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2401.09350 | 0 | 4 2 0 2
n a J 7 1 ] S D . s c [
1 v 0 5 3 9 0 . 1 0 4 2 : v i X r a
Sebastian Bruch
# Foundations of Vector Retrieval
# Preface
We are witness to a few years of remarkable developments in Artificial Intelligence with the use of advanced machine learning algorithms, and in particular, deep learning. Gargantuan, complex neural networks that can learn through self-supervisionâand quickly so with the aid of special- ized hardwareâtransformed the research landscape so dramatically that, incremental overnight it seems, many fields experienced not the usual, progress, but rather a leap forward. Machine translation, natural language understanding, information retrieval, recommender systems, and computer vision are but a few examples of research areas that have had to grapple with the shock. Countless other disciplines beyond computer science such as robotics, biology, and chemistry too have benefited from deep learning. | 2401.09350#0 | 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 | [] | [
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] | {
"authors": "Sebastian Bruch",
"chunk_id": 0,
"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": "4 2 0 2\nn a J 7 1 ] S D . s c [\n1 v 0 5 3 9 0 . 1 0 4 2 : v i X r a\nSebastian Bruch\n# Foundations of Vector Retrieval\n# Preface\nWe are witness to a few years of remarkable developments in Artificial Intelligence with the use of advanced machine learning algorithms, and in particular, deep learning. Gargantuan, complex neural networks that can learn through self-supervisionâand quickly so with the aid of special- ized hardwareâtransformed the research landscape so dramatically that, incremental overnight it seems, many fields experienced not the usual, progress, but rather a leap forward. Machine translation, natural language understanding, information retrieval, recommender systems, and computer vision are but a few examples of research areas that have had to grapple with the shock. Countless other disciplines beyond computer science such as robotics, biology, and chemistry too have benefited from deep learning.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} |
2401.09350 | 1 | These neural networks and their training algorithms may be complex, and the scope of their impact broad and wide, but nonetheless they are simply functions in a high-dimensional space. A trained neural network takes a vector as input, crunches and transforms it in various ways, and produces another vector, often in some other space. An image may thereby be turned into a vector, a song into a sequence of vectors, and a social network as a structured collection of vectors. It seems as though much of human knowledge, or at least what is expressed as text, audio, image, and video, has a vector representation in one form or another.
It should be noted that representing data as vectors is not unique to neural networks and deep learning. In fact, long before learnt vector representations of pieces of dataâwhat is commonly known as âembeddingsââcame along, data was often encoded as hand-crafted feature vectors. Each feature quanti- fied into continuous or discrete values some facet of the data that was deemed relevant to a particular task (such as classification or regression). Vectors of that form, too, reflect our understanding of a real-world object or concept. | 2401.09350#1 | 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 | [] | [
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] | {
"authors": "Sebastian Bruch",
"chunk_id": 1,
"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": "These neural networks and their training algorithms may be complex, and the scope of their impact broad and wide, but nonetheless they are simply functions in a high-dimensional space. A trained neural network takes a vector as input, crunches and transforms it in various ways, and produces another vector, often in some other space. An image may thereby be turned into a vector, a song into a sequence of vectors, and a social network as a structured collection of vectors. It seems as though much of human knowledge, or at least what is expressed as text, audio, image, and video, has a vector representation in one form or another.\nIt should be noted that representing data as vectors is not unique to neural networks and deep learning. In fact, long before learnt vector representations of pieces of dataâwhat is commonly known as âembeddingsââcame along, data was often encoded as hand-crafted feature vectors. Each feature quanti- fied into continuous or discrete values some facet of the data that was deemed relevant to a particular task (such as classification or regression). Vectors of that form, too, reflect our understanding of a real-world object or concept.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} |
2401.09350 | 2 | If new and old knowledge can be squeezed into a collection of learnt or hand-crafted vectors, what useful things does that enable us to do? A metaphor that might help us think about that question is this: An ever- evolving database full of such vectors that capture various pieces of data can
v
# vi
be understood as a memory of sorts. We can then recall information from this memory to answer questions, learn about past and present events, reason about new problems, generate new content, and more.
