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2401.09350	212	In the second case, suppose Î´(Âµ, u) < Î´(Âµ, v), so that v is on the surface of B and u is in its interior. Consider the function f (Ï) = Î´(v, Ï) â Î´(u, Ï). Clearly, f (v) < 0 and f (Âµ) > 0. Therefore, there must be a point w â B on the line segment Âµ + Î»(v â Âµ) for Î» â [0, 1] for which f (w) = 0. T...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 = Î´(q, u). By assumption we have that the ball centered at q with radius r, B(q, r), is non-empty because it must con...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 opti...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 Ì¸= 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 ...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 Î´. 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 grap...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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. T...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 Ì¸= â where â³ denotes the symmetric difference between sets. Let r = maxuâQ...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 necessari...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 Î´(u, w). Note that, the resulting graph may be directed, depending on the choice of Î´. We should mention, however, that researchers have explored ways of turning th...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 Ì¸= â, it is a convex cone that is the intersection of half-spaces tha...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 gra...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	228	Now define the collection Â¥ & 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 Â¥ is the same as its region given V. From before, we also know that q Â¢ R,. Considering the fact that RÂ¢ = Uvex Rv: q must belong to R,...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 Necess...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 distan...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 Â¢q : RÂ¢ + RÂ¢+â¢ for a collection X of m vectors as follows: Â¢a(u\) = u Â© (\/1 = lu \|[3)ecasi), where ul is the i-th data point i...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 Theo...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 g...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 no...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 t...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 Ph...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 Î± = 0, then every node v Ì¸= u in the graph has an equal chance of being the destination of a long-range edge from u. When Î± is large, the protocol bec...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 Ã m nodes using the proba- bilistic model above with Î± = 2 and l = 1. The probability that there exists a long-range edge between two nodes u, v â V is at least Î´(u, v)â...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 nod...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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-...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 el...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 i...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 tar...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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 l...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	248	dÎ¸ 2Ï ln(r + dr) â ln r ln Î´â â ln Î´â â dÎ¸ 2Ï dr/r ln â = 1 2Ï ln â rdÎ¸dr 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 + Î²)Î´(u, v). We can therefore see that the probability that the long-range end-point l...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
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, Î´(u, uâ)/6) is at least 1/98 ln â. As such, the total number of hops, X, from u to a poi...	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	250	93 ââ # ââ 94 6 Graph Algorithms # 6.3.3 Approximation The results of Beaumont et al. [2007a] are encouraging. In theory, so long as we can construct the Delaunay graph, we not only have the optimality guarantee, but we are also guaranteed to have a poly-logarithmic number of hops to reach the optimal answer. A...	2401.09350#250	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	251	We are back, then, to approximation with the help of heuristics. Beau- mont et al. [2007b] describe one such method in a follow-up study. Their method approximates the Voronoi regions of every node by resorting to a gossip protocol. In this procedure, every node has a list of 3d + 1 of its cur- rent neighbors, where d ...	2401.09350#251	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	252	Malkov et al. [2014] take a different approach. They simply permute the vectors in the collection X , and sequentially add each vector to the graph. Every time a vector is inserted into the graph, it is linked to its k nearest neighbors from the current snapshot of the graph. The intuition is that, as the graph grows, ...	2401.09350#252	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	253	In the preceding section, our starting point was the Delaunay graph. We aug- mented it with random long-range connections to improve the transmission 6.4 Neighborhood Graphs rate through the network. Because the resulting structure contains the Delau- nay graph, we get the optimality guarantee of Theorem 6.4 for free. ...	