1 + ϵ)â¥u â vâ¥2 2,
for all u, v â X .
This result has been studied extensively and further developed since its introduction. Using simple proofs, for example, it can be shown that the mapping Ï may be a linear transformation by a d⦠à d random matrix Φ drawn from a particular class of distributions. Such a matrix Φ is said to form a JL transform. | 2401.09350#359 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 359,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 10.2 Linear Sketching with the JL Transform\nLet us begin by repeating the well-known result due to Johnson and Linden- strauss [1984], which we refer to as the JL Lemma:\nLemma 10.1 For ϵ â (0, 1) and any set X of m points in Rd, and an integer d⦠= â¦(ϵâ2 ln m), there exists a Lipschitz mapping Ï : Rd â Rd⦠such that\n(1 â ϵ)â¥u â vâ¥2 2 ⤠â¥Ï(u) â Ï(v)â¥2 2 ⤠(1 + ϵ)â¥u â vâ¥2 2,\nfor all u, v â X .\nThis result has been studied extensively and further developed since its introduction. Using simple proofs, for example, it can be shown that the mapping Ï may be a linear transformation by a d⦠à d random matrix Φ drawn from a particular class of distributions. Such a matrix Φ is said to form a JL transform.",
"title": "Foundations of Vector Retrieval",
"year": 2024
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2401.09350 | 360 | Definition 10.1 A random matrix Φ â Rdâ¦Ãd forms a Johnson-Lindenstrauss transform with parameters (ϵ, δ, m), if with probability at least 1 â δ, for any m-element subset X â Rd, for all u, v â X it holds that |â¨Î¦u, Φvâ© â â¨u, vâ©| ⤠ϵâ¥uâ¥2â¥vâ¥2.
There are many constructions of Φ that form a JL transform. It is trivial to show that when the entries of Φ are independently drawn from N (0, 1 ), d⦠then Φ is a JL transform with parameters (ϵ, δ, m) if d⦠= â¦(ϵâ2 ln(m/δ)). In yet another construction, Φ = 1â R, where R â {±1}dâ¦Ãd is a matrix whose d⦠entries are independent Rademacher random variables.
We take the latter as an example due to its simplicity and analyze its properties. As before, we refer the reader to [Woodruff, 2014] for a far more detailed discussion of other (more efficient) constructions of the JL transform.
145
146 10 Sketching
# 10.2.1 Theoretical Analysis | 2401.09350#360 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 360,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Definition 10.1 A random matrix Φ â Rdâ¦Ãd forms a Johnson-Lindenstrauss transform with parameters (ϵ, δ, m), if with probability at least 1 â δ, for any m-element subset X â Rd, for all u, v â X it holds that |â¨Î¦u, Φvâ© â â¨u, vâ©| ⤠ϵâ¥uâ¥2â¥vâ¥2.\nThere are many constructions of Φ that form a JL transform. It is trivial to show that when the entries of Φ are independently drawn from N (0, 1 ), d⦠then Φ is a JL transform with parameters (ϵ, δ, m) if d⦠= â¦(ϵâ2 ln(m/δ)). In yet another construction, Φ = 1â R, where R â {±1}dâ¦Ãd is a matrix whose d⦠entries are independent Rademacher random variables.\nWe take the latter as an example due to its simplicity and analyze its properties. As before, we refer the reader to [Woodruff, 2014] for a far more detailed discussion of other (more efficient) constructions of the JL transform.\n145\n146 10 Sketching\n# 10.2.1 Theoretical Analysis",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 361 | 145
146 10 Sketching
# 10.2.1 Theoretical Analysis
We are interested in analyzing the transformation above in the context of inner product. Specifically, we wish to understand what we should expect if, instead of computing the inner product between two vectors u and v in Rd, we perform the operation â¨Ru, Rvâ© in the transformed space in Rd⦠. Is the outcome an unbiased estimate of the true inner product? How far off may this estimate be? The following result is a first step to answering these questions for two fixed vectors.
Theorem 10.1 Fix two vectors u and v â Rd. Define ZSketch = â¨Ï(u), Ï(v)â© as the random variable representing the inner product of sketches of size dâ¦, dâ¦}dâ¦Ãd being a prepared using the projection Ï(u) = Ru, with R â {±1/ random Rademacher matrix. ZSketch is an unbiased estimator of â¨u, vâ©. Its distribution tends to a Gaussian with variance:
1 = (\lulBliolld + (u,v) ay ei 2). | 2401.09350#361 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 361,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "145\n146 10 Sketching\n# 10.2.1 Theoretical Analysis\nWe are interested in analyzing the transformation above in the context of inner product. Specifically, we wish to understand what we should expect if, instead of computing the inner product between two vectors u and v in Rd, we perform the operation â¨Ru, Rvâ© in the transformed space in Rd⦠. Is the outcome an unbiased estimate of the true inner product? How far off may this estimate be? The following result is a first step to answering these questions for two fixed vectors.\nTheorem 10.1 Fix two vectors u and v â Rd. Define ZSketch = â¨Ï(u), Ï(v)â© as the random variable representing the inner product of sketches of size dâ¦, dâ¦}dâ¦Ãd being a prepared using the projection Ï(u) = Ru, with R â {±1/ random Rademacher matrix. ZSketch is an unbiased estimator of â¨u, vâ©. Its distribution tends to a Gaussian with variance:\n1 = (\\lulBliolld + (u,v) ay ei 2).",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 362 | 1 = (\lulBliolld + (u,v) ay ei 2).
Proof. Consider the random variable Z = ( y Rjuj)( Ye Reve), where R;,âs are Rademacher random wih oe 7 is clear that d.Z is the product of the sketch coordinate 7 (for any i): 6(u):6(v);# vale.
We can expand the expected value of Z as follows:
E[Z]=E [( x Ryuy)( » Rive) = = R?uj;v;] + E[ > Rj Reujve] j#k = Ss uv; E[R?] + Ss uj vz E[R; Rx] nn oc non = (u,v).
The variance of Z can be expressed as follows:
Var[Z] = E[Z?] â E[Z)? = E{( Le him)! (22 Rev) |- (u,v).
We have the following:
10.2 Linear Sketching with the JL Transform | 2401.09350#362 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 362,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "1 = (\\lulBliolld + (u,v) ay ei 2).\nProof. Consider the random variable Z = ( y Rjuj)( Ye Reve), where R;,âs are Rademacher random wih oe 7 is clear that d.Z is the product of the sketch coordinate 7 (for any i): 6(u):6(v);# vale.\nWe can expand the expected value of Z as follows:\nE[Z]=E [( x Ryuy)( » Rive) = = R?uj;v;] + E[ > Rj Reujve] j#k = Ss uv; E[R?] + Ss uj vz E[R; Rx] nn oc non = (u,v).\nThe variance of Z can be expressed as follows:\nVar[Z] = E[Z?] â E[Z)? = E{( Le him)! (22 Rev) |- (u,v).\nWe have the following:\n10.2 Linear Sketching with the JL Transform",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 363 | We have the following:
10.2 Linear Sketching with the JL Transform
Eo Ryu) (SS Ravn)â | =E (> w+ Ss RiRjuiuy)(S> ut > ReRivew)| ij k kAl = |lull3|loll3 +E p> ur) ReRivevi] i kAl 0 +E D> uz Ss RR; uits| +E p> Ri Rjuyuj Ss ReRivwe| . (10.1) k idj Al kAl 0
The last term can be decomposed as follows:
B| Ss RiRjReRiasujreri| ipjAkAl +E [ Ss RAR ReRiaiujrevi| i=k jAIVixk, j=l ) RAR RR iujujrev4] . i#j,i=k,j=lViFj,i=lj=k & +
The first two terms are 0 and the last term can be rewritten as follows:
2E ~ uivi( >> UjV;j â uiv)| = 2(u,v)? â 2 Ss uzv?. (10.2) + J
We now substitute the last term in Equation (10.1) with Equation (10.2)
to obtain: | 2401.09350#363 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 363,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "We have the following:\n10.2 Linear Sketching with the JL Transform\nEo Ryu) (SS Ravn)â | =E (> w+ Ss RiRjuiuy)(S> ut > ReRivew)| ij k kAl = |lull3|loll3 +E p> ur) ReRivevi] i kAl 0 +E D> uz Ss RR; uits| +E p> Ri Rjuyuj Ss ReRivwe| . (10.1) k idj Al kAl 0\nThe last term can be decomposed as follows:\nB| Ss RiRjReRiasujreri| ipjAkAl +E [ Ss RAR ReRiaiujrevi| i=k jAIVixk, j=l ) RAR RR iujujrev4] . i#j,i=k,j=lViFj,i=lj=k & +\nThe first two terms are 0 and the last term can be rewritten as follows:\n2E ~ uivi( >> UjV;j â uiv)| = 2(u,v)? â 2 Ss uzv?. (10.2) + J\nWe now substitute the last term in Equation (10.1) with Equation (10.2)\nto obtain:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 364 | We now substitute the last term in Equation (10.1) with Equation (10.2)
to obtain:
Var[Z] = â¥uâ¥2 2â¥vâ¥2 2 + â¨u, vâ©2 â 2 i v2 u2 i . i
i Ï(u)iÏ(v)i is the sum of independent, identically distributed random variables. Furthermore, for bounded vectors u and v, the variance is finite. By the application of the Central Limit The- orem, we can deduce that the distribution of ZSketch tends to a Gaussian distribution with the stated expected value. Noting that Var[ZSketch] = 1/d2 ââ â¦
Theorem 10.1 gives a clear model of the inner product error when two fixed vectors are transformed using our particular choice of the JL trans- form. We learnt that inner product of sketches is an ubiased estimator of the inner product between vectors, and have shown that the error follows a Gaussian distribution.
147
(10.1)
148
10 Sketching
Let us now position this result in the context of top-k retrieval where the query point is fixed, but the data points are random. To make the analysis more interesting, let us consider sparse vectors, where each coordinate may be 0 with a non-zero probability. | 2401.09350#364 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 364,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "We now substitute the last term in Equation (10.1) with Equation (10.2)\nto obtain:\nVar[Z] = â¥uâ¥2 2â¥vâ¥2 2 + â¨u, vâ©2 â 2 i v2 u2 i . i\ni Ï(u)iÏ(v)i is the sum of independent, identically distributed random variables. Furthermore, for bounded vectors u and v, the variance is finite. By the application of the Central Limit The- orem, we can deduce that the distribution of ZSketch tends to a Gaussian distribution with the stated expected value. Noting that Var[ZSketch] = 1/d2 ââ â¦\nTheorem 10.1 gives a clear model of the inner product error when two fixed vectors are transformed using our particular choice of the JL trans- form. We learnt that inner product of sketches is an ubiased estimator of the inner product between vectors, and have shown that the error follows a Gaussian distribution.\n147\n(10.1)\n148\n10 Sketching\nLet us now position this result in the context of top-k retrieval where the query point is fixed, but the data points are random. To make the analysis more interesting, let us consider sparse vectors, where each coordinate may be 0 with a non-zero probability.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 365 | Theorem 10.2 Fix a query vector q â Rd and let X be a random vector drawn according to the following probabilistic model. Coordinate i, Xi, is non- zero with probability pi > 0 and, if it is non-zero, draws its value from a dis- tribution with mean µ and variance Ï2 < â. Then, ZSketch = â¨Ï(q), Ï(X)â©, i piqi and with Ï(u) = Ru and R â {±1/ variance:
=] (2+ 0°)(Ilal3 op: - Yaa?) + (Sari)? -y fanâ) . i
Proof. It is easy to see that:
E[Zsxercu] = » Gi E[Xi] = = wd Pia
As for the variance, we start from Theorem 10.1 and arrive at the following expression:
7, (VilBBXIB) +Bl@.X)"] 2 @EX2)), 03)
where the expectation is with respect to X. Let us consider the terms inside the parentheses one by one. The first term becomes:
Walls Ell [3] = llall3 SS ELX?] i = lall3 +07) Sop. a
The second term reduces to: | 2401.09350#365 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 365,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Theorem 10.2 Fix a query vector q â Rd and let X be a random vector drawn according to the following probabilistic model. Coordinate i, Xi, is non- zero with probability pi > 0 and, if it is non-zero, draws its value from a dis- tribution with mean µ and variance Ï2 < â. Then, ZSketch = â¨Ï(q), Ï(X)â©, i piqi and with Ï(u) = Ru and R â {±1/ variance:\n=] (2+ 0°)(Ilal3 op: - Yaa?) + (Sari)? -y fanâ) . i\nProof. It is easy to see that:\nE[Zsxercu] = » Gi E[Xi] = = wd Pia\nAs for the variance, we start from Theorem 10.1 and arrive at the following expression:\n7, (VilBBXIB) +Bl@.X)\"] 2 @EX2)), 03)\nwhere the expectation is with respect to X. Let us consider the terms inside the parentheses one by one. The first term becomes:\nWalls Ell [3] = llall3 SS ELX?] i = lall3 +07) Sop. a\nThe second term reduces to:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 367 | Putting all these terms back into Equation (10.3) yields the desired ex- ââ
Let us consider a special case to better grasp the implications of Theo- rem 10.2. Suppose p; b/d for some constant ~ for all dimensions i. Further assume, without loss of generality, that the (fixed) query vector has uni norm: ||q||z2 = 1. We can observe that the variance of Zgxwrcn decomposes into a term that is (u? + 07)(1 â 1/d)q/do, and a second term that is a function of 1/d?. The mean, on the other hand, is a linear function of the non-zero coordinates in the query: (>>; qi)~/d. As d grows, the mean o: Zsxnron tends to 0 at a rate proportional to the sparsity rate (w/d), while its variance tends to (1? + 0â)/do. | 2401.09350#367 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 367,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Putting all these terms back into Equation (10.3) yields the desired ex- ââ\nLet us consider a special case to better grasp the implications of Theo- rem 10.2. Suppose p; b/d for some constant ~ for all dimensions i. Further assume, without loss of generality, that the (fixed) query vector has uni norm: ||q||z2 = 1. We can observe that the variance of Zgxwrcn decomposes into a term that is (u? + 07)(1 â 1/d)q/do, and a second term that is a function of 1/d?. The mean, on the other hand, is a linear function of the non-zero coordinates in the query: (>>; qi)~/d. As d grows, the mean o: Zsxnron tends to 0 at a rate proportional to the sparsity rate (w/d), while its variance tends to (1? + 0â)/do.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 368 | The above suggests that the ability of Ï(·) to preserve the inner product of a query point with a randomly drawn data point deteriorates as a function of the number of non-zero coordinates. For example, when the number of non-zero coordinates becomes larger, â¨Ï(q), Ï(X)â© for a fixed query q and a random point X becomes less reliable because the variance of the approxi- mation increases.
# 10.3 Asymmetric Sketching
Our second sketching algorithm is due to Bruch et al. [2023]. It is unusual in several ways. First, it is designed specifically for retrieval. That is, the objective of the sketching technique is not to preserve the inner product between points in a collection; in fact, as we will learn shortly, the sketch is not even an unbiased estimator. Instead, it is assumed that the setup is retrieval, where we receive a query and wish to rank data points in response. That brings us to its second unusual property: asymmetry. That means, only the data points are sketched while queries remain in the original space. With the help of an asymmetric distance function, however, we can easily compute an upper-bound on the query-data point inner product, using the raw query point and the sketch of a data point. | 2401.09350#368 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 368,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The above suggests that the ability of Ï(·) to preserve the inner product of a query point with a randomly drawn data point deteriorates as a function of the number of non-zero coordinates. For example, when the number of non-zero coordinates becomes larger, â¨Ï(q), Ï(X)â© for a fixed query q and a random point X becomes less reliable because the variance of the approxi- mation increases.\n# 10.3 Asymmetric Sketching\nOur second sketching algorithm is due to Bruch et al. [2023]. It is unusual in several ways. First, it is designed specifically for retrieval. That is, the objective of the sketching technique is not to preserve the inner product between points in a collection; in fact, as we will learn shortly, the sketch is not even an unbiased estimator. Instead, it is assumed that the setup is retrieval, where we receive a query and wish to rank data points in response. That brings us to its second unusual property: asymmetry. That means, only the data points are sketched while queries remain in the original space. With the help of an asymmetric distance function, however, we can easily compute an upper-bound on the query-data point inner product, using the raw query point and the sketch of a data point.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 369 | Finally, in its original construction as presented in [Bruch et al., 2023], the sketch was tailored specifically to sparse vectors. As we will show, however, it is trivial to modify the algorithm and adapt it to dense vectors.
In the rest of this section, we will first describe the sketching algorithm for sparse vectors, as well as its extension to dense vectors. We then describe how the distance between a query point in the original space and the sketch of any
149
150
10 Sketching
# Algorithm 5: Sketching of sparse vectors
Input: Sparse vector u â Rd. Requirements: h independent random mappings Ïo : [d] â [dâ¦/2]. Result: Sketch of u, {nz (u); u; u} consisting of the index of non-zero coordinates of u, the lower-bound sketch, and the upper-bound sketch. 1: Let u, u â Rdâ¦/2 be zero vectors 2: for all k â [ d⦠3: 4: 5: 6: end for 7: return {nz (u), u, u} 2 ] do I â {i â nz (u) | â o s.t. Ïo(i) = k} uk â maxiâI ui uk â miniâI ui | 2401.09350#369 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 369,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Finally, in its original construction as presented in [Bruch et al., 2023], the sketch was tailored specifically to sparse vectors. As we will show, however, it is trivial to modify the algorithm and adapt it to dense vectors.\nIn the rest of this section, we will first describe the sketching algorithm for sparse vectors, as well as its extension to dense vectors. We then describe how the distance between a query point in the original space and the sketch of any\n149\n150\n10 Sketching\n# Algorithm 5: Sketching of sparse vectors\nInput: Sparse vector u â Rd. Requirements: h independent random mappings Ïo : [d] â [dâ¦/2]. Result: Sketch of u, {nz (u); u; u} consisting of the index of non-zero coordinates of u, the lower-bound sketch, and the upper-bound sketch. 1: Let u, u â Rdâ¦/2 be zero vectors 2: for all k â [ d⦠3: 4: 5: 6: end for 7: return {nz (u), u, u} 2 ] do I â {i â nz (u) | â o s.t. Ïo(i) = k} uk â maxiâI ui uk â miniâI ui",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 370 | data point can be computed asymmetrically. Lastly, we review an analysis of the sketching algorithm.
# 10.3.1 The Sketching Algorithm
Algorithm 5 shows the logic behind the sketching of sparse vectors. It is assumed throughout that the sketch size, dâ¦, is even, so that dâ¦/2 is an integer. The algorithm also makes use of h independent random mappings Ïo : [d] â [dâ¦/2], where each Ïo(·) projects coordinates in the original space to an integer in the set [dâ¦/2] uniformly randomly.
Intuitively, the sketch of u â Rd is a data structure comprising of the index of its set of non-zero coordinates (i.e., nz (u)), along with an upper-bound sketch (u â Rdâ¦/2) and a lower-bound sketch (u â Rdâ¦/2) on the non-zero values of u. More precisely, the k-th coordinate of u (u) records the largest (smallest) value from the set of all non-zero coordinates in u that map into k according to at least one Ïo(·). | 2401.09350#370 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 370,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "data point can be computed asymmetrically. Lastly, we review an analysis of the sketching algorithm.\n# 10.3.1 The Sketching Algorithm\nAlgorithm 5 shows the logic behind the sketching of sparse vectors. It is assumed throughout that the sketch size, dâ¦, is even, so that dâ¦/2 is an integer. The algorithm also makes use of h independent random mappings Ïo : [d] â [dâ¦/2], where each Ïo(·) projects coordinates in the original space to an integer in the set [dâ¦/2] uniformly randomly.\nIntuitively, the sketch of u â Rd is a data structure comprising of the index of its set of non-zero coordinates (i.e., nz (u)), along with an upper-bound sketch (u â Rdâ¦/2) and a lower-bound sketch (u â Rdâ¦/2) on the non-zero values of u. More precisely, the k-th coordinate of u (u) records the largest (smallest) value from the set of all non-zero coordinates in u that map into k according to at least one Ïo(·).",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 371 | This sketching algorithm offers a great deal of flexibility. When data vec- tors are non-negative, we may drop the lower-bounds from the sketch, so that the sketch of u consists only of {nz (u), u}. When vectors are dense, the sketch clearly does not need to store the set of non-zero co- ordinates, so that the sketch of u becomes {u, u}. Finally, when vectors are dense and non-negative, the sketch of u simplifies to u.