# Vector Retrieval
Mathematically, ârecalling informationâ translates to finding vectors that are most similar to a query vector. The query vector represents what we wish to know more about, or recall information for. So, if we have a particular question in mind, the query is the vector representation of that question. If we wish to know more about an event, our query is that event expressed as a vector. If we wish to predict the function of a protein, perhaps we may learn a thing or two from known proteins that have a similar structure to the one in question, making a vector representation of the structure of our new protein a query. | 2401.09350#2 | 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 | [] | [
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] | {
"authors": "Sebastian Bruch",
"chunk_id": 2,
"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": "If new and old knowledge can be squeezed into a collection of learnt or hand-crafted vectors, what useful things does that enable us to do? A metaphor that might help us think about that question is this: An ever- evolving database full of such vectors that capture various pieces of data can\nv\n# vi\nbe understood as a memory of sorts. We can then recall information from this memory to answer questions, learn about past and present events, reason about new problems, generate new content, and more.\n# Vector Retrieval\nMathematically, ârecalling informationâ translates to finding vectors that are most similar to a query vector. The query vector represents what we wish to know more about, or recall information for. So, if we have a particular question in mind, the query is the vector representation of that question. If we wish to know more about an event, our query is that event expressed as a vector. If we wish to predict the function of a protein, perhaps we may learn a thing or two from known proteins that have a similar structure to the one in question, making a vector representation of the structure of our new protein a query.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} |
2401.09350 | 3 | Similarity is then a function of two vectors, quantifying how similar two vectors are. It may, for example, be based on the Euclidean distance between the query vector and a database vector, where similar vectors have a smaller distance. Or it may instead be based on the inner product between two vec- tors. Or their angle. Whatever function we use to measure similarity between pieces of data defines the structure of a database.
Finding k vectors from a database that have the highest similarity to a query vector is known as the top-k retrieval problem. When similarity is based on the Euclidean distance, the resulting problem is known as near- est neighbor search. Inner product for similarity leads to a problem known as maximum inner product search. Angular distance gives maximum cosine similarity search. These are mathematical formulations of the mechanism we called ârecalling information.â
The need to search for similar vectors from a large database arises in virtually every single one of our online transactions. Indeed, when we search the web for information about a topic, the search engine itself performs this similarity search over millions of web documents to find what may lexically or semantically match our query. Recommender systems find the most similar items to your browsing history by encoding items as vectors and, effectively, searching through a database of such items. Finding an old photo in a photo library, as another routine example, boils down to performing a similarity search over vector representations of images. | 2401.09350#3 | 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 | [] | [
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] | {
"authors": "Sebastian Bruch",
"chunk_id": 3,
"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": "Similarity is then a function of two vectors, quantifying how similar two vectors are. It may, for example, be based on the Euclidean distance between the query vector and a database vector, where similar vectors have a smaller distance. Or it may instead be based on the inner product between two vec- tors. Or their angle. Whatever function we use to measure similarity between pieces of data defines the structure of a database.\nFinding k vectors from a database that have the highest similarity to a query vector is known as the top-k retrieval problem. When similarity is based on the Euclidean distance, the resulting problem is known as near- est neighbor search. Inner product for similarity leads to a problem known as maximum inner product search. Angular distance gives maximum cosine similarity search. These are mathematical formulations of the mechanism we called ârecalling information.â\nThe need to search for similar vectors from a large database arises in virtually every single one of our online transactions. Indeed, when we search the web for information about a topic, the search engine itself performs this similarity search over millions of web documents to find what may lexically or semantically match our query. Recommender systems find the most similar items to your browsing history by encoding items as vectors and, effectively, searching through a database of such items. Finding an old photo in a photo library, as another routine example, boils down to performing a similarity search over vector representations of images.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} |
2401.09350 | 4 | A neural network that is trained to perform a general task such as question- answering, could conceivably augment its view of the world by ârecallingâ in- formation from such a database and finding answers to new questions. This is particularly useful for generative agents such as chatbots who would oth- erwise be frozen in time, and whose knowledge limited to what they were
Preface
# Preface
Preface
exposed to during their training. With a vector database on the side, how- ever, they would have access to real-time information and can deduce new observations about content that is new to them. This is, in fact, the cor- nerstone of what is known as retrieval-augmented generation, an emerging learning paradigm.