10.3 Asymmetric Sketching
# Algorithm 6: Asymmetric distance computation for sparse vectors
Input: Sparse query vector q â Rd; sketch of data point u: {nz (u), u, u} Requirements: h independent random mappings Ïo : [d] â [dâ¦/2]. Result: Upper-bound on â¨q, uâ©. 1: s â 0 2: for i â nz (q) â© nz (u) do J â {Ïo(i) | o â [h]} 3: if qi > 0 then 4: 5: 6: 7: 8: end if 9: end for 10: return s
# 10.3.2 Inner Product Approximation | 2401.09350#371 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 371,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "This sketching algorithm offers a great deal of flexibility. When data vec- tors are non-negative, we may drop the lower-bounds from the sketch, so that the sketch of u consists only of {nz (u), u}. When vectors are dense, the sketch clearly does not need to store the set of non-zero co- ordinates, so that the sketch of u becomes {u, u}. Finally, when vectors are dense and non-negative, the sketch of u simplifies to u.\n10.3 Asymmetric Sketching\n# Algorithm 6: Asymmetric distance computation for sparse vectors\nInput: Sparse query vector q â Rd; sketch of data point u: {nz (u), u, u} Requirements: h independent random mappings Ïo : [d] â [dâ¦/2]. Result: Upper-bound on â¨q, uâ©. 1: s â 0 2: for i â nz (q) â© nz (u) do J â {Ïo(i) | o â [h]} 3: if qi > 0 then 4: 5: 6: 7: 8: end if 9: end for 10: return s\n# 10.3.2 Inner Product Approximation",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 372 | # 10.3.2 Inner Product Approximation
Suppose that we are given a query point q â Rd and wish to obtain an estimate of the inner product â¨q, uâ© for some data vector u. We must do so using only the sketch of u as produced by Algorithm 5. Because the query point is not sketched and, instead, remains in the original d-dimensional space, while u is only known in its sketched form, we say this computation is asymmetric. This is not unlike the distance computation between a query point and a quantized data point, as seen in Chapter 9.
This asymmetric procedure is described in Algorithm 6. The algorithm iterates over the intersection of the non-zero coordinates of the query vector and the non-zero coordinates of the data point (which is included in the sketch). It goes without saying that, if the vectors are dense, we may simply iterate over all coordinates. When visiting the i-th coordinate, we first form the set of coordinates that i maps to according to the hash functions Ïoâs; that is the set J in the algorithm. | 2401.09350#372 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 372,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 10.3.2 Inner Product Approximation\nSuppose that we are given a query point q â Rd and wish to obtain an estimate of the inner product â¨q, uâ© for some data vector u. We must do so using only the sketch of u as produced by Algorithm 5. Because the query point is not sketched and, instead, remains in the original d-dimensional space, while u is only known in its sketched form, we say this computation is asymmetric. This is not unlike the distance computation between a query point and a quantized data point, as seen in Chapter 9.\nThis asymmetric procedure is described in Algorithm 6. The algorithm iterates over the intersection of the non-zero coordinates of the query vector and the non-zero coordinates of the data point (which is included in the sketch). It goes without saying that, if the vectors are dense, we may simply iterate over all coordinates. When visiting the i-th coordinate, we first form the set of coordinates that i maps to according to the hash functions Ïoâs; that is the set J in the algorithm.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 373 | The next step then depends on the sign of the query at that coordinate. When qi is positive, we find the least upper-bound on the value of ui from its upper-bound sketch. That can be determined by looking at uj for all j â J , and taking the minimum value among those sketch coordinates. When qi < 0, on the other hand, we find the greatest lower- bound instead. In this way, it is always guaranteed that the partial inner product is an upper-bound on the actual partial inner product, qiui, as stated in the next theorem.
Theorem 10.3 The quantity returned by Algorithm 6 is an upper-bound on the inner product of query and data vectors.
151
152
10 Sketching
# 10.3.3 Theoretical Analysis
Theorem 10.3 implies that Algorithm 6 always overestimates the inner prod- uct between query and data points. In other words, the inner product approx- imation error is non-negative. But what can be said about the probability that such an error occurs? How large is the overestimation error? We turn to these questions next. | 2401.09350#373 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 373,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The next step then depends on the sign of the query at that coordinate. When qi is positive, we find the least upper-bound on the value of ui from its upper-bound sketch. That can be determined by looking at uj for all j â J , and taking the minimum value among those sketch coordinates. When qi < 0, on the other hand, we find the greatest lower- bound instead. In this way, it is always guaranteed that the partial inner product is an upper-bound on the actual partial inner product, qiui, as stated in the next theorem.\nTheorem 10.3 The quantity returned by Algorithm 6 is an upper-bound on the inner product of query and data vectors.\n151\n152\n10 Sketching\n# 10.3.3 Theoretical Analysis\nTheorem 10.3 implies that Algorithm 6 always overestimates the inner prod- uct between query and data points. In other words, the inner product approx- imation error is non-negative. But what can be said about the probability that such an error occurs? How large is the overestimation error? We turn to these questions next.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 374 | Before we do so, however, we must agree on a probabilistic model of the data. We follow [Bruch et al., 2023] and assume that a random sparse vector X is drawn from the following distribution. All coordinates of X are mutually independent. Its i-th coordinate is inactive (i.e., zero) with probability 1 â pi. Otherwise, it is active and its value is a random variable, Xi, drawn iid from some distribution with probability density function (PDF) Ï and cumulative distribution function (CDF) Φ.
# 10.3.3.1 Probability of Error
Let us focus on the approximation error of a single active coordinate. Con- cretely, suppose we have a random vector X whose i-th coordinate is active: i â nz (X). We are interested in quantifying the likelihood that, if we esti- mated the value of Xi from the sketch, the estimated value, ËXi, overshoots or undershoots the actual value. | 2401.09350#374 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 374,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Before we do so, however, we must agree on a probabilistic model of the data. We follow [Bruch et al., 2023] and assume that a random sparse vector X is drawn from the following distribution. All coordinates of X are mutually independent. Its i-th coordinate is inactive (i.e., zero) with probability 1 â pi. Otherwise, it is active and its value is a random variable, Xi, drawn iid from some distribution with probability density function (PDF) Ï and cumulative distribution function (CDF) Φ.\n# 10.3.3.1 Probability of Error\nLet us focus on the approximation error of a single active coordinate. Con- cretely, suppose we have a random vector X whose i-th coordinate is active: i â nz (X). We are interested in quantifying the likelihood that, if we esti- mated the value of Xi from the sketch, the estimated value, ËXi, overshoots or undershoots the actual value.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 375 | Formally, we wish to model P[ ËXi ̸= Xi], Note that, depending on the sign of the queryâs i-th coordinate ËXi may be estimated from the upper- bound sketch (X), resulting in overestimation, or the lower-bound sketch (X), resulting in underestimation. Because the two cases are symmetric, we state the main result for the former case: When ËXi is the least upper-bound on Xi, estimated from X:
ËXi = min jâ{Ïo(i) | oâ[h]} X j. (10.4)
Theorem 10.4 For large values of dâ¦, an active Xi, and ËXi estimated using Equation (10.4),
P [X: > Xi] ~ | f âexp ( - ml - F(a) >») "blade, Ai
where Ï(·) and Φ(·) are the PDF and CDF of Xi.
Extending this result to the lower-bound sketch involves replacing 1âΦ(α) with Φ(α). When the distribution defined by Ï is symmetric, the probabilities of error too are symmetric for the upper-bound and lower-bound sketches.
Proof of Theorem 10.4. Recall that ËXi is estimated as follows:
10.3 Asymmetric Sketching | 2401.09350#375 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 375,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Formally, we wish to model P[ ËXi ̸= Xi], Note that, depending on the sign of the queryâs i-th coordinate ËXi may be estimated from the upper- bound sketch (X), resulting in overestimation, or the lower-bound sketch (X), resulting in underestimation. Because the two cases are symmetric, we state the main result for the former case: When ËXi is the least upper-bound on Xi, estimated from X:\nËXi = min jâ{Ïo(i) | oâ[h]} X j. (10.4)\nTheorem 10.4 For large values of dâ¦, an active Xi, and ËXi estimated using Equation (10.4),\nP [X: > Xi] ~ | f âexp ( - ml - F(a) >») \"blade, Ai\nwhere Ï(·) and Φ(·) are the PDF and CDF of Xi.\nExtending this result to the lower-bound sketch involves replacing 1âΦ(α) with Φ(α). When the distribution defined by Ï is symmetric, the probabilities of error too are symmetric for the upper-bound and lower-bound sketches.\nProof of Theorem 10.4. Recall that ËXi is estimated as follows:\n10.3 Asymmetric Sketching",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 376 | Proof of Theorem 10.4. Recall that ËXi is estimated as follows:
10.3 Asymmetric Sketching
ËXi = min jâ{Ïo(i) | oâ[h]} X j.
So we must look up X j for values of j produced by Ïoâs.
Suppose one such value is k (i.e., k = Ïo(i) for some o â [h]). The event that X k > Xi happens only when there exists another active coordinate Xj such that Xj > Xi and Ïo(j) = k for some Ïo.
To derive P[X k > Xi], it is easier to think in terms of complementary events: X k = Xi if every other active coordinate whose value is larger than Xi maps to a sketch coordinate except k. Clearly the probability that any arbitrary Xj maps to a sketch coordinate other than k is simply 1 â 2/dâ¦. Therefore, given a vector X, the probability that no active coordinate Xj larger than Xi maps to the k-th coordinate of the sketch, which we denote by âEvent A,â is:
2 P [Event A| x] =1-(1- a) es is activel x;>X;_ 0
# P | 2401.09350#376 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 376,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Proof of Theorem 10.4. Recall that ËXi is estimated as follows:\n10.3 Asymmetric Sketching\nËXi = min jâ{Ïo(i) | oâ[h]} X j.\nSo we must look up X j for values of j produced by Ïoâs.\nSuppose one such value is k (i.e., k = Ïo(i) for some o â [h]). The event that X k > Xi happens only when there exists another active coordinate Xj such that Xj > Xi and Ïo(j) = k for some Ïo.\nTo derive P[X k > Xi], it is easier to think in terms of complementary events: X k = Xi if every other active coordinate whose value is larger than Xi maps to a sketch coordinate except k. Clearly the probability that any arbitrary Xj maps to a sketch coordinate other than k is simply 1 â 2/dâ¦. Therefore, given a vector X, the probability that no active coordinate Xj larger than Xi maps to the k-th coordinate of the sketch, which we denote by âEvent A,â is:\n2 P [Event A| x] =1-(1- a) es is activel x;>X;_ 0\n# P",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 377 | 2 P [Event A| x] =1-(1- a) es is activel x;>X;_ 0
# P
Because d⦠is large by assumption, we can approximate eâ1 â (1 â 2/dâ¦)dâ¦/2 and rewrite the expression above as follows:
2h P [Event A | X] + 1â exp ( - > 1x, is active x,>x;): ° 5#i
Finally, we marginalize the expression above over Xjâs for j ̸= i to remove the dependence on all but the i-th coordinate of X. To simplify the expression, however, we take the expectation over the first-order Taylor expansion of the right hand side around 0. This results in the following approximation:
P [Event A | X; =a] ~ 1 exp (- 1 â 9(a)) 3). ° j#i
For ËXi to be larger than Xi, event A must take place for all h sketch coordinates. That probability, by the independence of random mappings, is:
~ 2h I P[X; > X; |X, = a ~ {1 â exp (= 7 â ®(a)) Â¥7p;)] â ° i#i | 2401.09350#377 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 377,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "2 P [Event A| x] =1-(1- a) es is activel x;>X;_ 0\n# P\nBecause d⦠is large by assumption, we can approximate eâ1 â (1 â 2/dâ¦)dâ¦/2 and rewrite the expression above as follows:\n2h P [Event A | X] + 1â exp ( - > 1x, is active x,>x;): ° 5#i\nFinally, we marginalize the expression above over Xjâs for j ̸= i to remove the dependence on all but the i-th coordinate of X. To simplify the expression, however, we take the expectation over the first-order Taylor expansion of the right hand side around 0. This results in the following approximation:\nP [Event A | X; =a] ~ 1 exp (- 1 â 9(a)) 3). ° j#i\nFor ËXi to be larger than Xi, event A must take place for all h sketch coordinates. That probability, by the independence of random mappings, is:\n~ 2h I P[X; > X; |X, = a ~ {1 â exp (= 7 â ®(a)) Â¥7p;)] â ° i#i",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 378 | ~ 2h I P[X; > X; |X, = a ~ {1 â exp (= 7 â ®(a)) Â¥7p;)] â ° i#i
In deriving the expression above, we conditioned the event on the value of Xi. Taking the marginal probability leads us to the following expression for the event that ËXi > Xi for any i, concluding the proof:
153
154
10 Sketching
P[X;>X,] [[:-©0(- 2h ®(a)) p))] PCa) xi & | [1 â exp ( - ma - 2(a)) pj) | da)da. ii v
ââ
Theorem 10.4 offers insights into the behavior of the upper-bound sketch. The first observation is that the sketching mechanism presented here is more suitable for distributions where larger values occur with a smaller probability such as sub-Gaussian variables. In such cases, the larger the value is, the smaller its chance of being overestimated by the upper-bound sketch. Regardless of the underlying distribution, empiri- cally, the largest value in a vector is always estimated exactly. | 2401.09350#378 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 378,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "~ 2h I P[X; > X; |X, = a ~ {1 â exp (= 7 â ®(a)) Â¥7p;)] â ° i#i\nIn deriving the expression above, we conditioned the event on the value of Xi. Taking the marginal probability leads us to the following expression for the event that ËXi > Xi for any i, concluding the proof:\n153\n154\n10 Sketching\nP[X;>X,] [[:-©0(- 2h ®(a)) p))] PCa) xi & | [1 â exp ( - ma - 2(a)) pj) | da)da. ii v\nââ\nTheorem 10.4 offers insights into the behavior of the upper-bound sketch. The first observation is that the sketching mechanism presented here is more suitable for distributions where larger values occur with a smaller probability such as sub-Gaussian variables. In such cases, the larger the value is, the smaller its chance of being overestimated by the upper-bound sketch. Regardless of the underlying distribution, empiri- cally, the largest value in a vector is always estimated exactly.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 379 | The second insight is that there is a sweet spot for h given a particular value of dâ¦: using more random mappings helps lower the probability of error until the sketch starts to saturate, at which point the error rate increases. This particular property is similar to the behavior of a Bloom filter [Bloom, 1970].
# 10.3.3.2 Distribution of Error
We have modeled the probability that the sketch of a vector overestimates a value. In this section, we examine the shape of the distribution of error in the form of its CDF. Formally, assuming Xi is active and ËXi is estimated using Equation (10.4), we wish to find an expression for P[| ËXi â Xi| < ϵ] for any ϵ > 0.
Theorem 10.5 Suppose Xi is active and draws its value from a distribution with PDF and CDF Ï and Φ. Suppose further that ËXi is the least upper-bound on Xi, obtained using Equation (10.4). Then:
h PIX; -Xi<qdwl -|[ [1 â exp ( - m1 â (a+ )) X»i)| o(a)da. ° j#i | 2401.09350#379 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 379,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The second insight is that there is a sweet spot for h given a particular value of dâ¦: using more random mappings helps lower the probability of error until the sketch starts to saturate, at which point the error rate increases. This particular property is similar to the behavior of a Bloom filter [Bloom, 1970].\n# 10.3.3.2 Distribution of Error\nWe have modeled the probability that the sketch of a vector overestimates a value. In this section, we examine the shape of the distribution of error in the form of its CDF. Formally, assuming Xi is active and ËXi is estimated using Equation (10.4), we wish to find an expression for P[| ËXi â Xi| < ϵ] for any ϵ > 0.\nTheorem 10.5 Suppose Xi is active and draws its value from a distribution with PDF and CDF Ï and Φ. Suppose further that ËXi is the least upper-bound on Xi, obtained using Equation (10.4). Then:\nh PIX; -Xi<qdwl -|[ [1 â exp ( - m1 â (a+ )) X»i)| o(a)da. ° j#i",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 380 | h PIX; -Xi<qdwl -|[ [1 â exp ( - m1 â (a+ )) X»i)| o(a)da. ° j#i
Proof. We begin by quantifying the conditional probability P[ ËXi â Xi ⤠ϵ | Xi = α]. Conceptually, the event in question happens when all values that collide with Xi are less than or equal to Xi+ϵ. This event can be characterized as the complement of the event that all h sketch coordinates that contain Xi collide with values greater than Xi + ϵ. Using this complementary event, we can write the conditional probability as follows:
10.3 Asymmetric Sketching
PLX; âX, <e| X; =a) =1-[1-(1- F yno-ter Esai)" ° h xe1- 1 (- Ha -a0+)En)| : Ft
We complete the proof by computing the marginal distribution over the sup- ââ port.
Given the CDF of ËXi â Xi and the fact that ËXi â Xi ⥠0, it follows that its expected value conditioned on Xi being active is:
Lemma 10.2 Under the conditions of Theorem 10.5:
5 h Bx â xy ~ [ [[1-o0(-Za-s0+9%n)| (a) da de. j#ft | 2401.09350#380 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 380,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "h PIX; -Xi<qdwl -|[ [1 â exp ( - m1 â (a+ )) X»i)| o(a)da. ° j#i\nProof. We begin by quantifying the conditional probability P[ ËXi â Xi ⤠ϵ | Xi = α]. Conceptually, the event in question happens when all values that collide with Xi are less than or equal to Xi+ϵ. This event can be characterized as the complement of the event that all h sketch coordinates that contain Xi collide with values greater than Xi + ϵ. Using this complementary event, we can write the conditional probability as follows:\n10.3 Asymmetric Sketching\nPLX; âX, <e| X; =a) =1-[1-(1- F yno-ter Esai)\" ° h xe1- 1 (- Ha -a0+)En)| : Ft\nWe complete the proof by computing the marginal distribution over the sup- ââ port.\nGiven the CDF of ËXi â Xi and the fact that ËXi â Xi ⥠0, it follows that its expected value conditioned on Xi being active is:\nLemma 10.2 Under the conditions of Theorem 10.5:\n5 h Bx â xy ~ [ [[1-o0(-Za-s0+9%n)| (a) da de. j#ft",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 381 | 5 h Bx â xy ~ [ [[1-o0(-Za-s0+9%n)| (a) da de. j#ft
# 10.3.3.3 Case Study: Gaussian Vectors
Let us make the analysis more concrete by applying the results to random Gaussian vectors. In other words, suppose all active Xiâs are drawn from a zero-mean, unit-variance Gaussian distribution. We can derive a closed-form expression for the overestimation probability as the following corollary shows.
Corollary 10.1 Suppose the probability that a coordinate is active, pi, is equal to p for all coordinates of the random vector X â Rd. When an ac- tive Xi, drawn from N (0, 1), is estimated using the upper-bound sketch with Equation (10.4), the overestimation probability is:
h ~ h . do _ 2kh(dâ P[X,> XJ x1+)5 () Osage aN). k=1
We begin by proving the special case where h = 1.
Lemma 10.3 Under the conditions of Corollary 10.1 with h = 1, the prob- ability that the upper-bound sketch overestimates the value of Xi is:
~ do _ 2(d=1)p P[X; > Xi] <1-sy-pt* a), | 2401.09350#381 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 381,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "5 h Bx â xy ~ [ [[1-o0(-Za-s0+9%n)| (a) da de. j#ft\n# 10.3.3.3 Case Study: Gaussian Vectors\nLet us make the analysis more concrete by applying the results to random Gaussian vectors. In other words, suppose all active Xiâs are drawn from a zero-mean, unit-variance Gaussian distribution. We can derive a closed-form expression for the overestimation probability as the following corollary shows.\nCorollary 10.1 Suppose the probability that a coordinate is active, pi, is equal to p for all coordinates of the random vector X â Rd. When an ac- tive Xi, drawn from N (0, 1), is estimated using the upper-bound sketch with Equation (10.4), the overestimation probability is:\nh ~ h . do _ 2kh(dâ P[X,> XJ x1+)5 () Osage aN). k=1\nWe begin by proving the special case where h = 1.\nLemma 10.3 Under the conditions of Corollary 10.1 with h = 1, the prob- ability that the upper-bound sketch overestimates the value of Xi is:\n~ do _ 2(d=1)p P[X; > Xi] <1-sy-pt* a),",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 382 | ~ do _ 2(d=1)p P[X; > Xi] <1-sy-pt* a),
Proof. From Theorem 10.4 we have that:
P[X, > Xj] ~ | [1 â BOPP)" g(a) het | [1 _ o GMENEY FB (Q),
155
156 10 Sketching
Given that Xiâs are drawn from a Gaussian distribution, and using the approximation above, we can rewrite the probability of error as:
# f°
2 $ 1 f° â24=0P (1 @(a))]_â 92 PIX > x= [ [l-e "% Jeâ = da. oo
We now break up the right hand side into the following three sums, replacing 2(d â 1)p/d⦠with β for brevity:
1 â 2Ï 1 â 2Ï 1 â 2Ï
~ xo] 2 P[X; > Xj] ~ | Tme 7 da (10.5) â0o
0 -|[ PO) 6 F doy (10.6) ~oo V2T
# ~oo
â eâβ(1âΦ(α))eâ α2 2 dα. (10.7) 0 | 2401.09350#382 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 382,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "~ do _ 2(d=1)p P[X; > Xi] <1-sy-pt* a),\nProof. From Theorem 10.4 we have that:\nP[X, > Xj] ~ | [1 â BOPP)\" g(a) het | [1 _ o GMENEY FB (Q),\n155\n156 10 Sketching\nGiven that Xiâs are drawn from a Gaussian distribution, and using the approximation above, we can rewrite the probability of error as:\n# f°\n2 $ 1 f° â24=0P (1 @(a))]_â 92 PIX > x= [ [l-e \"% Jeâ = da. oo\nWe now break up the right hand side into the following three sums, replacing 2(d â 1)p/d⦠with β for brevity:\n1 â 2Ï 1 â 2Ï 1 â 2Ï\n~ xo] 2 P[X; > Xj] ~ | Tme 7 da (10.5) â0o\n0 -|[ PO) 6 F doy (10.6) ~oo V2T\n# ~oo\nâ eâβ(1âΦ(α))eâ α2 2 dα. (10.7) 0",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 383 | # ~oo
â eâβ(1âΦ(α))eâ α2 2 dα. (10.7) 0
The sum in (10.5) is equal to the quantity 1. Let us turn to (10.7) first. We have that:
# a
a
As a result, we can write:
oo oo [ eee da = | 1 .-84-M@)) oF day 0 0 V20 x =e | 1 66r@),- da 0 «V20
By similar reasoning, and noting that:
aco 1 od 2 1â (a) 2+ 5 e- = dt, a 7 ee)" =a) S
we arrive at:
wit ) [ Le BU-#) oF da = Leâ 8(1âe- âe e⬠la = se âe ~oo V2T B
Plugging the results above into Equations (10.5), (10.6), and (10.7) results in:
10.3 Asymmetric Sketching
P(X > Xx 1- Fie 8) â Fe F(1 er!) =1-50-e (1404) =1- do (1 2) ~*~ 3dâ Dp © ,
which completes the proof.
Given the result above, the solution for the general case of h > 0 is straight- forward to obtain. | 2401.09350#383 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 383,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# ~oo\nâ eâβ(1âΦ(α))eâ α2 2 dα. (10.7) 0\nThe sum in (10.5) is equal to the quantity 1. Let us turn to (10.7) first. We have that:\n# a\na\nAs a result, we can write:\noo oo [ eee da = | 1 .-84-M@)) oF day 0 0 V20 x =e | 1 66r@),- da 0 «V20\nBy similar reasoning, and noting that:\naco 1 od 2 1â (a) 2+ 5 e- = dt, a 7 ee)\" =a) S\nwe arrive at:\nwit ) [ Le BU-#) oF da = Leâ 8(1âe- âe e⬠la = se âe ~oo V2T B\nPlugging the results above into Equations (10.5), (10.6), and (10.7) results in:\n10.3 Asymmetric Sketching\nP(X > Xx 1- Fie 8) â Fe F(1 er!) =1-50-e (1404) =1- do (1 2) ~*~ 3dâ Dp © ,\nwhich completes the proof.\nGiven the result above, the solution for the general case of h > 0 is straight- forward to obtain.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 384 | which completes the proof.
Given the result above, the solution for the general case of h > 0 is straight- forward to obtain.
Proof of Corollary 10.1. Using the binomial theorem, we have that:
P[X; > Xi] ~ | {1- et A-8()(4-DP) "Gq P(q) h => (I) [ce Betonemny tary k k=0
We rewrite the expression above for Gaussian variables to arrive at:
2 ~ 1 fk °° â2h(4=DP (1_9(a))\k 22 PIX > xX] ~ Fede k [ices yTe = da.
Following the proof of the previous lemma, we can expand the right hand side as follows:
h ~ 1 h OO 2kh(d=1)p 2 P[X; > X;] Â¥1+â= -1 â| ee OPN =F day Bio xder+ Fed (eof h do 2kh(dâ1)p h 1+ (1) te, a (;) 2kh(d â 1)p
which completes the proof.
Let us now consider the CDF of the overestimation error. | 2401.09350#384 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 384,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "which completes the proof.\nGiven the result above, the solution for the general case of h > 0 is straight- forward to obtain.\nProof of Corollary 10.1. Using the binomial theorem, we have that:\nP[X; > Xi] ~ | {1- et A-8()(4-DP) \"Gq P(q) h => (I) [ce Betonemny tary k k=0\nWe rewrite the expression above for Gaussian variables to arrive at:\n2 ~ 1 fk °° â2h(4=DP (1_9(a))\\k 22 PIX > xX] ~ Fede k [ices yTe = da.\nFollowing the proof of the previous lemma, we can expand the right hand side as follows:\nh ~ 1 h OO 2kh(d=1)p 2 P[X; > X;] Â¥1+â= -1 â| ee OPN =F day Bio xder+ Fed (eof h do 2kh(dâ1)p h 1+ (1) te, a (;) 2kh(d â 1)p\nwhich completes the proof.\nLet us now consider the CDF of the overestimation error.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 385 | which completes the proof.
Let us now consider the CDF of the overestimation error.
Corollary 10.2 Under the conditions of Corollary 10.1 the CDF of overes- timation error for an active coordinate Xi â¼ N (0, Ï) is:
h P[X; -Xi<qdei1- f â exp ( - Mae - â)| ;
where Φâ²(·) is the CDF of a zero-mean Gaussian with standard deviation Ï
157
ââ
ââ
158 10 Sketching
Proof. When the active values of a vector are drawn from a Gaussian dis- tribution, then the pairwise difference between any two coordinates has a 2. As such, Gaussian distribution with standard deviation we may estimate 1 â Φ(α + ϵ) by considering the probability that a pair of coordinates (one of which having value α) has a difference greater than ϵ: P[Xi â Xj > ϵ]. With that idea, we may thus write:
1 â Φ(α + ϵ) = 1 â Φâ²(ϵ).
The claim follows by using the above identity in Theorem 10.5.
Corollary 10.2 enables us to find a particular sketch configuration given a desired bound on the probability of error, as the following lemma shows. | 2401.09350#385 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 385,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "which completes the proof.\nLet us now consider the CDF of the overestimation error.\nCorollary 10.2 Under the conditions of Corollary 10.1 the CDF of overes- timation error for an active coordinate Xi â¼ N (0, Ï) is:\nh P[X; -Xi<qdei1- f â exp ( - Mae - â)| ;\nwhere Φâ²(·) is the CDF of a zero-mean Gaussian with standard deviation Ï\n157\nââ\nââ\n158 10 Sketching\nProof. When the active values of a vector are drawn from a Gaussian dis- tribution, then the pairwise difference between any two coordinates has a 2. As such, Gaussian distribution with standard deviation we may estimate 1 â Φ(α + ϵ) by considering the probability that a pair of coordinates (one of which having value α) has a difference greater than ϵ: P[Xi â Xj > ϵ]. With that idea, we may thus write:\n1 â Φ(α + ϵ) = 1 â Φâ²(ϵ).\nThe claim follows by using the above identity in Theorem 10.5.\nCorollary 10.2 enables us to find a particular sketch configuration given a desired bound on the probability of error, as the following lemma shows.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 386 | Corollary 10.2 enables us to find a particular sketch configuration given a desired bound on the probability of error, as the following lemma shows.
Lemma 10.4 Under the conditions of Corollary 10.2, and given a choice of ϵ, δ â (0, 1) and the number of random mappings h, P[ ËXi â Xi ⤠ϵ] with probability at least 1 â δ if:
d⦠> â 2h(d â 1)p(1 â Φâ²(ϵ)) log(1 â δ1/h) .
# 10.3.3.4 Error of Inner Product
We have thus far quantified the probability that a value estimated from the upper-bound sketch overestimates the original value of a randomly chosen coordinate. We also characterized the distribution of the overestimation error for a single coordinate and derived expressions for special distributions. In this section, we quantify the overestimation error when approximating the inner product between a fixed query point and a random data point using Algorithm 6. | 2401.09350#386 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 386,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Corollary 10.2 enables us to find a particular sketch configuration given a desired bound on the probability of error, as the following lemma shows.\nLemma 10.4 Under the conditions of Corollary 10.2, and given a choice of ϵ, δ â (0, 1) and the number of random mappings h, P[ ËXi â Xi ⤠ϵ] with probability at least 1 â δ if:\nd⦠> â 2h(d â 1)p(1 â Φâ²(ϵ)) log(1 â δ1/h) .\n# 10.3.3.4 Error of Inner Product\nWe have thus far quantified the probability that a value estimated from the upper-bound sketch overestimates the original value of a randomly chosen coordinate. We also characterized the distribution of the overestimation error for a single coordinate and derived expressions for special distributions. In this section, we quantify the overestimation error when approximating the inner product between a fixed query point and a random data point using Algorithm 6.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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0.008923425,
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2401.09350 | 387 | To make the notation less cluttered, however, let us denote by ËXi our estimate of Xi. The estimated quantity is 0 if i /â nz (X). Otherwise, it is estimated either from the upper-bound sketch or the lower-bound sketch, depending on the sign of qi. Finally denote by ËX a reconstruction of X where each ËXi is estimated as described above.
Consider the expected value of ËXi â Xi conditioned on Xi being activeâ that is a quantity we analyzed previously. Let µi = E[ ËXi â Xi; Xi is active]. Similarly denote by Ï2 i its variance when Xi is active. Given that Xi is active with probability pi and inactive with probability 1 â pi, it is easy to show that E[ ËXi â Xi] = piµi (note we have removed the condition on Xi being active) and that its variance Var[ ËXi â Xi] = piÏ2
With the above in mind, we state the following result.
Theorem 10.6 Suppose that q â Rd is a sparse vector. Suppose in a random sparse vector X â Rd, a coordinate Xi is active with probability pi and, when active, draws its value from some well-behaved distribution (i.e., with finite
ââ
10.3 Asymmetric Sketching | 2401.09350#387 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 387,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "To make the notation less cluttered, however, let us denote by ËXi our estimate of Xi. The estimated quantity is 0 if i /â nz (X). Otherwise, it is estimated either from the upper-bound sketch or the lower-bound sketch, depending on the sign of qi. Finally denote by ËX a reconstruction of X where each ËXi is estimated as described above.\nConsider the expected value of ËXi â Xi conditioned on Xi being activeâ that is a quantity we analyzed previously. Let µi = E[ ËXi â Xi; Xi is active]. Similarly denote by Ï2 i its variance when Xi is active. Given that Xi is active with probability pi and inactive with probability 1 â pi, it is easy to show that E[ ËXi â Xi] = piµi (note we have removed the condition on Xi being active) and that its variance Var[ ËXi â Xi] = piÏ2\nWith the above in mind, we state the following result.\nTheorem 10.6 Suppose that q â Rd is a sparse vector. Suppose in a random sparse vector X â Rd, a coordinate Xi is active with probability pi and, when active, draws its value from some well-behaved distribution (i.e., with finite\nââ\n10.3 Asymmetric Sketching",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 388 | ââ
10.3 Asymmetric Sketching
expectation, variance, and third moment). If µi = E[ ËXi â Xi; Xi is active] i = Var[ ËXi â Xi; Xi is active], then the random variable Z defined as and Ï2 follows:
. (a, X âX) = Vienz(q) GPibi Dieneo G (pio? + pi(1 â pi) ue?) ZA (10.8)
approximately tends to a standard Gaussian distribution as |nz (q)| grows.
Proof. Let us expand the inner product between q and ËX â X as follows:
(XÂ¥-X)= YO a(Xi-%). (10.9) ienz(q) x
The expected value of â¨q, ËX â Xâ© is:
El(a,XâX)}= SO aElZl= SD apie. iâ¬nz(q) iâ¬nz(q)
Its variance is:
Var[(q.X âX)]= Sa? VarlZi]= D0 a? (pio? + pi( â pide?) iâ¬nz(q) iâ¬nz(q) | 2401.09350#388 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 388,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "ââ\n10.3 Asymmetric Sketching\nexpectation, variance, and third moment). If µi = E[ ËXi â Xi; Xi is active] i = Var[ ËXi â Xi; Xi is active], then the random variable Z defined as and Ï2 follows:\n. (a, X âX) = Vienz(q) GPibi Dieneo G (pio? + pi(1 â pi) ue?) ZA (10.8)\napproximately tends to a standard Gaussian distribution as |nz (q)| grows.\nProof. Let us expand the inner product between q and ËX â X as follows:\n(XÂ¥-X)= YO a(Xi-%). (10.9) ienz(q) x\nThe expected value of â¨q, ËX â Xâ© is:\nEl(a,XâX)}= SO aElZl= SD apie. iâ¬nz(q) iâ¬nz(q)\nIts variance is:\nVar[(q.X âX)]= Sa? VarlZi]= D0 a? (pio? + pi( â pide?) iâ¬nz(q) iâ¬nz(q)",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 389 | Because we assumed that the distribution of Xi is well-behaved, we can conclude that Var[Zi] > 0 and that E[|Zi|3] < â. If we operated on the assumption that qiZiâs are independentâin reality, they are weakly dependentâalbeit not identically distributed, we can appeal to the Berry- ââ Esseen theorem to complete the proof.
# 10.3.4 Fixing the Sketch Size
It is often desirable for a sketching algorithm to produce a sketch with a con- stant size. That makes the size of a collection of sketches predictable, which is often required for resource allocation. Algorithm 5, however, produces a sketch whose size is variable. That is because the sketch contains the set of non-zero coordinates of the vector.
It is, however, straightforward to fix the sketch size. The key to that is the fact that Algorithm 6 uses nz (u) of a vector u only to ascertain if a queryâs non-zero coordinates are present in the vector u. In effect, all the sketch must provide is a mechanism to perform set membership tests. That is precisely what fixed-size signatures such as Bloom filters [Bloom, 1970] do, albeit probabilistically.
159
160
10 Sketching
Algorithm 7: Sketching with threshold sampling | 2401.09350#389 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 389,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Because we assumed that the distribution of Xi is well-behaved, we can conclude that Var[Zi] > 0 and that E[|Zi|3] < â. If we operated on the assumption that qiZiâs are independentâin reality, they are weakly dependentâalbeit not identically distributed, we can appeal to the Berry- ââ Esseen theorem to complete the proof.\n# 10.3.4 Fixing the Sketch Size\nIt is often desirable for a sketching algorithm to produce a sketch with a con- stant size. That makes the size of a collection of sketches predictable, which is often required for resource allocation. Algorithm 5, however, produces a sketch whose size is variable. That is because the sketch contains the set of non-zero coordinates of the vector.\nIt is, however, straightforward to fix the sketch size. The key to that is the fact that Algorithm 6 uses nz (u) of a vector u only to ascertain if a queryâs non-zero coordinates are present in the vector u. In effect, all the sketch must provide is a mechanism to perform set membership tests. That is precisely what fixed-size signatures such as Bloom filters [Bloom, 1970] do, albeit probabilistically.\n159\n160\n10 Sketching\nAlgorithm 7: Sketching with threshold sampling",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 390 | 159
160
10 Sketching
Algorithm 7: Sketching with threshold sampling
Input: Vector u â Rd. Requirements: a random mapping Ï : [d] â [0, 1]. Result: Sketch of u, {I, V, â¥uâ¥2 2} consisting of the index and value of sampled coordinates in I and V, and the squared norm of the vector. 1: I, V â â
2: for i â nz (u) do 3: θ â d⦠u2 i â¥uâ¥2 2 4: 5: 6: end if 7: end for 8: return {I, V, â¥uâ¥2 2} if Ï(i) ⤠θ then Append i to I, ui to V
# 10.4 Sketching by Sampling | 2401.09350#390 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 390,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "159\n160\n10 Sketching\nAlgorithm 7: Sketching with threshold sampling\nInput: Vector u â Rd. Requirements: a random mapping Ï : [d] â [0, 1]. Result: Sketch of u, {I, V, â¥uâ¥2 2} consisting of the index and value of sampled coordinates in I and V, and the squared norm of the vector. 1: I, V â â
2: for i â nz (u) do 3: θ â d⦠u2 i â¥uâ¥2 2 4: 5: 6: end if 7: end for 8: return {I, V, â¥uâ¥2 2} if Ï(i) ⤠θ then Append i to I, ui to V\n# 10.4 Sketching by Sampling",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 391 | # 10.4 Sketching by Sampling
Our final sketching algorithm is designed specifically for inner product and is due to Daliri et al. [2023]. The guiding principle is simple: coordinates with larger values contribute more heavily to inner product than coordinates with smaller values. That is an obvious fact that is a direct result of the linearity of inner product: (u,v) = 0, ujv;Daliri et al. [2023] use that insight as follows. When forming the sketch of vector u, they sample coordinates (without replacement) from u according to a distribution defined by the magnitude of each coordinate. Larger values are given a higher chance of being sampled, while smaller values are less likely to be selected. The sketch, in the end, is a data structure that is made up of the index of sampled coordinates, their values, and additional statistics.
The research question here concerns the sampling process: How must we sample coordinates such that any distance computed from the sketch is an unbiased estimate of the inner product itself? The answer to that question also depends, of course, on how we compute the distance from a pair of sketches. Considering the non-linearity of the sketch, distance computation can no longer be the inner product of sketches. | 2401.09350#391 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 391,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 10.4 Sketching by Sampling\nOur final sketching algorithm is designed specifically for inner product and is due to Daliri et al. [2023]. The guiding principle is simple: coordinates with larger values contribute more heavily to inner product than coordinates with smaller values. That is an obvious fact that is a direct result of the linearity of inner product: (u,v) = 0, ujv;Daliri et al. [2023] use that insight as follows. When forming the sketch of vector u, they sample coordinates (without replacement) from u according to a distribution defined by the magnitude of each coordinate. Larger values are given a higher chance of being sampled, while smaller values are less likely to be selected. The sketch, in the end, is a data structure that is made up of the index of sampled coordinates, their values, and additional statistics.\nThe research question here concerns the sampling process: How must we sample coordinates such that any distance computed from the sketch is an unbiased estimate of the inner product itself? The answer to that question also depends, of course, on how we compute the distance from a pair of sketches. Considering the non-linearity of the sketch, distance computation can no longer be the inner product of sketches.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 392 | In the remainder of this section, we review the sketching algorithm, de- scribe distance computation given sketches, and analyze the expected error. In our presentation, we focus on the simpler variant of the algorithm proposed by Daliri et al. [2023], dubbed âthreshold sampling.â
# 10.4.1 The Sketching Algorithm
Algorithm 7 presents the âthreshold samplingâ sketching technique by Daliri et al. [2023]. It is assumed throughout that the desired sketch size is dâ¦, and
10.4 Sketching by Sampling
Algorithm 8: Distance computation for threshold sampling 2} and {Iv, Vv, â¥vâ¥2
Input: Sketches of vectors u and v: {Iu, Vu, â¥uâ¥2 Result: An unbiased estimate of â¨u, vâ©. 1: s â 0 2: for i â Iu â© Iv do 3: 4: end for 5: return s s â s + uivi/ min(1, dâ¦u2 i /â¥uâ¥2 2, dâ¦v2 i /â¥vâ¥2 2) 2}.
that the algorithm has access to a random hash function Ï that maps integers in [d] to the unit interval. | 2401.09350#392 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 392,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "In the remainder of this section, we review the sketching algorithm, de- scribe distance computation given sketches, and analyze the expected error. In our presentation, we focus on the simpler variant of the algorithm proposed by Daliri et al. [2023], dubbed âthreshold sampling.â\n# 10.4.1 The Sketching Algorithm\nAlgorithm 7 presents the âthreshold samplingâ sketching technique by Daliri et al. [2023]. It is assumed throughout that the desired sketch size is dâ¦, and\n10.4 Sketching by Sampling\nAlgorithm 8: Distance computation for threshold sampling 2} and {Iv, Vv, â¥vâ¥2\nInput: Sketches of vectors u and v: {Iu, Vu, â¥uâ¥2 Result: An unbiased estimate of â¨u, vâ©. 1: s â 0 2: for i â Iu â© Iv do 3: 4: end for 5: return s s â s + uivi/ min(1, dâ¦u2 i /â¥uâ¥2 2, dâ¦v2 i /â¥vâ¥2 2) 2}.\nthat the algorithm has access to a random hash function Ï that maps integers in [d] to the unit interval.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 393 | that the algorithm has access to a random hash function Ï that maps integers in [d] to the unit interval.
The algorithm iterates over all non-zero coordinates of the input vector and makes a decision as to whether that coordinate should be added to the sketch. The decision is made based on the relative magnitude of the coordinate, as i /â¥uâ¥2 weighted by u2 i is large, coordinate i has a higher chance of being sampled, as desired.
Notice, however, that the target sketch size d⦠is realized in expectation only. In other words, we may end up with more than d⦠coordinates in the sketch, or we may have fewer entries. Daliri et al. [2023] propose a different variant of the algorithm that is guaranteed to give a fixed sketch size; we refer the reader to their work for details.
# 10.4.2 Inner Product Approximation
When sketching a vector using a JL transform, we simply get a vector in the dâ¦-dimensional Euclidean space, where inner product is well-defined. So if Ï(u) and Ï(v) are sketches of two d-dimensional vectors u and v, we ap- proximate â¨u, vâ© with â¨Ï(u), Ï(v)â©. It could not be more straightforward. | 2401.09350#393 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 393,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "that the algorithm has access to a random hash function Ï that maps integers in [d] to the unit interval.\nThe algorithm iterates over all non-zero coordinates of the input vector and makes a decision as to whether that coordinate should be added to the sketch. The decision is made based on the relative magnitude of the coordinate, as i /â¥uâ¥2 weighted by u2 i is large, coordinate i has a higher chance of being sampled, as desired.\nNotice, however, that the target sketch size d⦠is realized in expectation only. In other words, we may end up with more than d⦠coordinates in the sketch, or we may have fewer entries. Daliri et al. [2023] propose a different variant of the algorithm that is guaranteed to give a fixed sketch size; we refer the reader to their work for details.\n# 10.4.2 Inner Product Approximation\nWhen sketching a vector using a JL transform, we simply get a vector in the dâ¦-dimensional Euclidean space, where inner product is well-defined. So if Ï(u) and Ï(v) are sketches of two d-dimensional vectors u and v, we ap- proximate â¨u, vâ© with â¨Ï(u), Ï(v)â©. It could not be more straightforward.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 394 | A sketch produced by Algorithm 7, however, is not as nice. Approximating â¨u, vâ© from their sketches requires a custom distance function defined for the sketch. That is precisely what Algorithm 8 outlines.