Finding the most similar vectors to a query vector is easy when the database is small or when time is not of the essence: We can simply com- pare every vector in the database with the query and sort them by similarity. When the database grows large and the time budget is limited, as is often the case in practice, a na¨ıve, exhaustive comparison of a query with database vectors is no longer realistic. That is where vector retrieval algorithms become relevant. | 2401.09350#4 | 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 | [] | [
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] | {
"authors": "Sebastian Bruch",
"chunk_id": 4,
"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 neural network that is trained to perform a general task such as question- answering, could conceivably augment its view of the world by ârecallingâ in- formation from such a database and finding answers to new questions. This is particularly useful for generative agents such as chatbots who would oth- erwise be frozen in time, and whose knowledge limited to what they were\nPreface\n# Preface\nPreface\nexposed to during their training. With a vector database on the side, how- ever, they would have access to real-time information and can deduce new observations about content that is new to them. This is, in fact, the cor- nerstone of what is known as retrieval-augmented generation, an emerging learning paradigm.\nFinding the most similar vectors to a query vector is easy when the database is small or when time is not of the essence: We can simply com- pare every vector in the database with the query and sort them by similarity. When the database grows large and the time budget is limited, as is often the case in practice, a na¨ıve, exhaustive comparison of a query with database vectors is no longer realistic. That is where vector retrieval algorithms become relevant.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} |
2401.09350 | 5 | For decades now, research on vector retrieval has sought to improve the efficiency of search over large vector databases. The resulting literature is rich with solutions ranging from heavily theoretical results to performant empir- ical heuristics. Many of the proposed algorithms have undergone rigorous benchmarking and have been challenged in competitions at major confer- ences. Technology giants and startups alike have invested heavily in devel- oping open-source libraries and managed infrastructure that offer fast and scalable vector retrieval.
That is not the end of that story, however. Research continues to date. In fact, how we do vector retrieval today faces a stress-test as databases grow orders of magnitude larger than ever before. None of the existing methods, for example, proves easy to scale to a database of billions of high-dimensional vectors, or a database whose records change frequently.
# About This Monograph
The need to conduct more research underlines the importance of making the existing literature more readily available and the research area more inviting. That is partially fulfilled with existing surveys that report the state of the art at various points in time. However, these publications are typically focused on a single class of vector retrieval algorithms, and compare and contrast published methods by their empirical performance alone. Importantly, no manuscript has yet summarized major algorithmic milestones in the vast vector retrieval literature, or has been prepared to serve as a reference for new and established researchers. | 2401.09350#5 | 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 | [] | [
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] | {
"authors": "Sebastian Bruch",
"chunk_id": 5,
"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": "For decades now, research on vector retrieval has sought to improve the efficiency of search over large vector databases. The resulting literature is rich with solutions ranging from heavily theoretical results to performant empir- ical heuristics. Many of the proposed algorithms have undergone rigorous benchmarking and have been challenged in competitions at major confer- ences. Technology giants and startups alike have invested heavily in devel- oping open-source libraries and managed infrastructure that offer fast and scalable vector retrieval.\nThat is not the end of that story, however. Research continues to date. In fact, how we do vector retrieval today faces a stress-test as databases grow orders of magnitude larger than ever before. None of the existing methods, for example, proves easy to scale to a database of billions of high-dimensional vectors, or a database whose records change frequently.\n# About This Monograph\nThe need to conduct more research underlines the importance of making the existing literature more readily available and the research area more inviting. That is partially fulfilled with existing surveys that report the state of the art at various points in time. However, these publications are typically focused on a single class of vector retrieval algorithms, and compare and contrast published methods by their empirical performance alone. Importantly, no manuscript has yet summarized major algorithmic milestones in the vast vector retrieval literature, or has been prepared to serve as a reference for new and established researchers.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} |
2401.09350 | 6 | That gap is what this monograph intends to close. With the goal of present- ing the fundamentals of vector retrieval as a sub-discipline, this manuscript delves into important data structures and algorithms that have emerged in the literature to solve the vector retrieval problem efficiently and effectively.
vii
# viii
viii
Preface
# Structure
This monograph is divided into four parts. The first part introduces the prob- lem of vector retrieval and formalizes the concepts involved. The second part delves into retrieval algorithms that help solve the vector retrieval problem efficiently and effectively. Part three is devoted to vector compression. Fi- nally, the fourth part presents a review of background material in a series of appendices.