In the algorithm, it is understood that ui and vi corresponding to i â Iu â© Iv are present in Vu and Vv, respectively. These quantities, along with d⦠and the norms of the vectors are used to weight each partial inner product. The final quantity, as we will learn shortly, is an unbiased estimate of the inner product between u and v.
161
162
10 Sketching
# 10.4.3 Theoretical Analysis
Theorem 10.7 Algorithm 7 produces sketches that consist of at most d⦠coordinates in expectation.
Proof. The number of sampled coordinates is |I|. That quantity can be ex- pressed as follows:
d [Z| = Ss lier. i=1
Taking expectation of both sides and using the linearity of expectation, we obtain the following:
2 Ui E|Z|] = » Efliez] = » min(1,do7âb5) < do. lull
ââ
Theorem 10.8 Algorithm 8 yields an unbiased estimate of inner product. | 2401.09350#394 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 394,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "A sketch produced by Algorithm 7, however, is not as nice. Approximating â¨u, vâ© from their sketches requires a custom distance function defined for the sketch. That is precisely what Algorithm 8 outlines.\nIn the algorithm, it is understood that ui and vi corresponding to i â Iu â© Iv are present in Vu and Vv, respectively. These quantities, along with d⦠and the norms of the vectors are used to weight each partial inner product. The final quantity, as we will learn shortly, is an unbiased estimate of the inner product between u and v.\n161\n162\n10 Sketching\n# 10.4.3 Theoretical Analysis\nTheorem 10.7 Algorithm 7 produces sketches that consist of at most d⦠coordinates in expectation.\nProof. The number of sampled coordinates is |I|. That quantity can be ex- pressed as follows:\nd [Z| = Ss lier. i=1\nTaking expectation of both sides and using the linearity of expectation, we obtain the following:\n2 Ui E|Z|] = » Efliez] = » min(1,do7âb5) < do. lull\nââ\nTheorem 10.8 Algorithm 8 yields an unbiased estimate of inner product.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 395 | ââ
Theorem 10.8 Algorithm 8 yields an unbiased estimate of inner product.
Proof. From the proof of the previous theorem, we know that coordinate i of an arbitrary vector u is included in the sketch with probability equal to:
min(1, d⦠u2 i â¥uâ¥2 2 ).
As such, the odds that i â Iu â© Iv is:
pi = min(1, d⦠u2 i â¥uâ¥2 2 , d⦠v2 i â¥vâ¥2 2 ).
Algorithm 8 gives us a weighted sum of the coordinates that are present in Iu â© Iv. We can rewrite that sum using indicator functions as follows:
a U. y liez.nz. i=l iVi Pi
In expectation, then:
d UzU, {Ui E [ lier, nz, i=1 Pi fu; iVi = y Pi = (u,v), Di (u,v) i=1 â
as required.
ââ
Theorem 10.9 If S is the output of Algorithm 8 for sketches of vectors u and v, then:
10.4 Sketching by Sampling
2 5 Var[] < > max (|leg|/3 13, [ul3|}v- (13). ° | 2401.09350#395 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 395,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "ââ\nTheorem 10.8 Algorithm 8 yields an unbiased estimate of inner product.\nProof. From the proof of the previous theorem, we know that coordinate i of an arbitrary vector u is included in the sketch with probability equal to:\nmin(1, d⦠u2 i â¥uâ¥2 2 ).\nAs such, the odds that i â Iu â© Iv is:\npi = min(1, d⦠u2 i â¥uâ¥2 2 , d⦠v2 i â¥vâ¥2 2 ).\nAlgorithm 8 gives us a weighted sum of the coordinates that are present in Iu â© Iv. We can rewrite that sum using indicator functions as follows:\na U. y liez.nz. i=l iVi Pi\nIn expectation, then:\nd UzU, {Ui E [ lier, nz, i=1 Pi fu; iVi = y Pi = (u,v), Di (u,v) i=1 â\nas required.\nââ\nTheorem 10.9 If S is the output of Algorithm 8 for sketches of vectors u and v, then:\n10.4 Sketching by Sampling\n2 5 Var[] < > max (|leg|/3 13, [ul3|}v- (13). °",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 396 | 10.4 Sketching by Sampling
2 5 Var[] < > max (|leg|/3 13, [ul3|}v- (13). °
where uâ and vâ are the vectors u and v restricted to the set of non-zero coordinates common to both vectors (i.e., â = {i | ui ̸= 0 ⧠vi ̸= 0}).
Proof. We use the same proof strategy as in the previous theorem. In partic- ular, we write:
UiVi UV; Var[S] = Var ~ liez,nz, > â| = So Var [Liez.az, De i ] tex v tex 2,2 U;ZU; = y + Var [Liezunz. |. ice Pi
Turning to the term inside the sum, we obtain:
Var [liez.or,] = Pi â Pi
which is 0 if pi = 1 and less than pi otherwise. Putting everything together, we complete the proof: | 2401.09350#396 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 396,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "10.4 Sketching by Sampling\n2 5 Var[] < > max (|leg|/3 13, [ul3|}v- (13). °\nwhere uâ and vâ are the vectors u and v restricted to the set of non-zero coordinates common to both vectors (i.e., â = {i | ui ̸= 0 ⧠vi ̸= 0}).\nProof. We use the same proof strategy as in the previous theorem. In partic- ular, we write:\nUiVi UV; Var[S] = Var ~ liez,nz, > â| = So Var [Liez.az, De i ] tex v tex 2,2 U;ZU; = y + Var [Liezunz. |. ice Pi\nTurning to the term inside the sum, we obtain:\nVar [liez.or,] = Pi â Pi\nwhich is 0 if pi = 1 and less than pi otherwise. Putting everything together, we complete the proof:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 397 | Var [liez.or,] = Pi â Pi
which is 0 if pi = 1 and less than pi otherwise. Putting everything together, we complete the proof:
: UPUF nay no (uf /llell3) (v?/Mell3) Var[S] < Ss = llullallulla Ss : z 22 z ipa Pi Ce) lu3lhvl3 = SO max (u?/|/ull3,0? /llell3) ° iâ¬x, ppAl elles uy 1 uF d. «jul * wie ilu 2B (el ; esl) t ; do Wells Mell 1 p 5 = (hee l3llol3 + leul3the-(13) o 2 5 SG max (ee [Sle llellalle-|l2)- o
Theorem 10.9 tells us that, if we estimated â¨u, vâ© for two vectors u and v using Algorithm 8, then the variance of our estimate will be bounded by factors that depend on the non-zero coordinates that u and v have in common. Because nz (u) â© nz (v) has at most d entries, estimates of inner product based on Threshold Sampling should generally be more accurate than those obtained from JL sketches. This is particularly the case when u and v are sparse.
163 | 2401.09350#397 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 397,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Var [liez.or,] = Pi â Pi\nwhich is 0 if pi = 1 and less than pi otherwise. Putting everything together, we complete the proof:\n: UPUF nay no (uf /llell3) (v?/Mell3) Var[S] < Ss = llullallulla Ss : z 22 z ipa Pi Ce) lu3lhvl3 = SO max (u?/|/ull3,0? /llell3) ° iâ¬x, ppAl elles uy 1 uF d. «jul * wie ilu 2B (el ; esl) t ; do Wells Mell 1 p 5 = (hee l3llol3 + leul3the-(13) o 2 5 SG max (ee [Sle llellalle-|l2)- o\nTheorem 10.9 tells us that, if we estimated â¨u, vâ© for two vectors u and v using Algorithm 8, then the variance of our estimate will be bounded by factors that depend on the non-zero coordinates that u and v have in common. Because nz (u) â© nz (v) has at most d entries, estimates of inner product based on Threshold Sampling should generally be more accurate than those obtained from JL sketches. This is particularly the case when u and v are sparse.\n163",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 398 | 163
ââ
164
10 Sketching
# References
B. H. Bloom. Space/time trade-offs in hash coding with allowable errors. Commun. ACM, 13(7):422â426, jul 1970.
S. Bruch, F. M. Nardini, A. Ingber, and E. Liberty. An approximate algorithm for maximum inner product search over streaming sparse vectors. ACM Transactions on Information Systems, 42(2), nov 2023.
M. Daliri, J. Freire, C. Musco, A. Santos, and H. Zhang. Sampling methods for inner product sketching, 2023.
W. B. Johnson and J. Lindenstrauss. Extensions of lipschitz mappings into hilbert space. Contemporary Mathematics, 26:189â206, 1984.
D. P. Woodruff. Sketching as a tool for numerical linear algebra. Foun- dations and Trends in Theoretical Computer Science, 10(1â2):1â157, Oct 2014. ISSN 1551-305X.
Part IV Appendices
# Appendix A Collections | 2401.09350#398 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 398,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "163\nââ\n164\n10 Sketching\n# References\nB. H. Bloom. Space/time trade-offs in hash coding with allowable errors. Commun. ACM, 13(7):422â426, jul 1970.\nS. Bruch, F. M. Nardini, A. Ingber, and E. Liberty. An approximate algorithm for maximum inner product search over streaming sparse vectors. ACM Transactions on Information Systems, 42(2), nov 2023.\nM. Daliri, J. Freire, C. Musco, A. Santos, and H. Zhang. Sampling methods for inner product sketching, 2023.\nW. B. Johnson and J. Lindenstrauss. Extensions of lipschitz mappings into hilbert space. Contemporary Mathematics, 26:189â206, 1984.\nD. P. Woodruff. Sketching as a tool for numerical linear algebra. Foun- dations and Trends in Theoretical Computer Science, 10(1â2):1â157, Oct 2014. ISSN 1551-305X.\nPart IV Appendices\n# Appendix A Collections",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 399 | Part IV Appendices
# Appendix A Collections
Abstract This appendix gives a description of the vector collections used in experiments throughout this monograph. These collections demonstrate different operating points in a typical use-case. For example, some consist of dense vectors, others of sparse vectors; some have few dimensions and others are in much higher dimensions; some are relatively small while others contain a large number of points.
Table A.1 gives a description of the dense vector collections used through- out this monograph and summarizes their key statistics.
Table A.1: Dense collections used in this monograph along with select statis- tics.
Collection Vector Count Query Count Dimensions GloVe-25 [Pennington et al., 2014] GloVe-50 GloVe-100 GloVe-200 Deep1b [Yandex and Lempitsky, 2016] MS Turing [Zhang et al., 2019] Sift [Lowe, 2004] Gist [Oliva and Torralba, 2001] 1.18M 1.18M 1.18M 1.18M 9.99M 10M 1M 1M 10,000 10,000 10,000 10,000 10,000 100,000 10,000 1,000 25 50 100 200 96 100 128 960 | 2401.09350#399 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 399,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Part IV Appendices\n# Appendix A Collections\nAbstract This appendix gives a description of the vector collections used in experiments throughout this monograph. These collections demonstrate different operating points in a typical use-case. For example, some consist of dense vectors, others of sparse vectors; some have few dimensions and others are in much higher dimensions; some are relatively small while others contain a large number of points.\nTable A.1 gives a description of the dense vector collections used through- out this monograph and summarizes their key statistics.\nTable A.1: Dense collections used in this monograph along with select statis- tics.\nCollection Vector Count Query Count Dimensions GloVe-25 [Pennington et al., 2014] GloVe-50 GloVe-100 GloVe-200 Deep1b [Yandex and Lempitsky, 2016] MS Turing [Zhang et al., 2019] Sift [Lowe, 2004] Gist [Oliva and Torralba, 2001] 1.18M 1.18M 1.18M 1.18M 9.99M 10M 1M 1M 10,000 10,000 10,000 10,000 10,000 100,000 10,000 1,000 25 50 100 200 96 100 128 960",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 400 | In addition to the vector collections above, we convert a few text collections into vectors using various embedding models. These collections are described in Table A.2. Please see [Bajaj et al., 2018] for a complete description of the MS MARCO v1 collection and [Thakur et al., 2021] for the others.
When transforming the text collections of Table A.2 into vectors, we use the following embedding models:
167
168
A Collections
Table A.2: Text collections along with key statistics. The rightmost two columns report the average number of non-zero entries in data points and, in parentheses, queries for sparse vector representations of the collections.
Collection Vector Count Query Count Splade Efficient Splade MS Marco Passage NQ Quora HotpotQA Fever DBPedia 8.8M 2.68M 523K 5.23M 5.42M 4.63M 6,980 3,452 10,000 7,405 6,666 400 127 (49) 153 (51) 68 (65) 131 (59) 145 (67) 134 (49) 185 (5.9) 212 (8) 68 (8.9) 125 (13) 140 (8.6) 131 (5.9)
⢠AllMiniLM-l6-v2:1 Projects text documents into 384-dimensional dense vectors for retrieval with angular distance. | 2401.09350#400 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 400,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "In addition to the vector collections above, we convert a few text collections into vectors using various embedding models. These collections are described in Table A.2. Please see [Bajaj et al., 2018] for a complete description of the MS MARCO v1 collection and [Thakur et al., 2021] for the others.\nWhen transforming the text collections of Table A.2 into vectors, we use the following embedding models:\n167\n168\nA Collections\nTable A.2: Text collections along with key statistics. The rightmost two columns report the average number of non-zero entries in data points and, in parentheses, queries for sparse vector representations of the collections.\nCollection Vector Count Query Count Splade Efficient Splade MS Marco Passage NQ Quora HotpotQA Fever DBPedia 8.8M 2.68M 523K 5.23M 5.42M 4.63M 6,980 3,452 10,000 7,405 6,666 400 127 (49) 153 (51) 68 (65) 131 (59) 145 (67) 134 (49) 185 (5.9) 212 (8) 68 (8.9) 125 (13) 140 (8.6) 131 (5.9)\n⢠AllMiniLM-l6-v2:1 Projects text documents into 384-dimensional dense vectors for retrieval with angular distance.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 401 | ⢠AllMiniLM-l6-v2:1 Projects text documents into 384-dimensional dense vectors for retrieval with angular distance.
Tas-B [Hofst¨atter et al., 2021]: A bi-encoder model that was trained using supervision from a cross-encoder and a ColBERT [Khattab and Zaharia, 2020] model, and produces 768-dimensional dense vectors that are meant for MIPS. The checkpoint used in this work is available on HuggingFace.2 ⢠Splade [Formal et al., 2022]:3 Produces sparse representations for text. The vectors have roughly 30,000 dimensions, where each dimension cor- responds to a term in the BERT [Devlin et al., 2019] WordPiece [Wu et al., 2016] vocabulary. Non-zero entries in a vector reflect learnt term importance weights.
⢠Efficient Splade [Lassance and Clinchant, 2022]:4 This model produces queries that have far fewer non-zero entries than the original Splade model, but documents that may have a larger number of non-zero entries.
# References | 2401.09350#401 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 401,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "⢠AllMiniLM-l6-v2:1 Projects text documents into 384-dimensional dense vectors for retrieval with angular distance.\nTas-B [Hofst¨atter et al., 2021]: A bi-encoder model that was trained using supervision from a cross-encoder and a ColBERT [Khattab and Zaharia, 2020] model, and produces 768-dimensional dense vectors that are meant for MIPS. The checkpoint used in this work is available on HuggingFace.2 ⢠Splade [Formal et al., 2022]:3 Produces sparse representations for text. The vectors have roughly 30,000 dimensions, where each dimension cor- responds to a term in the BERT [Devlin et al., 2019] WordPiece [Wu et al., 2016] vocabulary. Non-zero entries in a vector reflect learnt term importance weights.\n⢠Efficient Splade [Lassance and Clinchant, 2022]:4 This model produces queries that have far fewer non-zero entries than the original Splade model, but documents that may have a larger number of non-zero entries.\n# References",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 402 | # References
P. Bajaj, D. Campos, N. Craswell, L. Deng, J. Gao, X. Liu, R. Majumder, A. McNamara, B. Mitra, T. Nguyen, M. Rosenberg, X. Song, A. Stoica, S. Tiwary, and T. Wang. Ms marco: A human generated machine reading comprehension dataset, 2018.
# 1 Available at https://huggingface.co/sentence-transformers/all-MiniLM-L6-v2 2
Available at https://huggingface.co/sentence-transformers/
msmarco-distilbert-base-tas-b 3 Pre-trained checkpoint from HuggingFace available at https://huggingface.co/naver/ splade-cocondenser-ensembledistil 4 Pre-trained checkpoints for document and query encoders were obtained from https:// huggingface.co/naver/efficient-splade-V-large-doc and https://huggingface.co/ naver/efficient-splade-V-large-query, respectively.
# References
References | 2401.09350#402 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 402,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# References\nP. Bajaj, D. Campos, N. Craswell, L. Deng, J. Gao, X. Liu, R. Majumder, A. McNamara, B. Mitra, T. Nguyen, M. Rosenberg, X. Song, A. Stoica, S. Tiwary, and T. Wang. Ms marco: A human generated machine reading comprehension dataset, 2018.\n# 1 Available at https://huggingface.co/sentence-transformers/all-MiniLM-L6-v2 2\nAvailable at https://huggingface.co/sentence-transformers/\nmsmarco-distilbert-base-tas-b 3 Pre-trained checkpoint from HuggingFace available at https://huggingface.co/naver/ splade-cocondenser-ensembledistil 4 Pre-trained checkpoints for document and query encoders were obtained from https:// huggingface.co/naver/efficient-splade-V-large-doc and https://huggingface.co/ naver/efficient-splade-V-large-query, respectively.\n# References\nReferences",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 403 | # References
References
J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova. BERT: Pre-training of deep bidirectional transformers for language understanding. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 4171â4186, June 2019.
T. Formal, C. Lassance, B. Piwowarski, and S. Clinchant. From distillation to hard negative sampling: Making sparse neural ir models more effective. In Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval, page 2353â2359, 2022.
S. Hofst¨atter, S.-C. Lin, J.-H. Yang, J. Lin, and A. Hanbury. Efficiently teaching an effective dense retriever with balanced topic aware sampling. In Proceedings of the 44th International ACM SIGIR Conference on Research and Development in Information Retrieval, page 113â122, 2021.
O. Khattab and M. Zaharia. Colbert: Efficient and effective passage search via contextualized late interaction over bert. In Proceedings of the 43rd International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 39â48, 2020. | 2401.09350#403 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 403,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# References\nReferences\nJ. Devlin, M.-W. Chang, K. Lee, and K. Toutanova. BERT: Pre-training of deep bidirectional transformers for language understanding. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 4171â4186, June 2019.\nT. Formal, C. Lassance, B. Piwowarski, and S. Clinchant. From distillation to hard negative sampling: Making sparse neural ir models more effective. In Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval, page 2353â2359, 2022.\nS. Hofst¨atter, S.-C. Lin, J.-H. Yang, J. Lin, and A. Hanbury. Efficiently teaching an effective dense retriever with balanced topic aware sampling. In Proceedings of the 44th International ACM SIGIR Conference on Research and Development in Information Retrieval, page 113â122, 2021.\nO. Khattab and M. Zaharia. Colbert: Efficient and effective passage search via contextualized late interaction over bert. In Proceedings of the 43rd International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 39â48, 2020.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 404 | In Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval, page 2220â2226, 2022.
D. G. Lowe. Distinctive image features from scale-invariant keypoints. In- ternational Journal of Computer Vision, 60:91â110, 2004.
A. Oliva and A. Torralba. Modeling the shape of the scene: A holistic rep- International Journal of Computer resentation of the spatial envelope. Vision, 42:145â175, 2001.
J. Pennington, R. Socher, and C. Manning. GloVe: Global vectors for word representation. In Proceedings of the 2014 Conference on Empirical Meth- ods in Natural Language Processing, pages 1532â1543, Oct. 2014.
N. Thakur, N. Reimers, A. R¨uckl´e, A. Srivastava, and I. Gurevych. BEIR: A heterogeneous benchmark for zero-shot evaluation of information retrieval models. In 35th Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 2), 2021. | 2401.09350#404 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 404,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "In Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval, page 2220â2226, 2022.\nD. G. Lowe. Distinctive image features from scale-invariant keypoints. In- ternational Journal of Computer Vision, 60:91â110, 2004.\nA. Oliva and A. Torralba. Modeling the shape of the scene: A holistic rep- International Journal of Computer resentation of the spatial envelope. Vision, 42:145â175, 2001.\nJ. Pennington, R. Socher, and C. Manning. GloVe: Global vectors for word representation. In Proceedings of the 2014 Conference on Empirical Meth- ods in Natural Language Processing, pages 1532â1543, Oct. 2014.\nN. Thakur, N. Reimers, A. R¨uckl´e, A. Srivastava, and I. Gurevych. BEIR: A heterogeneous benchmark for zero-shot evaluation of information retrieval models. In 35th Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 2), 2021.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 405 | Y. Wu, M. Schuster, Z. Chen, Q. V. Le, M. Norouzi, W. Macherey, M. Krikun, Y. Cao, Q. Gao, K. Macherey, J. Klingner, A. Shah, M. Johnson, X. Liu, L. Kaiser, S. Gouws, Y. Kato, T. Kudo, H. Kazawa, K. Stevens, G. Kurian, N. Patil, W. Wang, C. Young, J. Smith, J. Riesa, A. Rudnick, O. Vinyals, G. Corrado, M. Hughes, and J. Dean. Googleâs neural machine translation system: Bridging the gap between human and machine translation, 2016. A. B. Yandex and V. Lempitsky. Efficient indexing of billion-scale datasets of deep descriptors. In 2016 IEEE Conference on Computer Vision and Pattern Recognition, pages 2055-2063, 2016.
H. Zhang, X. Song, C. Xiong, C. Rosset, P. N. Bennett, N. Craswell, and S. Tiwary. Generic intent representation in web search. In Proceedings of the 42nd International ACM SIGIR Conference on Research and Develop- ment in Information Retrieval, pages 65â74, 2019.
169 | 2401.09350#405 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 405,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Y. Wu, M. Schuster, Z. Chen, Q. V. Le, M. Norouzi, W. Macherey, M. Krikun, Y. Cao, Q. Gao, K. Macherey, J. Klingner, A. Shah, M. Johnson, X. Liu, L. Kaiser, S. Gouws, Y. Kato, T. Kudo, H. Kazawa, K. Stevens, G. Kurian, N. Patil, W. Wang, C. Young, J. Smith, J. Riesa, A. Rudnick, O. Vinyals, G. Corrado, M. Hughes, and J. Dean. Googleâs neural machine translation system: Bridging the gap between human and machine translation, 2016. A. B. Yandex and V. Lempitsky. Efficient indexing of billion-scale datasets of deep descriptors. In 2016 IEEE Conference on Computer Vision and Pattern Recognition, pages 2055-2063, 2016.\nH. Zhang, X. Song, C. Xiong, C. Rosset, P. N. Bennett, N. Craswell, and S. Tiwary. Generic intent representation in web search. In Proceedings of the 42nd International ACM SIGIR Conference on Research and Develop- ment in Information Retrieval, pages 65â74, 2019.\n169",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 406 | 169
Appendix B Probability Review
Abstract We briefly review key concepts in probability in this appendix.
# B.1 Probability
We identify a probability space denoted by (â¦, F, P) with an outcome space, an events set, and a probability measure. The outcome space, â¦, is the set of all possible outcomes. For example, when flipping a two-sided coin, the outcome space is simply {0, 1}. When rolling a six-sided die, it is instead the set [6] = {1, 2, . . . , 6}.
The events set F is a set of subsets of ⦠that includes ⦠as a member and is closed under complementation and countable unions. That is, if E â F, then we must have that EâF. Furthermore, the union of countably many events Eiâs in F is itself in F: âªiEi â F. A set F that satisfies these properties is called a Ï-algebra. | 2401.09350#406 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 406,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "169\nAppendix B Probability Review\nAbstract We briefly review key concepts in probability in this appendix.\n# B.1 Probability\nWe identify a probability space denoted by (â¦, F, P) with an outcome space, an events set, and a probability measure. The outcome space, â¦, is the set of all possible outcomes. For example, when flipping a two-sided coin, the outcome space is simply {0, 1}. When rolling a six-sided die, it is instead the set [6] = {1, 2, . . . , 6}.\nThe events set F is a set of subsets of ⦠that includes ⦠as a member and is closed under complementation and countable unions. That is, if E â F, then we must have that EâF. Furthermore, the union of countably many events Eiâs in F is itself in F: âªiEi â F. A set F that satisfies these properties is called a Ï-algebra.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 407 | Finally, a function P : F > R is a probability measure if it satisfies the following conditions: P[Q] = 1; P[E] > 0 for any event E ⬠F; P[E®] = 1 â P[E); and, finally, for countably many disjoint events E;âs: P[U;E;] = Â¥, PIE. We should note that, P is also known as a âprobability distributionâ or simply a âdistribution.â The pair (2,F) is called a measurable space, and the elements of F are known as a measurable sets. The reason they are called âmeasurableâ is because they can be âmeasuredâ with P: The function P assigns values to them.
In many of the discussions throughout this monograph, we omit the out- come space and events set because that information is generally clear from context. However, a more formal treatment of our arguments requires a com- plete definition of the probability space.
171
172
B Probability Review
# B.2 Random Variables | 2401.09350#407 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 407,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Finally, a function P : F > R is a probability measure if it satisfies the following conditions: P[Q] = 1; P[E] > 0 for any event E ⬠F; P[E®] = 1 â P[E); and, finally, for countably many disjoint events E;âs: P[U;E;] = Â¥, PIE. We should note that, P is also known as a âprobability distributionâ or simply a âdistribution.â The pair (2,F) is called a measurable space, and the elements of F are known as a measurable sets. The reason they are called âmeasurableâ is because they can be âmeasuredâ with P: The function P assigns values to them.\nIn many of the discussions throughout this monograph, we omit the out- come space and events set because that information is generally clear from context. However, a more formal treatment of our arguments requires a com- plete definition of the probability space.\n171\n172\nB Probability Review\n# B.2 Random Variables",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 408 | 171
172
B Probability Review
# B.2 Random Variables
A random variable on a measurable space (â¦, F) is a measurable function X : ⦠â R. It is measurable in the sense that the preimage of any Borel set B â B is an event: X â1(B) = {Ï â ⦠| X(Ï) â B} â F.
A random variable X generates a Ï-algebra that comprises of the preimage of all Borel sets. It is denoted by Ï(X) and formally defined as Ï(X) = {X â1(B) | B â B}.
Random variables are typically categorized as discrete or continuous. X is discrete when it maps ⦠to a discrete set. In that case, its probability mass function is defined as P[X = x] for some x in its range. A continuous random variable is often associated with a probability density function, fX , such that:
b Pla< X <dj= i fx(x)dx.
Consider, for instance, the following probability density function over the real line for parameters µ â R and Ï > 0:
f (x) = â 1 2ÏÏ2 eâ (xâµ)2 2Ï2 . | 2401.09350#408 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 408,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "171\n172\nB Probability Review\n# B.2 Random Variables\nA random variable on a measurable space (â¦, F) is a measurable function X : ⦠â R. It is measurable in the sense that the preimage of any Borel set B â B is an event: X â1(B) = {Ï â ⦠| X(Ï) â B} â F.\nA random variable X generates a Ï-algebra that comprises of the preimage of all Borel sets. It is denoted by Ï(X) and formally defined as Ï(X) = {X â1(B) | B â B}.\nRandom variables are typically categorized as discrete or continuous. X is discrete when it maps ⦠to a discrete set. In that case, its probability mass function is defined as P[X = x] for some x in its range. A continuous random variable is often associated with a probability density function, fX , such that:\nb Pla< X <dj= i fx(x)dx.\nConsider, for instance, the following probability density function over the real line for parameters µ â R and Ï > 0:\nf (x) = â 1 2ÏÏ2 eâ (xâµ)2 2Ï2 .",
"title": "Foundations of Vector Retrieval",
"year": 2024
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] |
2401.09350 | 409 | f (x) = â 1 2ÏÏ2 eâ (xâµ)2 2Ï2 .
A random variable with the density function above is said to follow a Gaussian distribution with mean µ and variance Ï2, denoted by X â¼ N (µ, Ï2). When µ = 0 and Ï2 = 1, the resulting distribution is called the standard Normal distribution.
Gaussian random variables have attractive properties. For example, the sum of two independent Gaussian random variables is itself a Gaussian vari- able. Concretely, X1 â¼ N (µ1, Ï2 2), then X1 + X2 â¼ N (µ1 + µ2, Ï2 1 + Ï2 2). The sum of the squares of m independent Gaussian ran- dom variables, on the other hand, follows a Ï2-distribution with m degrees of freedom.
# B.3 Conditional Probability
Conditional probabilities give us a way to model how the probability of an event changes in the presence of extra information, such as partial knowledge about a random outcome. Concretely, if (â¦, F, P) is a probability space and A, B â F such that P[B] > 0, then the conditional probability of A given the event B is denoted by P[A | B] and defined as follows: | 2401.09350#409 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 409,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "f (x) = â 1 2ÏÏ2 eâ (xâµ)2 2Ï2 .\nA random variable with the density function above is said to follow a Gaussian distribution with mean µ and variance Ï2, denoted by X â¼ N (µ, Ï2). When µ = 0 and Ï2 = 1, the resulting distribution is called the standard Normal distribution.\nGaussian random variables have attractive properties. For example, the sum of two independent Gaussian random variables is itself a Gaussian vari- able. Concretely, X1 â¼ N (µ1, Ï2 2), then X1 + X2 â¼ N (µ1 + µ2, Ï2 1 + Ï2 2). The sum of the squares of m independent Gaussian ran- dom variables, on the other hand, follows a Ï2-distribution with m degrees of freedom.\n# B.3 Conditional Probability\nConditional probabilities give us a way to model how the probability of an event changes in the presence of extra information, such as partial knowledge about a random outcome. Concretely, if (â¦, F, P) is a probability space and A, B â F such that P[B] > 0, then the conditional probability of A given the event B is denoted by P[A | B] and defined as follows:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 410 | P[A | B] = P[A â© B] P[B] .
B.4 Independence
We use a number of helpful results concerning conditional probabilities in proofs throughout the monograph. One particularly useful inequality is what is known as the union bound and is stated as follows:
P[âªiAi] ⤠P[Ai]. i
Another fundamental property is the law of total probability. It states that, for mutually disjoint events Aiâs such that ⦠= âªAi, the probability of any event B can be expanded as follows:
P[B] = P[B | Ai] P[Ai]. i
This is easy to verify: the summand is by definition equal to P[B â© Ai] and, considering the events (B â© Ai)âs are mutually disjoint, their sum is equal to P[B â© (âªAi)] = P[B].
# B.4 Independence
Another tool that reflects the effect (or lack thereof) of additional knowledge on probabilities is the concept of independence. Two events A and B are said to be independent if P[A â© B] = P[A] Ã P[B]. Equivalently, we say that A is independent of B if and only if P[A | B] = P[A] when P[B] > 0. | 2401.09350#410 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 410,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "P[A | B] = P[A â© B] P[B] .\nB.4 Independence\nWe use a number of helpful results concerning conditional probabilities in proofs throughout the monograph. One particularly useful inequality is what is known as the union bound and is stated as follows:\nP[âªiAi] ⤠P[Ai]. i\nAnother fundamental property is the law of total probability. It states that, for mutually disjoint events Aiâs such that ⦠= âªAi, the probability of any event B can be expanded as follows:\nP[B] = P[B | Ai] P[Ai]. i\nThis is easy to verify: the summand is by definition equal to P[B â© Ai] and, considering the events (B â© Ai)âs are mutually disjoint, their sum is equal to P[B â© (âªAi)] = P[B].\n# B.4 Independence\nAnother tool that reflects the effect (or lack thereof) of additional knowledge on probabilities is the concept of independence. Two events A and B are said to be independent if P[A â© B] = P[A] à P[B]. Equivalently, we say that A is independent of B if and only if P[A | B] = P[A] when P[B] > 0.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 411 | Independence between two random variables is defined similarly but re- quires a bit more care. If X and Y are two random variables and Ï(X) and Ï(Y ) denote the Ï-algebras generated by them, then X is independent of Y if all events A â Ï(X) and B â Ï(Y ) are independent.
When a sequence of random variables are mutually independent and are drawn from the same distribution (i.e., have the same probability density function), we say the random variables are drawn tid: independent and identically-distributed. We stress that mutual independence is a stronger re- striction than pairwise independence: m events {E;}", are mutually inde- pendent if P(N;£;] = []; P[Ei).
We typically assume that data and query points are drawn iid from some (unknown) distribution. This is a standard and often necessary assumption that eases analysis.
173
174
B Probability Review
# B.5 Expectation and Variance
The expected value of a discrete random variable X is denoted by E[X] and defined as follows:
E[X] = x P[X = x]. x
When X is continuous, its expected value is based on the following Lebesgue integral:
E[X] = Xd P . ⦠| 2401.09350#411 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 411,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Independence between two random variables is defined similarly but re- quires a bit more care. If X and Y are two random variables and Ï(X) and Ï(Y ) denote the Ï-algebras generated by them, then X is independent of Y if all events A â Ï(X) and B â Ï(Y ) are independent.\nWhen a sequence of random variables are mutually independent and are drawn from the same distribution (i.e., have the same probability density function), we say the random variables are drawn tid: independent and identically-distributed. We stress that mutual independence is a stronger re- striction than pairwise independence: m events {E;}\", are mutually inde- pendent if P(N;£;] = []; P[Ei).\nWe typically assume that data and query points are drawn iid from some (unknown) distribution. This is a standard and often necessary assumption that eases analysis.\n173\n174\nB Probability Review\n# B.5 Expectation and Variance\nThe expected value of a discrete random variable X is denoted by E[X] and defined as follows:\nE[X] = x P[X = x]. x\nWhen X is continuous, its expected value is based on the following Lebesgue integral:\nE[X] = Xd P . â¦",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 412 | When X is continuous, its expected value is based on the following Lebesgue integral:
E[X] = Xd P . â¦
So when a random variable has probability density function fX , its expected value becomes:
E[X] = xfX (x)dx.
For a nonnegative random variable X, it is sometimes more convenient to unpack E X as follows instead:
E[X] = i P[X > aldz.
A fundamental property of expectation is that it is a linear operator. For- mally, E[X + Y ] = E[X] + E[Y ] for two random variables X and Y . We use this property often in proofs.
We state another important property for independent random variables that is easy to prove. If X and Y are independent, then E[XY ] = E[X] E[Y ].
The variance of a random variable is defined as follows:
Var[X] = E [(X - E[X])*] = ELX} - B[Xâ).
Unlike expectation, variance is not linear unless the random variables involved are independent. It is also easy to see that Var[aX] = a2 Var[X] for a constant a.
# B.6 Central Limit Theorem | 2401.09350#412 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 412,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "When X is continuous, its expected value is based on the following Lebesgue integral:\nE[X] = Xd P . â¦\nSo when a random variable has probability density function fX , its expected value becomes:\nE[X] = xfX (x)dx.\nFor a nonnegative random variable X, it is sometimes more convenient to unpack E X as follows instead:\nE[X] = i P[X > aldz.\nA fundamental property of expectation is that it is a linear operator. For- mally, E[X + Y ] = E[X] + E[Y ] for two random variables X and Y . We use this property often in proofs.\nWe state another important property for independent random variables that is easy to prove. If X and Y are independent, then E[XY ] = E[X] E[Y ].\nThe variance of a random variable is defined as follows:\nVar[X] = E [(X - E[X])*] = ELX} - B[Xâ).\nUnlike expectation, variance is not linear unless the random variables involved are independent. It is also easy to see that Var[aX] = a2 Var[X] for a constant a.\n# B.6 Central Limit Theorem",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 413 | # B.6 Central Limit Theorem
The result known as the Central Limit Theorem is one of the most useful tools in probability. Informally, it states that the average of iid random variables with finite mean and variance converges to a Gaussian distribution. There are several variants of this result that extend the claim to, for example, independent but not identically distributed variables. Below we repeat the formal result for the iid case.
B.6 Central Limit Theorem
Theorem B.1 Let Xiâs be a sequence of n iid random variables with finite mean µ and variance Ï2. Then, for any x â R:
ny * tim p | AMX X) He ) 1 Fut, o?/n oo V2T : noo Z
implying that Z â¼ N (0, 1).
175
Appendix C Concentration of Measure
Abstract By the strong law of large numbers, we know that the average of a sequence of m iid random variables with mean µ converges to µ with probability 1 as m tends to infinity. But how far is that average from µ when m is finite? Concentration inequalities helps us answer that question quantitatively. This appendix reviews important inequalities that are used in the proofs and arguments throughout this monograph.
# C.1 Markovâs Inequality
Lemma C.1 For a nonnegative random variable X and a nonnegative con- stant a ⥠0: | 2401.09350#413 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 413,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# B.6 Central Limit Theorem\nThe result known as the Central Limit Theorem is one of the most useful tools in probability. Informally, it states that the average of iid random variables with finite mean and variance converges to a Gaussian distribution. There are several variants of this result that extend the claim to, for example, independent but not identically distributed variables. Below we repeat the formal result for the iid case.\nB.6 Central Limit Theorem\nTheorem B.1 Let Xiâs be a sequence of n iid random variables with finite mean µ and variance Ï2. Then, for any x â R:\nny * tim p | AMX X) He ) 1 Fut, o?/n oo V2T : noo Z\nimplying that Z â¼ N (0, 1).\n175\nAppendix C Concentration of Measure\nAbstract By the strong law of large numbers, we know that the average of a sequence of m iid random variables with mean µ converges to µ with probability 1 as m tends to infinity. But how far is that average from µ when m is finite? Concentration inequalities helps us answer that question quantitatively. This appendix reviews important inequalities that are used in the proofs and arguments throughout this monograph.\n# C.1 Markovâs Inequality\nLemma C.1 For a nonnegative random variable X and a nonnegative con- stant a ⥠0:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 414 | # C.1 Markovâs Inequality
Lemma C.1 For a nonnegative random variable X and a nonnegative con- stant a ⥠0:
E[X] a P[X ⥠a] ⤠.
Proof. Recall that the expectation of a nonnegative random variable X can be written as:
E[X] = P[X ⥠x]dx. 0
Because P[X ⥠x] is monotonically nonincreasing, we can expand the above as follows to complete the proof:
E[X] > [ax > alder > [ax > aldx = aP[X > al.
ââ
177
178
C Concentration of Measure
# C.2 Chebyshevâs Inequality
Lemma C.2 For a random variable X and a constant a > 0:
Var[X] P [|X -E[X]| >a] <=.
# P
Proof.
= 2 2 Var[X] P [|x - ELx]]| > a| =P|(X -E[X]) >a| <=.
where the last step follows by the application of Markovâs inequality.
Lemma C.3 Let {Xi}n µ < â and variance Ï2 < â. For δ â (0, 1), with probability 1 â δ:
exo lz nia n | 2401.09350#414 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 414,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# C.1 Markovâs Inequality\nLemma C.1 For a nonnegative random variable X and a nonnegative con- stant a ⥠0:\nE[X] a P[X ⥠a] ⤠.\nProof. Recall that the expectation of a nonnegative random variable X can be written as:\nE[X] = P[X ⥠x]dx. 0\nBecause P[X ⥠x] is monotonically nonincreasing, we can expand the above as follows to complete the proof:\nE[X] > [ax > alder > [ax > aldx = aP[X > al.\nââ\n177\n178\nC Concentration of Measure\n# C.2 Chebyshevâs Inequality\nLemma C.2 For a random variable X and a constant a > 0:\nVar[X] P [|X -E[X]| >a] <=.\n# P\nProof.\n= 2 2 Var[X] P [|x - ELx]]| > a| =P|(X -E[X]) >a| <=.\nwhere the last step follows by the application of Markovâs inequality.\nLemma C.3 Let {Xi}n µ < â and variance Ï2 < â. For δ â (0, 1), with probability 1 â δ:\nexo lz nia n",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 415 | exo lz nia n
Proof. By Lemma C.2, for any a > 0:
i< o/n P |= X,- |= < ;
Setting the right-hand-side to δ, we obtain:
Ï2 na2 = δ =â a = Ï2 δn ,
which completes the proof.
# C.3 Chernoff Bounds
Lemma C.4 Let {X;}"_, be independent Bernoulli variables with success probability p;. Define X = 30, X; and up = E[X] = 30, p;. Then:
30, X; and up = E[X] = [x > (1+ )n| < eT MOH,
P ⤠eâh(δ)µ,
where,
h(t) = (1 + t) log(1 + t) â t.
Proof. Using Markovâs inequality of Lemma C.1 we can write the following for any t > 0:
# a
ââ
C.4 Hoeffdingâs Inequality
tx P Bs >(d+ 5)n| =P [e'* > fra] < Fle
Expanding the expectation, we obtain: | 2401.09350#415 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 415,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "exo lz nia n\nProof. By Lemma C.2, for any a > 0:\ni< o/n P |= X,- |= < ;\nSetting the right-hand-side to δ, we obtain:\nÏ2 na2 = δ =â a = Ï2 δn ,\nwhich completes the proof.\n# C.3 Chernoff Bounds\nLemma C.4 Let {X;}\"_, be independent Bernoulli variables with success probability p;. Define X = 30, X; and up = E[X] = 30, p;. Then:\n30, X; and up = E[X] = [x > (1+ )n| < eT MOH,\nP ⤠eâh(δ)µ,\nwhere,\nh(t) = (1 + t) log(1 + t) â t.\nProof. Using Markovâs inequality of Lemma C.1 we can write the following for any t > 0:\n# a\nââ\nC.4 Hoeffdingâs Inequality\ntx P Bs >(d+ 5)n| =P [e'* > fra] < Fle\nExpanding the expectation, we obtain:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 416 | tx P Bs >(d+ 5)n| =P [e'* > fra] < Fle
Expanding the expectation, we obtain:
E [e'*] =E [eee] =E Te] = [Ele] i i = Il (pie! +(1 ~ pi) i =[[(@+pi(e'- 1) < Tew = el De, by (l+t <e') i
Putting all this together gives us:
e(e De P(x > (1+5)u] < Sate (C.1)
This bound holds for any value t > 0, and in particular a value of t that minimizes the right-hand-side. To find such a t, we may differentiate the right-hand-side, set it to 0, and solve for t to obtain:
µete(etâ1)µ et(1+δ)µ â µ(1 + δ) e(etâ1)µ et(1+δ)µ = 0 =â µet = µ(1 + δ) =â t = log(1 + δ).
Substituting t into Equation (C.1) gives the desired result.