# Introduction
We start with a thorough introduction to the problem itself in Chapter 1 where we define the various flavors of vector retrieval. We then elaborate what is so difficult about the problem in high-dimensional spaces in Chapter 2.
In fact, sometimes high-dimensional spaces are hopeless. However, in re- ality data often lie on some low-dimensional space, even though their na¨ıve vector representations are in high dimensions. In those cases, it turns out, we can do much better. Exactly how we characterize this low âintrinsicâ dimensionality is the topic of Chapter 3.
# Retrieval Algorithms | 2401.09350#6 | 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 | [] | [
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] | {
"authors": "Sebastian Bruch",
"chunk_id": 6,
"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 gap is what this monograph intends to close. With the goal of present- ing the fundamentals of vector retrieval as a sub-discipline, this manuscript delves into important data structures and algorithms that have emerged in the literature to solve the vector retrieval problem efficiently and effectively.\nvii\n# viii\nviii\nPreface\n# Structure\nThis monograph is divided into four parts. The first part introduces the prob- lem of vector retrieval and formalizes the concepts involved. The second part delves into retrieval algorithms that help solve the vector retrieval problem efficiently and effectively. Part three is devoted to vector compression. Fi- nally, the fourth part presents a review of background material in a series of appendices.\n# Introduction\nWe start with a thorough introduction to the problem itself in Chapter 1 where we define the various flavors of vector retrieval. We then elaborate what is so difficult about the problem in high-dimensional spaces in Chapter 2.\nIn fact, sometimes high-dimensional spaces are hopeless. However, in re- ality data often lie on some low-dimensional space, even though their na¨ıve vector representations are in high dimensions. In those cases, it turns out, we can do much better. Exactly how we characterize this low âintrinsicâ dimensionality is the topic of Chapter 3.\n# Retrieval Algorithms",
"title": "Foundations of Vector Retrieval",
"year": 2024
} |
2401.09350 | 7 | # Retrieval Algorithms
With that foundation in place and the question clearly formulated, the second part of the monograph explores the different classes of existing solutions in great depth. We close each chapter with a summary of algorithmic insights. There, we will also discuss what remains challenging and explore future re- search directions.
We start with branch-and-bound algorithms in Chapter 4. The high- level idea is to lay a hierarchical mesh over the space, then given a query point navigate the hierarchy to the cell that likely contains the solution. We will see, however, that in high dimensions, the basic forms of these methods become highly inefficient to the point where an exhaustive search likely performs much better. | 2401.09350#7 | 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 | [] | [
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] | {
"authors": "Sebastian Bruch",
"chunk_id": 7,
"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": "# Retrieval Algorithms\nWith that foundation in place and the question clearly formulated, the second part of the monograph explores the different classes of existing solutions in great depth. We close each chapter with a summary of algorithmic insights. There, we will also discuss what remains challenging and explore future re- search directions.\nWe start with branch-and-bound algorithms in Chapter 4. The high- level idea is to lay a hierarchical mesh over the space, then given a query point navigate the hierarchy to the cell that likely contains the solution. We will see, however, that in high dimensions, the basic forms of these methods become highly inefficient to the point where an exhaustive search likely performs much better.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} |
2401.09350 | 8 | "Alternatively, instead of laying a mesh over the space, we may define a fixed number of buckets and(...TRUNCATED) | 2401.09350#8 | Foundations of Vector Retrieval | "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of th(...TRUNCATED) | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | [-0.04254426,-0.0010715789,0.075284086,0.013443586,0.042369183,-0.04654607,0.011355141,-0.017345352,(...TRUNCATED) | {"authors":"Sebastian Bruch","chunk_id":8,"doc_id":"2401.09350","primary_category":"cs.DS","publishe(...TRUNCATED) |
2401.09350 | 9 | "As we examine vector retrieval algorithms, it is inevitable that we must ink in extra pages to disc(...TRUNCATED) | 2401.09350#9 | Foundations of Vector Retrieval | "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of th(...TRUNCATED) | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | [0.0054342416,0.030341903,0.019559814,0.010010294,0.022036469,0.011392613,0.03418936,0.027600303,-0.(...TRUNCATED) | {"authors":"Sebastian Bruch","chunk_id":9,"doc_id":"2401.09350","primary_category":"cs.DS","publishe(...TRUNCATED) |
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