# C.4 Hoeffdingâs Inequality | 2401.09350#416 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 416,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "tx P Bs >(d+ 5)n| =P [e'* > fra] < Fle\nExpanding the expectation, we obtain:\nE [e'*] =E [eee] =E Te] = [Ele] i i = Il (pie! +(1 ~ pi) i =[[(@+pi(e'- 1) < Tew = el De, by (l+t <e') i\nPutting all this together gives us:\ne(e De P(x > (1+5)u] < Sate (C.1)\nThis bound holds for any value t > 0, and in particular a value of t that minimizes the right-hand-side. To find such a t, we may differentiate the right-hand-side, set it to 0, and solve for t to obtain:\nµete(etâ1)µ et(1+δ)µ â µ(1 + δ) e(etâ1)µ et(1+δ)µ = 0 =â µet = µ(1 + δ) =â t = log(1 + δ).\nSubstituting t into Equation (C.1) gives the desired result.\n# C.4 Hoeffdingâs Inequality",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 212 | In the second case, suppose δ(µ, u) < δ(µ, v), so that v is on the surface of B and u is in its interior. Consider the function f (Ï) = δ(v, Ï) â δ(u, Ï). Clearly, f (v) < 0 and f (µ) > 0. Therefore, there must be a point w â B on the line segment µ + λ(v â µ) for λ â [0, 1] for which f (w) = 0. That implies that δ(w, u) = δ(w, v). Furthermore, v is the closest point on the surface of B to w, so that the ball centered at w with radius δ(w, v) is entirely contained in B. This is illustrated in Figure 6.3.
Importantly, no other point in X is closer to w than u and v. So w rests ââ
Proof of Theorem 6.1. We prove the result for the case where δ is the Eu- clidean distance and leave the proof of the more general case as an exercise. (Hint: To prove the general case you should make the line segment argument as in the proof of Lemma 6.1.) | 2401.09350#212 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 212,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "In the second case, suppose δ(µ, u) < δ(µ, v), so that v is on the surface of B and u is in its interior. Consider the function f (Ï) = δ(v, Ï) â δ(u, Ï). Clearly, f (v) < 0 and f (µ) > 0. Therefore, there must be a point w â B on the line segment µ + λ(v â µ) for λ â [0, 1] for which f (w) = 0. That implies that δ(w, u) = δ(w, v). Furthermore, v is the closest point on the surface of B to w, so that the ball centered at w with radius δ(w, v) is entirely contained in B. This is illustrated in Figure 6.3.\nImportantly, no other point in X is closer to w than u and v. So w rests ââ\nProof of Theorem 6.1. We prove the result for the case where δ is the Eu- clidean distance and leave the proof of the more general case as an exercise. (Hint: To prove the general case you should make the line segment argument as in the proof of Lemma 6.1.)",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 213 | Suppose the greedy search for q stops at some local optimum u that is different from the global optimum, uâ, and that (u, uâ) /â Eâotherwise, the algorithm must terminate at uâ instead. Let r = δ(q, u).
By assumption we have that the ball centered at q with radius r, B(q, r), is non-empty because it must contain uâ whose distance to q is less than r. Let v be the point in this ball that is closest to u. Consider now the ball B((u + v)/2, δ(u, v)/2). This ball is empty: otherwise v would not be the closest point to u. By Lemma 6.1, we must have that (u, v) â E. This is a ââ contradiction because the greedy search cannot stop at u.
79
80
6 Graph Algorithms
Notice that Theorem 6.1 holds for any graph that contains the De- launay graph. The next theorem strengthens this result to show that the Delaunay graph represents the minimal edge set that guarantees an optimal solution through greedy traversal.
Theorem 6.2 The Delaunay graph is the minimal graph over which the best- first-search algorithm gives the optimal solution to the top-1 retrieval problem. | 2401.09350#213 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 213,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Suppose the greedy search for q stops at some local optimum u that is different from the global optimum, uâ, and that (u, uâ) /â Eâotherwise, the algorithm must terminate at uâ instead. Let r = δ(q, u).\nBy assumption we have that the ball centered at q with radius r, B(q, r), is non-empty because it must contain uâ whose distance to q is less than r. Let v be the point in this ball that is closest to u. Consider now the ball B((u + v)/2, δ(u, v)/2). This ball is empty: otherwise v would not be the closest point to u. By Lemma 6.1, we must have that (u, v) â E. This is a ââ contradiction because the greedy search cannot stop at u.\n79\n80\n6 Graph Algorithms\nNotice that Theorem 6.1 holds for any graph that contains the De- launay graph. The next theorem strengthens this result to show that the Delaunay graph represents the minimal edge set that guarantees an optimal solution through greedy traversal.\nTheorem 6.2 The Delaunay graph is the minimal graph over which the best- first-search algorithm gives the optimal solution to the top-1 retrieval problem.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 214 | Theorem 6.2 The Delaunay graph is the minimal graph over which the best- first-search algorithm gives the optimal solution to the top-1 retrieval problem.
In other words, if a graph does not contain the Delaunay graph, then we can find queries for which the greedy traversal from an entry point does not produce the optimal top-1 solution.
Proof of Theorem 6.2. Suppose that the data points X are in general posi- tion. Suppose further that G = (V, E) is a graph built from X , and that u and v are two nodes in the graph. Suppose further that (v, u) /â E but that that edge exists in the Delaunay graph of X .
If we could sample a query point q such that δ(q, u) < δ(q, v) but δ(q, w) > max(δ(q, u), δ(q, v)) for all w ̸= u, v, then we are done. That is because, if we entered the graph through v, then v is a local optimum in its neighborhood: all other points that are connected to v have a distance larger than δ(q, v). But v is not the globally optimal solution, so that the greedy traversal does not converge to the optimal solution. | 2401.09350#214 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 214,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Theorem 6.2 The Delaunay graph is the minimal graph over which the best- first-search algorithm gives the optimal solution to the top-1 retrieval problem.\nIn other words, if a graph does not contain the Delaunay graph, then we can find queries for which the greedy traversal from an entry point does not produce the optimal top-1 solution.\nProof of Theorem 6.2. Suppose that the data points X are in general posi- tion. Suppose further that G = (V, E) is a graph built from X , and that u and v are two nodes in the graph. Suppose further that (v, u) /â E but that that edge exists in the Delaunay graph of X .\nIf we could sample a query point q such that δ(q, u) < δ(q, v) but δ(q, w) > max(δ(q, u), δ(q, v)) for all w ̸= u, v, then we are done. That is because, if we entered the graph through v, then v is a local optimum in its neighborhood: all other points that are connected to v have a distance larger than δ(q, v). But v is not the globally optimal solution, so that the greedy traversal does not converge to the optimal solution.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 215 | It remains to show that such a point q always exists. Suppose it did not. That is, for any point that is in the Voronoi region of u, there is a data point w ̸= v that is closer to it than v. If that were the case, then no ball whose boundary passes through u and v can be empty, which contradicts Lemma 6.1 ââ (the âempty-circleâ property of the Delaunay graph).
As a final remark on the Delaunay graph and its use in top-1 retrieval, we note that the Delaunay graph only makes sense if we have precise knowledge of the structure of the space (i.e., the metric). It is not enough to have just pairwise distances between points in a collection X . In fact, Navarro [2002] showed that if pairwise distances are all we know about a collection of points, then the only sensible graph that contains the Delaunay graph and is amenable to greedy search is the complete graph. This is stated as the following theorem. | 2401.09350#215 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 215,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "It remains to show that such a point q always exists. Suppose it did not. That is, for any point that is in the Voronoi region of u, there is a data point w ̸= v that is closer to it than v. If that were the case, then no ball whose boundary passes through u and v can be empty, which contradicts Lemma 6.1 ââ (the âempty-circleâ property of the Delaunay graph).\nAs a final remark on the Delaunay graph and its use in top-1 retrieval, we note that the Delaunay graph only makes sense if we have precise knowledge of the structure of the space (i.e., the metric). It is not enough to have just pairwise distances between points in a collection X . In fact, Navarro [2002] showed that if pairwise distances are all we know about a collection of points, then the only sensible graph that contains the Delaunay graph and is amenable to greedy search is the complete graph. This is stated as the following theorem.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 216 | Theorem 6.3 Suppose the structure of the metric space is unknown, but we have pairwise distances between the points in a collection X , due to an arbitrary, but proper distance function δ. For every choice of u, v â X , there is a choice of the metric space such that (u, v) â E, where G = (V, E) is a Delaunay graph for X .
Proof. The idea behind the proof is to assume (u, v) /â E, then construct a query point that necessitates the existence of an edge between u and v. To
6.2 The Delaunay Graph
that end, consider a query point q such that its distance to u is C + ϵ for some constant C and ϵ > 0, its distance to v is C, and its distance to every other point in X is C + 2ϵ.
This is a valid arrangement if we choose ϵ such that ϵ ⤠1/2 minx,yâX δ(x, y) and C such that C ⥠1/2 maxx,yâX δ(x, y). It is easy to verify that, if those conditions hold, a point q with the prescribed distances can exist as the distances do not violate any of the triangle inequalities. | 2401.09350#216 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 216,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Theorem 6.3 Suppose the structure of the metric space is unknown, but we have pairwise distances between the points in a collection X , due to an arbitrary, but proper distance function δ. For every choice of u, v â X , there is a choice of the metric space such that (u, v) â E, where G = (V, E) is a Delaunay graph for X .\nProof. The idea behind the proof is to assume (u, v) /â E, then construct a query point that necessitates the existence of an edge between u and v. To\n6.2 The Delaunay Graph\nthat end, consider a query point q such that its distance to u is C + ϵ for some constant C and ϵ > 0, its distance to v is C, and its distance to every other point in X is C + 2ϵ.\nThis is a valid arrangement if we choose ϵ such that ϵ ⤠1/2 minx,yâX δ(x, y) and C such that C ⥠1/2 maxx,yâX δ(x, y). It is easy to verify that, if those conditions hold, a point q with the prescribed distances can exist as the distances do not violate any of the triangle inequalities.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 217 | Consider then a search starting from node u. If (u, v) /â E, then for the search algorithm to walk from u to the optimal solution, v, it must first get farther from q. But we know by the properties of the Delaunay graph that such an event implies that u (which would be the local optimum) must be the global optimum. That is clearly not true. So we must have that (u, v) â E, ââ giving the claim.
# 6.2.4 Top-k Retrieval
Let us now consider the general case of top-k retrieval over the Delaunay graph. The following result states that Algorithm 3 is correct if executed on any graph that contains the Delaunay graph, in the sense that it returns the optimal solution to top-k retrieval.
Theorem 6.4 Let G = (V, E) be a graph that contains the Delaunay graph of m vectors X â Rd. Algorithm 3 over G gives the optimal solution to the top-k retrieval problem for any arbitrary query q if δ(·, ·) is proper.
Proof. As with the proof of Theorem 6.1, we show the result for the case where δ is the Euclidean distance and leave the proof of the more general case as an exercise. | 2401.09350#217 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 217,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Consider then a search starting from node u. If (u, v) /â E, then for the search algorithm to walk from u to the optimal solution, v, it must first get farther from q. But we know by the properties of the Delaunay graph that such an event implies that u (which would be the local optimum) must be the global optimum. That is clearly not true. So we must have that (u, v) â E, ââ giving the claim.\n# 6.2.4 Top-k Retrieval\nLet us now consider the general case of top-k retrieval over the Delaunay graph. The following result states that Algorithm 3 is correct if executed on any graph that contains the Delaunay graph, in the sense that it returns the optimal solution to top-k retrieval.\nTheorem 6.4 Let G = (V, E) be a graph that contains the Delaunay graph of m vectors X â Rd. Algorithm 3 over G gives the optimal solution to the top-k retrieval problem for any arbitrary query q if δ(·, ·) is proper.\nProof. As with the proof of Theorem 6.1, we show the result for the case where δ is the Euclidean distance and leave the proof of the more general case as an exercise.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 218 | The proof is similar to the proof of Theorem 6.1 but the argument needs a little more care when k > 1. Suppose Algorithm 3 for q stops at some local optimum set Q that is different from the global optimum, Qâ. In other words, Q â³ Qâ ̸= â
where â³ denotes the symmetric difference between sets.
Let r = maxuâQ δ(q, u) and consider the ball B(q, r). Because Q â³ Qâ ̸= â
, there must be at least k points in the interior of this ball. Let v /â Q be a point in the interior and suppose u â Q is its closest point in the ball. Clearly, the ball B((u + v)/2, δ(u, v)/2) is empty: otherwise v would not be the closest point to u. By Lemma 6.1, we must have that (u, v) â E. This is a contradiction because Algorithm 3 would, before termination, place v in Q ââ to replace the node that is on the surface of the ball.
81
82
6 Graph Algorithms | 2401.09350#218 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 218,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The proof is similar to the proof of Theorem 6.1 but the argument needs a little more care when k > 1. Suppose Algorithm 3 for q stops at some local optimum set Q that is different from the global optimum, Qâ. In other words, Q â³ Qâ ̸= â
where â³ denotes the symmetric difference between sets.\nLet r = maxuâQ δ(q, u) and consider the ball B(q, r). Because Q â³ Qâ ̸= â
, there must be at least k points in the interior of this ball. Let v /â Q be a point in the interior and suppose u â Q is its closest point in the ball. Clearly, the ball B((u + v)/2, δ(u, v)/2) is empty: otherwise v would not be the closest point to u. By Lemma 6.1, we must have that (u, v) â E. This is a contradiction because Algorithm 3 would, before termination, place v in Q ââ to replace the node that is on the surface of the ball.\n81\n82\n6 Graph Algorithms",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 220 | Fig. 6.4: Comparison of the Delaunay graph (a) with the k-NN graph for k = 2 (b) for an example collection in R2. In the illustration of the directed k-NN graph, edges that go in both directions are rendered as lines without arrow heads. Notice that, the top left node cannot be reached from the rest of the graph.
# 6.2.5 The k-NN Graph
From our discussion of Voronoi diagrams and Delaunay graphs, it appears as though we have found the graph we have been looking for. Indeed, the Delaunay graph of a collection of vectors gives us the exact solution to top- k queries, using such a strikingly simple search algorithm. Sadly, the story does not end there and, as usual, the relentless curse of dimensionality poses a serious challenge.
The first major obstacle in high dimensions actually concerns the construc- tion of the Delaunay graph itself. While there are many algorithms [Edels- brunner and Shah, 1992, Guibas et al., 1992, Guibas and Stolfi, 1985] that can be used to construct the Delaunay graphâor, to be more precise, to perform Delaunay triangulationâall suffer from an exponential dependence on the number of dimensions d. So building the graph itself seems infeasible when d is too large. | 2401.09350#220 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 220,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Fig. 6.4: Comparison of the Delaunay graph (a) with the k-NN graph for k = 2 (b) for an example collection in R2. In the illustration of the directed k-NN graph, edges that go in both directions are rendered as lines without arrow heads. Notice that, the top left node cannot be reached from the rest of the graph.\n# 6.2.5 The k-NN Graph\nFrom our discussion of Voronoi diagrams and Delaunay graphs, it appears as though we have found the graph we have been looking for. Indeed, the Delaunay graph of a collection of vectors gives us the exact solution to top- k queries, using such a strikingly simple search algorithm. Sadly, the story does not end there and, as usual, the relentless curse of dimensionality poses a serious challenge.\nThe first major obstacle in high dimensions actually concerns the construc- tion of the Delaunay graph itself. While there are many algorithms [Edels- brunner and Shah, 1992, Guibas et al., 1992, Guibas and Stolfi, 1985] that can be used to construct the Delaunay graphâor, to be more precise, to perform Delaunay triangulationâall suffer from an exponential dependence on the number of dimensions d. So building the graph itself seems infeasible when d is too large.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 221 | Even if we were able to quickly construct the Delaunay graph for a large collection of points, we would face a second debilitating issue: The graph is close to complete! While exact bounds on the expected number of edges in the graph surely depend on the data distribution, in high dimensions the graph becomes necessarily more dense. Consider, for example, vectors that are independent and identically-distributed in each dimension. Recall from our discussion from Chapter 2, that in such an arrangement of points, the distance between any pair of points tends to concentrate sharply. As a result, the Delaunay graph has an edge between almost every pair of nodes.
6.2 The Delaunay Graph
These two problems are rather serious, rendering the guarantees of the De- launay graph for top-k retrieval mainly of theoretical interest. These same dif- ficulties motivated research to approximate the Delaunay graph. One promi- nent method is known as the k-NN graph [Ch´avez and Tellez, 2010, Hajebi et al., 2011, Fu and Cai, 2016]. | 2401.09350#221 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 221,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Even if we were able to quickly construct the Delaunay graph for a large collection of points, we would face a second debilitating issue: The graph is close to complete! While exact bounds on the expected number of edges in the graph surely depend on the data distribution, in high dimensions the graph becomes necessarily more dense. Consider, for example, vectors that are independent and identically-distributed in each dimension. Recall from our discussion from Chapter 2, that in such an arrangement of points, the distance between any pair of points tends to concentrate sharply. As a result, the Delaunay graph has an edge between almost every pair of nodes.\n6.2 The Delaunay Graph\nThese two problems are rather serious, rendering the guarantees of the De- launay graph for top-k retrieval mainly of theoretical interest. These same dif- ficulties motivated research to approximate the Delaunay graph. One promi- nent method is known as the k-NN graph [Ch´avez and Tellez, 2010, Hajebi et al., 2011, Fu and Cai, 2016].",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 222 | The k-NN graph is simply a k-regular graph where every node (i.e., vector) is connected to its k closest nodes. So (u, v) â E if v â arg min(k) wâX δ(u, w). Note that, the resulting graph may be directed, depending on the choice of δ. We should mention, however, that researchers have explored ways of turning the k-NN graph into an undirected graph [Ch´avez and Tellez, 2010]. An example is depicted in Figure 6.4.
We must remark on two important properties of the k-NN graph. First, the graph itself is far more efficient to construct than the Delaunay graph [Chen et al., 2009, Vaidya, 1989, Connor and Kumar, 2010, Dong et al., 2011]. The second point concerns the connectivity of the graph. As Brito et al. [1997] show, under mild conditions governing the distribution of the vectors and with k large enough, the resulting graph has a high probability of being connected. When k is too small, on the other hand, the resulting graph may become too sparse, leading the greedy search algorithm to get stuck in local minima. | 2401.09350#222 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 222,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The k-NN graph is simply a k-regular graph where every node (i.e., vector) is connected to its k closest nodes. So (u, v) â E if v â arg min(k) wâX δ(u, w). Note that, the resulting graph may be directed, depending on the choice of δ. We should mention, however, that researchers have explored ways of turning the k-NN graph into an undirected graph [Ch´avez and Tellez, 2010]. An example is depicted in Figure 6.4.\nWe must remark on two important properties of the k-NN graph. First, the graph itself is far more efficient to construct than the Delaunay graph [Chen et al., 2009, Vaidya, 1989, Connor and Kumar, 2010, Dong et al., 2011]. The second point concerns the connectivity of the graph. As Brito et al. [1997] show, under mild conditions governing the distribution of the vectors and with k large enough, the resulting graph has a high probability of being connected. When k is too small, on the other hand, the resulting graph may become too sparse, leading the greedy search algorithm to get stuck in local minima.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 223 | Finally, at the risk of stating the obvious, the k-NN graph does not enjoy any of the guarantees of the Delaunay graph in the context of top-k retrieval. That is simply because the k-NN graph is likely only a sub-graph of the Delaunay graph, while Theorems 6.1 and 6.4 are provable only for super- graphs of the Delaunay graph. Despite these deficiencies, the k-NN graph remains an important component of advanced graph-based, approximate top- k retrieval algorithms.
# 6.2.6 The Case of Inner Product
Everything we have stated so far about the Voronoi diagrams and its duality with the Delaunay graph was contingent on δ(·, ·) being proper. In particular, the proof of the optimality guarantees implicitly require non-negativity and the triangle inequality. As a result, none of the results apply to MIPS prima facie. As it turns out, however, we could extend the definition of Voronoi regions and the Delaunay graph to inner product, and present guarantees for MIPS (with k = 1, but not with k > 1). That is the proposal by Morozov and Babenko [2018].
83
84
6 Graph Algorithms | 2401.09350#223 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 223,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Finally, at the risk of stating the obvious, the k-NN graph does not enjoy any of the guarantees of the Delaunay graph in the context of top-k retrieval. That is simply because the k-NN graph is likely only a sub-graph of the Delaunay graph, while Theorems 6.1 and 6.4 are provable only for super- graphs of the Delaunay graph. Despite these deficiencies, the k-NN graph remains an important component of advanced graph-based, approximate top- k retrieval algorithms.\n# 6.2.6 The Case of Inner Product\nEverything we have stated so far about the Voronoi diagrams and its duality with the Delaunay graph was contingent on δ(·, ·) being proper. In particular, the proof of the optimality guarantees implicitly require non-negativity and the triangle inequality. As a result, none of the results apply to MIPS prima facie. As it turns out, however, we could extend the definition of Voronoi regions and the Delaunay graph to inner product, and present guarantees for MIPS (with k = 1, but not with k > 1). That is the proposal by Morozov and Babenko [2018].\n83\n84\n6 Graph Algorithms",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 224 | # (a) Euclidean Delaunay
# (b) Inner Product Voronoi
# (c) IP-Delaunay
Fig. 6.5: Comparison of the Voronoi diagrams and Delaunay graphs for the same set of points but according to Euclidean distance versus inner prod- uct. Note that, for the non-metric distance function based on inner prod- uct, the Voronoi regions are convex cones determined by the intersection of half-spaces passing through the origin. Observe additionally that the inner product-induced Voronoi region of a point (those in white) may be an empty set. Such points can never be the solution to the 1-MIPS problem.
# 6.2.6.1 The IP-Delaunay Graph
Let us begin by characterizing the Voronoi regions for inner product. The Voronoi region Ru of a vector u â X comprises of the set of points for which u is the maximizer of inner product:
Ru = {x â Rd | u = arg max â¨x, vâ©}. vâX
This definition is essentially the same as how we defined the Voronoi region for a proper δ, and, indeed, the resulting Voronoi diagram is a partitioning of the whole space. The properties of the resulting Voronoi regions, however, could not be more different. | 2401.09350#224 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 224,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# (a) Euclidean Delaunay\n# (b) Inner Product Voronoi\n# (c) IP-Delaunay\nFig. 6.5: Comparison of the Voronoi diagrams and Delaunay graphs for the same set of points but according to Euclidean distance versus inner prod- uct. Note that, for the non-metric distance function based on inner prod- uct, the Voronoi regions are convex cones determined by the intersection of half-spaces passing through the origin. Observe additionally that the inner product-induced Voronoi region of a point (those in white) may be an empty set. Such points can never be the solution to the 1-MIPS problem.\n# 6.2.6.1 The IP-Delaunay Graph\nLet us begin by characterizing the Voronoi regions for inner product. The Voronoi region Ru of a vector u â X comprises of the set of points for which u is the maximizer of inner product:\nRu = {x â Rd | u = arg max â¨x, vâ©}. vâX\nThis definition is essentially the same as how we defined the Voronoi region for a proper δ, and, indeed, the resulting Voronoi diagram is a partitioning of the whole space. The properties of the resulting Voronoi regions, however, could not be more different.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 225 | First, recall from Section 1.3.3 that inner product does not even enjoy what we called coincidence. That is, in general, u = arg maxvâX â¨u, vâ© is not guaranteed. So it is very much possible that Ru is empty for some u â X . Second, when Ru ̸= â
, it is a convex cone that is the intersection of half-spaces that pass through the origin. So the Voronoi regions have a substantially different geometry. Figure 6.5(b) visualizes this phenomenon.
Moving on to the Delaunay graph, Morozov and Babenko [2018] construct the graph in much the same way as before and call the resulting graph the IP- Delaunay graph. Two nodes u, v â V in the IP-Delaunay graph are connected if their Voronoi regions intersect: Ru â© Rv ̸= â
. Note that, by the reasoning above, the nodes whose Voronoi regions are empty will be isolated in the
6.2 The Delaunay Graph
graph. These nodes represent vectors that can never be the solution to MIPS for any queryâremember that we are only considering k = 1. So it would be inconsequential if we removed these nodes from the graph. This is also illustrated in Figure 6.5(c). | 2401.09350#225 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 225,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "First, recall from Section 1.3.3 that inner product does not even enjoy what we called coincidence. That is, in general, u = arg maxvâX â¨u, vâ© is not guaranteed. So it is very much possible that Ru is empty for some u â X . Second, when Ru ̸= â
, it is a convex cone that is the intersection of half-spaces that pass through the origin. So the Voronoi regions have a substantially different geometry. Figure 6.5(b) visualizes this phenomenon.\nMoving on to the Delaunay graph, Morozov and Babenko [2018] construct the graph in much the same way as before and call the resulting graph the IP- Delaunay graph. Two nodes u, v â V in the IP-Delaunay graph are connected if their Voronoi regions intersect: Ru â© Rv ̸= â
. Note that, by the reasoning above, the nodes whose Voronoi regions are empty will be isolated in the\n6.2 The Delaunay Graph\ngraph. These nodes represent vectors that can never be the solution to MIPS for any queryâremember that we are only considering k = 1. So it would be inconsequential if we removed these nodes from the graph. This is also illustrated in Figure 6.5(c).",
"title": "Foundations of Vector Retrieval",
"year": 2024
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2401.09350 | 226 | Considering the data structure above for inner product, Morozov and Babenko [2018] prove the following result to give optimality guarantee for the greedy search algorithm for 1-MIPS (granted we enter the graph from a non-isolated node). Nothing, however, may be said about k-MIPS.
Theorem 6.5 Suppose G = (V, E) is a graph that contains the IP-Delaunay graph for a collection X minus the isolated nodes. Invoking Algorithm 3 with k = 1 and δ(·, ·) = ââ¨Â·, ·⩠gives the optimal solution to the top-1 MIPS problem. | 2401.09350#226 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 226,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Considering the data structure above for inner product, Morozov and Babenko [2018] prove the following result to give optimality guarantee for the greedy search algorithm for 1-MIPS (granted we enter the graph from a non-isolated node). Nothing, however, may be said about k-MIPS.\nTheorem 6.5 Suppose G = (V, E) is a graph that contains the IP-Delaunay graph for a collection X minus the isolated nodes. Invoking Algorithm 3 with k = 1 and δ(·, ·) = ââ¨Â·, ·⩠gives the optimal solution to the top-1 MIPS problem.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 227 | Proof. If we showed that a local optimum is necessarily the global optimum, then we are done. To that end, consider a query q for which Algorithm 3 terminates when it reaches node u â X which is distinct from the globally optimal solution uâ /â N (u). In other words, we have that â¨q, uâ© > â¨q, vâ© for all v â N (u), but â¨q, uââ© > â¨q, uâ© and (u, uâ) /â E. If that is true, then q /â Ru, the Voronoi region of u, but instead we must have that q â Ruâ . | 2401.09350#227 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 227,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Proof. If we showed that a local optimum is necessarily the global optimum, then we are done. To that end, consider a query q for which Algorithm 3 terminates when it reaches node u â X which is distinct from the globally optimal solution uâ /â N (u). In other words, we have that â¨q, uâ© > â¨q, vâ© for all v â N (u), but â¨q, uââ© > â¨q, uâ© and (u, uâ) /â E. If that is true, then q /â Ru, the Voronoi region of u, but instead we must have that q â Ruâ .",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 228 | Now define the collection ¥ & N(u) U {u}, and consider the Voronoi diagram of the resulting collection. It is easy to show that the Voronoi region of u in the presence of points in ¥ is the same as its region given V. From before, we also know that q ¢ R,. Considering the fact that R¢ = Uvex Rv: q must belong to R, for some v ⬠X with v #u. That implies that (g,v) > (q,u) for some v ⬠¥ \u. But because v ⬠N(u) (by construction), the last inequality poses a contradiction to our premise that u was locally optimal. O
In addition to the fact that the IP-Delaunay graph does not answer top-k queries, it also suffers from the same deficiencies we noted for the Euclidean Delaunay graph earlier in this section. Naturally then, to make the data struc- ture more practical in high-dimensional regimes, we must resort to heuristics and approximations, which in their simplest form may be the k-MIPS graph (i.e., a k-NN graph where the distance function for finding the top-k nodes is inner product). This is the general direction Morozov and Babenko [2018] and a few other works have explored [Liu et al., 2019, Zhou et al., 2019]. | 2401.09350#228 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 228,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Now define the collection ¥ & N(u) U {u}, and consider the Voronoi diagram of the resulting collection. It is easy to show that the Voronoi region of u in the presence of points in ¥ is the same as its region given V. From before, we also know that q ¢ R,. Considering the fact that R¢ = Uvex Rv: q must belong to R, for some v ⬠X with v #u. That implies that (g,v) > (q,u) for some v ⬠¥ \\u. But because v ⬠N(u) (by construction), the last inequality poses a contradiction to our premise that u was locally optimal. O\nIn addition to the fact that the IP-Delaunay graph does not answer top-k queries, it also suffers from the same deficiencies we noted for the Euclidean Delaunay graph earlier in this section. Naturally then, to make the data struc- ture more practical in high-dimensional regimes, we must resort to heuristics and approximations, which in their simplest form may be the k-MIPS graph (i.e., a k-NN graph where the distance function for finding the top-k nodes is inner product). This is the general direction Morozov and Babenko [2018] and a few other works have explored [Liu et al., 2019, Zhou et al., 2019].",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 229 | As in the case of metric distance functions, none of the guarantees stated above port over to these approximate graphs. But, once again, empirical evidence gathered from a variety of datasets show that these graphs perform reasonably well in practice, even for top-k with k > 1.
# 6.2.6.2 Is the IP-Delaunay Graph Necessary?
Morozov and Babenko [2018] justify the need for developing the IP-Delaunay graph by comparing its structure with the following alternative: First, apply
85
86
6 Graph Algorithms
a MIPS-to-NN asymmetric transformation [Bachrach et al., 2014] from R4 to R¢+1. This involves transforming a data point u with da(u) = [u; V/1 â |[ull3] and a query point q with ¢,(v) = [v;0]. Next, construct the standard (Eu- clidean) Delaunay graph over the transformed vectors.
What happens if we form the Delaunay graph on the transformed collection oa(X)? Observe the Euclidean distance between ¢a(u) and ¢q(v) for two vectors u,u ⬠¥: | 2401.09350#229 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 229,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "As in the case of metric distance functions, none of the guarantees stated above port over to these approximate graphs. But, once again, empirical evidence gathered from a variety of datasets show that these graphs perform reasonably well in practice, even for top-k with k > 1.\n# 6.2.6.2 Is the IP-Delaunay Graph Necessary?\nMorozov and Babenko [2018] justify the need for developing the IP-Delaunay graph by comparing its structure with the following alternative: First, apply\n85\n86\n6 Graph Algorithms\na MIPS-to-NN asymmetric transformation [Bachrach et al., 2014] from R4 to R¢+1. This involves transforming a data point u with da(u) = [u; V/1 â |[ull3] and a query point q with ¢,(v) = [v;0]. Next, construct the standard (Eu- clidean) Delaunay graph over the transformed vectors.\nWhat happens if we form the Delaunay graph on the transformed collection oa(X)? Observe the Euclidean distance between ¢a(u) and ¢q(v) for two vectors u,u ⬠¥:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 230 | IIda(u) â bar) [I3 = ea(u)|l3 + [Ibalw)|3 â 2a), balv)) (\lull +1 â lula) + (loll + 1 â [|l3) â (u,v) â 2y/ (1 = |lel|3) (1 = IIe 113).
Should we use these distances to construct the Delaunay graph, the resulting structure will have nothing to do with the original MIPS problem. That is because the L2 distance between a pair of transformed data points is not rank-equivalent to the inner product between the original data points. For this reason, Morozov and Babenko [2018] argue that the IP-Delaunay graph is a more sensible choice.
However, we note that their argument rests heavily on their particular choice of MIPS-to-NN transformation. The transformation they chose makes sense in contexts where we only care about preserving the inner product between query-data point pairs. But when forming the Delaunay graph, pre- serving inner product between pairs of data points, too, is imperative. That is the reason why we lose rank-equivalence between L2 in Rd+1 and inner product in Rd. | 2401.09350#230 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 230,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "IIda(u) â bar) [I3 = ea(u)|l3 + [Ibalw)|3 â 2a), balv)) (\\lull +1 â lula) + (loll + 1 â [|l3) â (u,v) â 2y/ (1 = |lel|3) (1 = IIe 113).\nShould we use these distances to construct the Delaunay graph, the resulting structure will have nothing to do with the original MIPS problem. That is because the L2 distance between a pair of transformed data points is not rank-equivalent to the inner product between the original data points. For this reason, Morozov and Babenko [2018] argue that the IP-Delaunay graph is a more sensible choice.\nHowever, we note that their argument rests heavily on their particular choice of MIPS-to-NN transformation. The transformation they chose makes sense in contexts where we only care about preserving the inner product between query-data point pairs. But when forming the Delaunay graph, pre- serving inner product between pairs of data points, too, is imperative. That is the reason why we lose rank-equivalence between L2 in Rd+1 and inner product in Rd.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 231 | There are, in fact, MIPS-to-NN transformations that are more appropriate for this problem and would invalidate the argument for the need for the IP- Delaunay graph. Consider for example ¢q : R¢ + R¢+⢠for a collection X of m vectors as follows: ¢a(u\) = u © (\/1 = lu |[3)ecasi), where ul is the i-th data point in the collection, and e; is the j-th standard basis vector. In other words, the i-th d-dimensional data point is augmented with an m-dimensional sparse vector whose i-th coordinate is non-zero. The query transformation is simply $,(q) = q@0, where 0 ⬠R⢠is a vector of m zeros. Despite the dependence on m, the transformation is remarkably easy to manage: the sparse subspace of every vector has at most one non-zero coor- dinate, making the doubling dimension of the sparse subspace O(log m) by Lemma 3.5. Distance computation between the transformed vectors, too, has negligible overhead. Crucially, we regain rank-equivalence between L2 dis- tance in R¢+⢠and inner product in R@ not only for query-data point pairs, but also for pairs of data points: | 2401.09350#231 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 231,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "There are, in fact, MIPS-to-NN transformations that are more appropriate for this problem and would invalidate the argument for the need for the IP- Delaunay graph. Consider for example ¢q : R¢ + R¢+⢠for a collection X of m vectors as follows: ¢a(u\\) = u © (\\/1 = lu |[3)ecasi), where ul is the i-th data point in the collection, and e; is the j-th standard basis vector. In other words, the i-th d-dimensional data point is augmented with an m-dimensional sparse vector whose i-th coordinate is non-zero. The query transformation is simply $,(q) = q@0, where 0 ⬠R⢠is a vector of m zeros. Despite the dependence on m, the transformation is remarkably easy to manage: the sparse subspace of every vector has at most one non-zero coor- dinate, making the doubling dimension of the sparse subspace O(log m) by Lemma 3.5. Distance computation between the transformed vectors, too, has negligible overhead. Crucially, we regain rank-equivalence between L2 dis- tance in R¢+⢠and inner product in R@ not only for query-data point pairs, but also for pairs of data points:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 232 | â¥Ïd(u) â Ïd(v)â¥2 2 = â¥Ïd(u)â¥2 2 + â¥Ïd(v)â¥2 2 â 2â¨Ïd(u), Ïd(v)â© = 2 â 2â¨u, vâ©.
6.3 The Small World Phenomenon
Finally, unlike the IP-Delaunay graph, the standard Delaunay graph in Rd+m over the transformed vector collection has optimality guarantee for the top-k retrieval problem per Theorem 6.4. It is, as such, unclear if the IP-Delaunay graph is even necessary as a theoretical tool.
In other words, suppose we are given a collection of points X and inner product as the similarity function. Consider a graph index where the presence of an edge is decided based on the inner product between data points. Take another graph index built for the transformed X using the transformation described above from Rd to Rd+m, and where the edge set is formed on the basis of the Euclidean distance between two (transformed) data points. The two graphs are equivalent. | 2401.09350#232 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 232,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "â¥Ïd(u) â Ïd(v)â¥2 2 = â¥Ïd(u)â¥2 2 + â¥Ïd(v)â¥2 2 â 2â¨Ïd(u), Ïd(v)â© = 2 â 2â¨u, vâ©.\n6.3 The Small World Phenomenon\nFinally, unlike the IP-Delaunay graph, the standard Delaunay graph in Rd+m over the transformed vector collection has optimality guarantee for the top-k retrieval problem per Theorem 6.4. It is, as such, unclear if the IP-Delaunay graph is even necessary as a theoretical tool.\nIn other words, suppose we are given a collection of points X and inner product as the similarity function. Consider a graph index where the presence of an edge is decided based on the inner product between data points. Take another graph index built for the transformed X using the transformation described above from Rd to Rd+m, and where the edge set is formed on the basis of the Euclidean distance between two (transformed) data points. The two graphs are equivalent.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 233 | The larger point is that, MIPS over m points in Rd is equivalent to NN over a transformation of the points in Rd+m. While the transforma- tion increases the apparent dimensionality, the intrinsic dimensionality of the data only increases by O(log m).
# 6.3 The Small World Phenomenon
Consider, once again, the Delaunay graph but, for the moment, set aside the fact that it is a prohibitively-expensive data structure to maintain for high dimensional vectors. By construction, every node in the graph is only connected to its Voronoi neighbors (i.e., nodes whose Voronoi region intersects with the current nodeâs). We showed that such a topology affords navigability, in the sense that the greedy procedure in Algorithm 3 can traverse the graph only based on information about immediate neighbors of a node and yet arrive at the globally optimal solution to the top-k retrieval problem.
Let us take a closer look at the traversal algorithm for the case of k = 1. It is clear that, navigating from the entry node to the solution takes us through every Voronoi region along the path. That is, we cannot âskipâ a Voronoi region that lies between the entry node and the answer. This implies that the running time of Algorithm 3 is directly affected by the diameter of the graph (in addition to the average degree of nodes). | 2401.09350#233 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 233,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The larger point is that, MIPS over m points in Rd is equivalent to NN over a transformation of the points in Rd+m. While the transforma- tion increases the apparent dimensionality, the intrinsic dimensionality of the data only increases by O(log m).\n# 6.3 The Small World Phenomenon\nConsider, once again, the Delaunay graph but, for the moment, set aside the fact that it is a prohibitively-expensive data structure to maintain for high dimensional vectors. By construction, every node in the graph is only connected to its Voronoi neighbors (i.e., nodes whose Voronoi region intersects with the current nodeâs). We showed that such a topology affords navigability, in the sense that the greedy procedure in Algorithm 3 can traverse the graph only based on information about immediate neighbors of a node and yet arrive at the globally optimal solution to the top-k retrieval problem.\nLet us take a closer look at the traversal algorithm for the case of k = 1. It is clear that, navigating from the entry node to the solution takes us through every Voronoi region along the path. That is, we cannot âskipâ a Voronoi region that lies between the entry node and the answer. This implies that the running time of Algorithm 3 is directly affected by the diameter of the graph (in addition to the average degree of nodes).",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 234 | Can we enhance this topology by adding long-range edges between non- Voronoi neighbors, so that we may skip over a fraction of Voronoi regions? After all, Theorem 6.4 guarantees navigability so long as the graph contains the Delaunay graph. Starting with the Delaunay graph and inserting long- range edges, then, will not take away any of the guarantees. But, what is the
87
88
6 Graph Algorithms
(a) (b)
© © e®@e 0 © ©, ©),@ © 0 * eoâ 6SeGe e@ oe o) oe ee 0 eee © e© eee e
9â9â9â9â-9â9 @âeâoâo-.eâ0d 1 Le I @âoâe;=9â9â6 oâoâ@;-9â9â9 acs ae oâeâoâoâoâo
Fig. 6.6: Example graphs generated by the probabilistic model introduced by Kleinberg [2000]. (a) illustrates the directed edges from u for the following configuration: r = 2, l = 0. (b) renders the regular structure for r = 1, where edges without arrows are bi-directional, and the long-range edges for node u with configuration l = 2. | 2401.09350#234 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 234,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Can we enhance this topology by adding long-range edges between non- Voronoi neighbors, so that we may skip over a fraction of Voronoi regions? After all, Theorem 6.4 guarantees navigability so long as the graph contains the Delaunay graph. Starting with the Delaunay graph and inserting long- range edges, then, will not take away any of the guarantees. But, what is the\n87\n88\n6 Graph Algorithms\n(a) (b) \n© © e®@e 0 © ©, ©),@ © 0 * eoâ 6SeGe e@ oe o) oe ee 0 eee © e© eee e\n9â9â9â9â-9â9 @âeâoâo-.eâ0d 1 Le I @âoâe;=9â9â6 oâoâ@;-9â9â9 acs ae oâeâoâoâoâo\nFig. 6.6: Example graphs generated by the probabilistic model introduced by Kleinberg [2000]. (a) illustrates the directed edges from u for the following configuration: r = 2, l = 0. (b) renders the regular structure for r = 1, where edges without arrows are bi-directional, and the long-range edges for node u with configuration l = 2.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 235 | right number of long-range edges and how do we determine which remote nodes should be connected? This section reviews the theoretical results that help answer these questions.
# 6.3.1 Lattice Networks
Let us begin with a simple topology that is relatively easy to reason aboutâ we will see later how the results from this section can be generalized to the Delaunay graph. The graph we have in mind is a lattice network where m à m nodes are laid on a two-dimensional grid. Define the distance between two nodes as their lattice (Manhattan) distance (i.e., the minimal number of horizontal and vertical hops that connect two nodes). That is the network examined by Kleinberg [2000] in a seminal paper that studied the effect of long-range edges on the time complexity of Algorithm 3.
We should take a brief detour and note that, Kleinberg [2000], in fact, studied the problem of transmitting a message from a source to a known tar- get using the best-first-search algorithm, and quantified the average number of hops required to do that in the presence of a variety of classes of long- range edges. That, in turn, was inspired by a social phenomenon colloquially known as the âsmall-world phenomenonâ: The empirical observation that two strangers are linked by a short chain of acquaintances [Milgram, 1967, Jeffrey Travers, 1969]. | 2401.09350#235 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 235,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "right number of long-range edges and how do we determine which remote nodes should be connected? This section reviews the theoretical results that help answer these questions.\n# 6.3.1 Lattice Networks\nLet us begin with a simple topology that is relatively easy to reason aboutâ we will see later how the results from this section can be generalized to the Delaunay graph. The graph we have in mind is a lattice network where m à m nodes are laid on a two-dimensional grid. Define the distance between two nodes as their lattice (Manhattan) distance (i.e., the minimal number of horizontal and vertical hops that connect two nodes). That is the network examined by Kleinberg [2000] in a seminal paper that studied the effect of long-range edges on the time complexity of Algorithm 3.\nWe should take a brief detour and note that, Kleinberg [2000], in fact, studied the problem of transmitting a message from a source to a known tar- get using the best-first-search algorithm, and quantified the average number of hops required to do that in the presence of a variety of classes of long- range edges. That, in turn, was inspired by a social phenomenon colloquially known as the âsmall-world phenomenonâ: The empirical observation that two strangers are linked by a short chain of acquaintances [Milgram, 1967, Jeffrey Travers, 1969].",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 236 | In particular, Kleinberg [2000] was interested in explaining why and un- der what types of long-range edges should our greedy algorithm be able to navigate to the optimal solution, by only utilizing information about im- mediate neighbors. To investigate this question, Kleinberg [2000] introduced
6.3 The Small World Phenomenon
the following probabilistic model of the lattice topology as an abstraction of individuals and their social connections.
# 6.3.1.1 The Probabilistic Model
Every node in the graph has a (directed) edge with every other node within lattice distance r, for some fixed hyperparameter r ⥠1. These connections make up the regular structure of the graph. Overlaid with this structure is a set of random, long-range edges that are generated according to the following probabilistic model. For fixed constants l ⥠0 and α ⥠0, we insert a directed edge between every node u and l other nodes, where a node v ̸= u is selected with probability proportional to δ(u, v)âα where δ(u, v) = â¥u â vâ¥1 is the lattice distance. Example graphs generated by this process are depicted in Figure 6.6. | 2401.09350#236 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 236,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "In particular, Kleinberg [2000] was interested in explaining why and un- der what types of long-range edges should our greedy algorithm be able to navigate to the optimal solution, by only utilizing information about im- mediate neighbors. To investigate this question, Kleinberg [2000] introduced\n6.3 The Small World Phenomenon\nthe following probabilistic model of the lattice topology as an abstraction of individuals and their social connections.\n# 6.3.1.1 The Probabilistic Model\nEvery node in the graph has a (directed) edge with every other node within lattice distance r, for some fixed hyperparameter r ⥠1. These connections make up the regular structure of the graph. Overlaid with this structure is a set of random, long-range edges that are generated according to the following probabilistic model. For fixed constants l ⥠0 and α ⥠0, we insert a directed edge between every node u and l other nodes, where a node v ̸= u is selected with probability proportional to δ(u, v)âα where δ(u, v) = â¥u â vâ¥1 is the lattice distance. Example graphs generated by this process are depicted in Figure 6.6.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 237 | The model above is reasonably powerful as it can express a variety of topologies. For example, when l = 0, the resulting graph has no long-range edges. When l > 0 and α = 0, then every node v ̸= u in the graph has an equal chance of being the destination of a long-range edge from u. When α is large, the protocol becomes more biased to forming a long-range connection from u to nodes closer to it.
# 6.3.1.2 The Claim
Kleinberg [2000] shows that, when 0 < a < 2, the best-first-search algorithm must visit at least O,,;,.(m@-/9) nodes. When a > 2, the number of nodes visited is at least O,.1,4(mâ°â?)/(¢-))) instead. But, rather uniquely, when a = 2 and r =/=1, we visit a number of nodes that is at most poly-logarithmic in m.
Theorem 6.6 states this result formally. But before we present the theorem, we state a useful lemma. | 2401.09350#237 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 237,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The model above is reasonably powerful as it can express a variety of topologies. For example, when l = 0, the resulting graph has no long-range edges. When l > 0 and α = 0, then every node v ̸= u in the graph has an equal chance of being the destination of a long-range edge from u. When α is large, the protocol becomes more biased to forming a long-range connection from u to nodes closer to it.\n# 6.3.1.2 The Claim\nKleinberg [2000] shows that, when 0 < a < 2, the best-first-search algorithm must visit at least O,,;,.(m@-/9) nodes. When a > 2, the number of nodes visited is at least O,.1,4(mâ°â?)/(¢-))) instead. But, rather uniquely, when a = 2 and r =/=1, we visit a number of nodes that is at most poly-logarithmic in m.\nTheorem 6.6 states this result formally. But before we present the theorem, we state a useful lemma.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 238 | Theorem 6.6 states this result formally. But before we present the theorem, we state a useful lemma.
Lemma 6.2 Generate a lattice G = (V, E) of m à m nodes using the proba- bilistic model above with α = 2 and l = 1. The probability that there exists a long-range edge between two nodes u, v â V is at least δ(u, v)â2/4 ln(6m).
Proof. u chooses v # u as its long-range destination with the following prob- ability: 6(u,v)~?/ Cwdu 5(u,w)~?. Let us first bound the denominator as follows:
89
90
6 Graph Algorithms
In the expression above, we derived the first inequality by iterating over all possible (lattice) distances between m2 nodes on a two-dimensional grid (ranging from 1 to 2m â 2 if u and w are at diagonally opposite corners), and noticing that there are at most 4i nodes at distance i from node u. From this we infer that the probability that node (u, v) â E is at least δ(u, v)â2/4 ln(6m). ââ | 2401.09350#238 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 238,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Theorem 6.6 states this result formally. But before we present the theorem, we state a useful lemma.\nLemma 6.2 Generate a lattice G = (V, E) of m à m nodes using the proba- bilistic model above with α = 2 and l = 1. The probability that there exists a long-range edge between two nodes u, v â V is at least δ(u, v)â2/4 ln(6m).\nProof. u chooses v # u as its long-range destination with the following prob- ability: 6(u,v)~?/ Cwdu 5(u,w)~?. Let us first bound the denominator as follows:\n89\n90\n6 Graph Algorithms\nIn the expression above, we derived the first inequality by iterating over all possible (lattice) distances between m2 nodes on a two-dimensional grid (ranging from 1 to 2m â 2 if u and w are at diagonally opposite corners), and noticing that there are at most 4i nodes at distance i from node u. From this we infer that the probability that node (u, v) â E is at least δ(u, v)â2/4 ln(6m). ââ",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 239 | Theorem 6.6 Generate a lattice G = bilistic model above with a = 2 andr = beginning from any arbitrary node and O(log? m) nodes on average. (V,â¬) of mxm nodes using the proba- l=1. The best-first-search algorithm ending in a target node visits at most
Proof. Define a sequence of sets Ai, where each Ai consists of nodes whose distance to the target uâ is greater than 2i and at most 2i+1. Formally, Ai = {v â V | 2i < δ(uâ, v) ⤠2i+1}. Suppose that the algorithm is currently in node u and that log m ⤠δ(u, uâ) < m, so that u â Ai for some log log m ⤠i < log m. What is the probability that the algorithm exits the set Ai in the next step?
That happens when one of uâs neighbors has a distance to uâ that is at most j=0 Aj. 2i. In other words, u must have a neighbor that is in the set A<i = âªj=iâ1 The number of nodes in A<i is at least:
2 2 1+) os=1+ s=1 "(2' +1) Ss Qi, 2 | 2401.09350#239 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 239,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Theorem 6.6 Generate a lattice G = bilistic model above with a = 2 andr = beginning from any arbitrary node and O(log? m) nodes on average. (V,â¬) of mxm nodes using the proba- l=1. The best-first-search algorithm ending in a target node visits at most\nProof. Define a sequence of sets Ai, where each Ai consists of nodes whose distance to the target uâ is greater than 2i and at most 2i+1. Formally, Ai = {v â V | 2i < δ(uâ, v) ⤠2i+1}. Suppose that the algorithm is currently in node u and that log m ⤠δ(u, uâ) < m, so that u â Ai for some log log m ⤠i < log m. What is the probability that the algorithm exits the set Ai in the next step?\nThat happens when one of uâs neighbors has a distance to uâ that is at most j=0 Aj. 2i. In other words, u must have a neighbor that is in the set A<i = âªj=iâ1 The number of nodes in A<i is at least:\n2 2 1+) os=1+ s=1 \"(2' +1) Ss Qi, 2",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 240 | 2 2 1+) os=1+ s=1 "(2' +1) Ss Qi, 2
How likely is it that (u, v) â E if v â A<i? We apply Lemma 6.2, noting that the distance of each of the nodes in A<i with u is at most 2i+1+2i < 2i+2. We obtain that, the probability that u is connected to a node in A<i is at least 22iâ1(2i+2)â2/4 ln(6m) = 1/128 ln(6m).
Next, consider the total number of nodes in Ai that are visited by the algorithm, and denote it by Xi. In expectation, we have the following:
E[X\] j=l j=l SPM 2 <d (1- 1 â 128 In(6m). 128In(6m) 8In(6m)
We obtain the same bound if we repeat the arguments for i = log m. When 0 ⤠i < log log m, the algorithm visits at most log m nodes, so that the bound above is trivially true.
Denoting by X the total number o . wos log f nodes visited, X = og in Li=0
j=0 Xj, we conclude that:
6.3 The Small World Phenomenon | 2401.09350#240 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 240,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "2 2 1+) os=1+ s=1 \"(2' +1) Ss Qi, 2\nHow likely is it that (u, v) â E if v â A<i? We apply Lemma 6.2, noting that the distance of each of the nodes in A<i with u is at most 2i+1+2i < 2i+2. We obtain that, the probability that u is connected to a node in A<i is at least 22iâ1(2i+2)â2/4 ln(6m) = 1/128 ln(6m).\nNext, consider the total number of nodes in Ai that are visited by the algorithm, and denote it by Xi. In expectation, we have the following:\nE[X\\] j=l j=l SPM 2 <d (1- 1 â 128 In(6m). 128In(6m) 8In(6m)\nWe obtain the same bound if we repeat the arguments for i = log m. When 0 ⤠i < log log m, the algorithm visits at most log m nodes, so that the bound above is trivially true.\nDenoting by X the total number o . wos log f nodes visited, X = og in Li=0\nj=0 Xj, we conclude that:\n6.3 The Small World Phenomenon",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 241 | j=0 Xj, we conclude that:
6.3 The Small World Phenomenon
E[X] < (1 + logm)(128In(6m)) = O(log? m),
thereby completing the proof.
The argument made by Kleinberg [2000] is that, in a lattice network where each node is connected to its (at most four) nearest neighbors within unit distance, and where every node has a long-range edge to one other node with probability that is proportional to 1/δ(·, ·)2, then the greedy algorithm visits at most a poly-logarithmic number of nodes. Translating this result to the case of top-1 retrieval using Algorithm 3 over the same network, we can state that the time complexity of the algorithm is O(log2 m), because the total number of neighbors per node is O(1).
While this result is significant, it only holds for the lattice network with the lattice distance. It has thus no bearing on the time complexity of top-1 retrieval over the Delaunay graph with the Euclidean distance. In the next section, we will see how Beaumont et al. [2007a] close this gap.
# 6.3.2 Extension to the Delaunay Graph | 2401.09350#241 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 241,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "j=0 Xj, we conclude that:\n6.3 The Small World Phenomenon\nE[X] < (1 + logm)(128In(6m)) = O(log? m),\nthereby completing the proof.\nThe argument made by Kleinberg [2000] is that, in a lattice network where each node is connected to its (at most four) nearest neighbors within unit distance, and where every node has a long-range edge to one other node with probability that is proportional to 1/δ(·, ·)2, then the greedy algorithm visits at most a poly-logarithmic number of nodes. Translating this result to the case of top-1 retrieval using Algorithm 3 over the same network, we can state that the time complexity of the algorithm is O(log2 m), because the total number of neighbors per node is O(1).\nWhile this result is significant, it only holds for the lattice network with the lattice distance. It has thus no bearing on the time complexity of top-1 retrieval over the Delaunay graph with the Euclidean distance. In the next section, we will see how Beaumont et al. [2007a] close this gap.\n# 6.3.2 Extension to the Delaunay Graph",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 242 | # 6.3.2 Extension to the Delaunay Graph
We saw in the preceding section that, the secret to creating a provably nav- igable graph where the best-first-search algorithm visits a poly-logarithmic number of nodes in the lattice network, was the highly specific distribution from which long-range edges were sampled. That element turns out to be the key ingredient when extending the results to the Delaunay graph too, as Beaumont et al. [2007a] argue.
We will describe the algorithm for data in the two-dimensional unit square. That is, we assume that the collection of data points X and query points are in [0, 1]2. That the vectors are bounded is not a limitation per seâas we discussed previously, we can always normalize vectors into the hypercube without loss of generality. That the algorithm does not naturally extend to high dimensions is a serious limitation, but then again, that is not surprising considering the Delaunay graph is expensive to construct. However, in the next section, we will review heuristics that take the idea to higher dimensions.
# 6.3.2.1 The Probabilistic Model | 2401.09350#242 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 242,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 6.3.2 Extension to the Delaunay Graph\nWe saw in the preceding section that, the secret to creating a provably nav- igable graph where the best-first-search algorithm visits a poly-logarithmic number of nodes in the lattice network, was the highly specific distribution from which long-range edges were sampled. That element turns out to be the key ingredient when extending the results to the Delaunay graph too, as Beaumont et al. [2007a] argue.\nWe will describe the algorithm for data in the two-dimensional unit square. That is, we assume that the collection of data points X and query points are in [0, 1]2. That the vectors are bounded is not a limitation per seâas we discussed previously, we can always normalize vectors into the hypercube without loss of generality. That the algorithm does not naturally extend to high dimensions is a serious limitation, but then again, that is not surprising considering the Delaunay graph is expensive to construct. However, in the next section, we will review heuristics that take the idea to higher dimensions.\n# 6.3.2.1 The Probabilistic Model",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 243 | # 6.3.2.1 The Probabilistic Model
Much like the lattice network, we assume there is a base graph and a num- ber of randomly generated long-range edges between nodes. For the base graph, Beaumont et al. [2007a] take the Delaunay graph.1 As for the long- range edges, each node has a directed edge to one other node that is selected
1 Beaumont et al. [2007a] additionally connect all nodes that are within δMin â 1/m distance from each other, where δMin is chosen such that the expected number of uniformly91
ââ
92
6 Graph Algorithms | 2401.09350#243 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 243,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 6.3.2.1 The Probabilistic Model\nMuch like the lattice network, we assume there is a base graph and a num- ber of randomly generated long-range edges between nodes. For the base graph, Beaumont et al. [2007a] take the Delaunay graph.1 As for the long- range edges, each node has a directed edge to one other node that is selected\n1 Beaumont et al. [2007a] additionally connect all nodes that are within δMin â 1/m distance from each other, where δMin is chosen such that the expected number of uniformly91\nââ\n92\n6 Graph Algorithms",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 245 | We already know from Theorem 6.4 that, because the network above con- tains the Delaunay graph, it is navigable by Algorithm 3. What remains to be investigated is what type of long-range edges could reduce the number of hops (i.e., the number of nodes the algorithm must visit as it navigates from an entry node to a target node). Because at each hop the algorithm needs to evaluate distances with O(1) neighbors, improving the number of steps directly improves the time complexity of Algorithm 3 (for the case of k = 1). Beaumont et al. [2007a] show that, if long-range edges are chosen according to the following protocol, then the number of hops is poly-logarithmic in m. The protocol is simple: For a node u in the graph, first sample α uniformly from [ln δâ, ln δâ], where δâ = minv,wâX δ(v, w) and δâ = maxv,wâX δ(v, w). Then choose θ â¼ [0, 2Ï], to finally obtain uâ² = u + z where z is the vector [eα cos θ, eα sin θ]. Let us | 2401.09350#245 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 245,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "We already know from Theorem 6.4 that, because the network above con- tains the Delaunay graph, it is navigable by Algorithm 3. What remains to be investigated is what type of long-range edges could reduce the number of hops (i.e., the number of nodes the algorithm must visit as it navigates from an entry node to a target node). Because at each hop the algorithm needs to evaluate distances with O(1) neighbors, improving the number of steps directly improves the time complexity of Algorithm 3 (for the case of k = 1). Beaumont et al. [2007a] show that, if long-range edges are chosen according to the following protocol, then the number of hops is poly-logarithmic in m. The protocol is simple: For a node u in the graph, first sample α uniformly from [ln δâ, ln δâ], where δâ = minv,wâX δ(v, w) and δâ = maxv,wâX δ(v, w). Then choose θ â¼ [0, 2Ï], to finally obtain uâ² = u + z where z is the vector [eα cos θ, eα sin θ]. Let us",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 247 | # 6.3.2.2 The Claim
Given the resulting graph, Beaumont et al. [2007a] state and prove that the average number of hops taken by the best-first-search algorithm is poly- logarithmic. Before we discuss the claim, however, let us state a useful lemma.
Lemma 6.3 The probability that the long-range end-point from a node u lands in a ball centered at another node v with radius βδ(u, v) for some small
distributed points in a ball of radius δMin is 1. We reformulate their method without δMin in the present monograph to simplify their result.
6.3 The Small World Phenomenon
β â [0, 1] is at least Kβ2/(1 + β)2 where K = (2 ln â)â1 and â = δâ/δâ is the aspect ratio.
Proof. The probability that the long-range end-point lands in an area dS that covers the distance [r, r + dr] and angle [θ, θ + dθ], for small dr and dθ, is: | 2401.09350#247 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 247,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 6.3.2.2 The Claim\nGiven the resulting graph, Beaumont et al. [2007a] state and prove that the average number of hops taken by the best-first-search algorithm is poly- logarithmic. Before we discuss the claim, however, let us state a useful lemma.\nLemma 6.3 The probability that the long-range end-point from a node u lands in a ball centered at another node v with radius βδ(u, v) for some small\ndistributed points in a ball of radius δMin is 1. We reformulate their method without δMin in the present monograph to simplify their result.\n6.3 The Small World Phenomenon\nβ â [0, 1] is at least Kβ2/(1 + β)2 where K = (2 ln â)â1 and â = δâ/δâ is the aspect ratio.\nProof. The probability that the long-range end-point lands in an area dS that covers the distance [r, r + dr] and angle [θ, θ + dθ], for small dr and dθ, is:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 248 | dθ 2Ï ln(r + dr) â ln r ln δâ â ln δâ â dθ 2Ï dr/r ln â = 1 2Ï ln â rdθdr r2 â dS 2Ï ln âr2
.
Observe now that the distance between a point u and any point in the ball described in the lemma is at most (1 + β)δ(u, v). We can therefore see that the probability that the long-range end-point lands in B(v, βδ(u, v)) is at least:
Ïβ2δ(u, v)2 2Ï ln â(1 + β)2δ(u, v)2 = β2 2 ln(â)(1 + β)2 ,
as required.
Theorem 6.7 Generate a graph G = (V, E) according to the probabilistic model described above, for vectors in [0, 1]2 equipped with the Euclidean dis- tance δ(·, ·). The number of nodes visited by the best-first-search algorithm starting from any arbitrary node and ending at a target node is O(log2 â). | 2401.09350#248 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 248,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "dθ 2Ï ln(r + dr) â ln r ln δâ â ln δâ â dθ 2Ï dr/r ln â = 1 2Ï ln â rdθdr r2 â dS 2Ï ln âr2\n.\nObserve now that the distance between a point u and any point in the ball described in the lemma is at most (1 + β)δ(u, v). We can therefore see that the probability that the long-range end-point lands in B(v, βδ(u, v)) is at least:\nÏβ2δ(u, v)2 2Ï ln â(1 + β)2δ(u, v)2 = β2 2 ln(â)(1 + β)2 ,\nas required.\nTheorem 6.7 Generate a graph G = (V, E) according to the probabilistic model described above, for vectors in [0, 1]2 equipped with the Euclidean dis- tance δ(·, ·). The number of nodes visited by the best-first-search algorithm starting from any arbitrary node and ending at a target node is O(log2 â).",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 249 | Proof. The proof follows the same reasoning as in the proof of Theorem 6.6. Suppose we are currently at node u and that uâ is our target node. By Lemma 6.3, the probability that the long-range end-point of u lands in B(uâ, δ(u, uâ)/6) is at least 1/98 ln â. As such, the total number of hops, X, from u to a point in B(uâ, δ(u, uâ)/6) has the following expectation:
E[X] = )OPIX >i <> (1- shy) 1 = 98InA.
Every time the algorithm moves from the current node u to some other node in B(uâ, δ(u, uâ)/6), the distance is shrunk by a factor of 6/5. As such, the total number of hops in expectation is at most:
(ines 4) x (98 In 4) = O(log? A).
We highlight that, Beaumont et al. [2007a] choose the interval from which α is sampled differently. Indeed, α in their work is chosen uniformly from the 2. Substituting that configuration into Theorem 6.7 range δMin â 1/m and gives an expected number of hops that is O(log2 m).
93 | 2401.09350#249 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 249,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Proof. The proof follows the same reasoning as in the proof of Theorem 6.6. Suppose we are currently at node u and that uâ is our target node. By Lemma 6.3, the probability that the long-range end-point of u lands in B(uâ, δ(u, uâ)/6) is at least 1/98 ln â. As such, the total number of hops, X, from u to a point in B(uâ, δ(u, uâ)/6) has the following expectation:\nE[X] = )OPIX >i <> (1- shy) 1 = 98InA.\nEvery time the algorithm moves from the current node u to some other node in B(uâ, δ(u, uâ)/6), the distance is shrunk by a factor of 6/5. As such, the total number of hops in expectation is at most:\n(ines 4) x (98 In 4) = O(log? A).\nWe highlight that, Beaumont et al. [2007a] choose the interval from which α is sampled differently. Indeed, α in their work is chosen uniformly from the 2. Substituting that configuration into Theorem 6.7 range δMin â 1/m and gives an expected number of hops that is O(log2 m).\n93",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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