2401.09350#253	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	254	In this section, we do the opposite. Instead of adding edges to the Delaunay graph and then resorting to heuristics to create a completely different graph, we prune the edges of the Delaunay graph to find a structure that is its sub- graph. Indeed, we cannot say anything meaningful about the optimality of exact top-k r...	2401.09350#254	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	255	The RNG was shown to contain the Minimum Spanning Tree [Toussaint, 1980], so that it is guaranteed to be connected. It is also provably contained in the Delaunay graph [OâRourke, 1982] in any metric space and in any number of dimensions. As a final property, it is not hard to see that such a graph G comes with a weak...	2401.09350#255	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	256	Later, Arya and Mount [1993] proposed a directed variant of the RNG, which they call the Sparse Neighborhood Graph (SNG) that is arguably more suitable for top-k retrieval. For every node u â V, we apply the following procedure: Let U = V \ {u}. Sort the nodes in U in increasing distance to u. Then, remove the closes...	2401.09350#256	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	257	Fig. 6.8: Examples of Î±-SNGs on a dataset of 20 points drawn uniformly from [0, 1]2 (blue circles). When Î± = 1, we recover the standard SNG. As Î± becomes larger, the resulting graph becomes more dense. Neighborhood graphs are the backbone of many graph algorithms for top- k retrieval [Malkov et al., 2014, Malkov and...	2401.09350#257	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	258	Fig. 6.9: The sets Bi and rings Ri in the proof of Theorem 6.8. # 6.4.1 From SNG to Î±-SNG Jayaram Subramanya et al. [2019] introduce a subtle adjustment to the SNG construction. In particular, suppose we are processing a node u, have already extracted the node v whose distance to u is minimal among the nodes in U (i.e...	2401.09350#258	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	259	That is what Indyk and Xu [2023] later refer to as an Î±-shortcut reachable graph. They define Î±-shortcut reachability as the property where, for any node u, we have that every other node w is either the target of an edge from u (so that (u, w) â E), or that there is a node v such that (u, v) â E and Î´(u, w) â¥ ...	2401.09350#259	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	260	Indyk and Xu [2023] present an analysis of the Î±-SNG for a collection of vectors X with doubling dimension dâ¦ as defined in Definition 3.2. For collections with a fixed doubling constant, Indyk and Xu [2023] state two bounds. One gives a bound on the degree of every node in an a-SNG. The other tells us the expected ...	2401.09350#260	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	261	Because Â¥ has a constant doubling dimension, we can cover each R; with O((4a)?) balls of radius 6,/a2'?. By construction, two points in each of these cover balls are at most 6,/a2't! apart. At the same time, the distance from u to any point in a cover ball is at least 6,/2â++. By construction, all points in a cove...	2401.09350#261	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	262	Proof. Suppose q is a query point and uâ = arg minuâX Î´(q, u). Further as- sume that the best-first-search algorithm is currently in node vi with distance Î´(q, vi) to the query. We make the following observations: â¢ By triangle inequality, we know that Î´(vi, uâ) â¤ Î´(vi, q) + Î´(q, uâ); and, â¢ By const...	2401.09350#262	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	263	There are three cases to consider. Case 1: When Î´(s, q) > 2Î´â, then by triangle inequality, Î´(q, uâ) > Î´(s, q) â Î´(s, uâ) > Î´(s, q) â Î´â > Î´(s, q)/2. Plugging this into Equation (6.1) yields: 6.4 Neighborhood Graphs Paw) OAT ig, 5(vi, g) < 2 d(q,u*) ~ a 6(v;,q) < ai As such, for any â¬ > 0, the alg...	2401.09350#263	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	264	Î´(vi, q) Î´(q, uâ) â¤ â¤ 2Î´â Î±iÎ´(q, uâ) 8(Î± + 1)Î´â Î±i(Î± â 1)Î´â + + Î± + 1 Î± â 1 Î± + 1 Î± â 1 . . As such, the number of steps to reach the approximation level is log, Sete which is O(log, A/(a â 1)e). 4(Î±+1) Î´â. Suppose vi Ì¸= uâ. Observe that: (a) Î´(vi, uâ) â¥ Î´â; (b) Î´(vi,...	2401.09350#264	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	265	2Î´â Î±i + =â Î±i â¤ 8â =â i â¤ logÎ± 8â. The three cases together give the desired result. In addition to the bounds above, Indyk and Xu [2023] present negative re- sults for other major SNG-based graph algorithms by proving (via contrived examples) linear-time lower-bounds on their performance. These resu...	2401.09350#265	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	266	99 ââ 100 6 Graph Algorithms it begins by searching the current snapshot of the graph for the top L nodes for the query point u, using Algorithm 3. Denote the returned set of nodes by S. It then performs the pruning algorithm by setting U = S \ {u}, rather than U = V \ {u}. That is the gist of the modified construc...	2401.09350#266	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	267	There is good reason for the uptick in research activity. Graph algorithms are among the most successful algorithms there are for top-k vector retrieval. They are often remarkably fast during retrieval and produce accurate solution sets. That success makes it all the more enticing to improve their other charac- teristi...	2401.09350#267	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	268	2 We have omitted minor but important details of the procedure in our prose. We refer the interested reader to [Jayaram Subramanya et al., 2019] for a description of the full algorithm. # References References is live and dynamic, the graph index must naturally handle deletions and insertions in real-time [Singh et al....	2401.09350#268	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	269	Y. Bachrach, Y. Finkelstein, R. Gilad-Bachrach, L. Katzir, N. Koenigstein, N. Nice, and U. Paquet. Speeding up the xbox recommender system using a euclidean transformation for inner-product spaces. In Proceedings of the 8th ACM Conference on Recommender Systems, page 257â264, 2014. O. Beaumont, A.-M. Kermarrec, L. Ma...	2401.09350#269	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	270	E. ChÂ´avez and E. S. Tellez. Navigating k-nearest neighbor graphs to solve nearest neighbor searches. In Proceedings of the 2nd Mexican Conference on Pattern Recognition: Advances in Pattern Recognition, pages 270â280, 2010. J. Chen, H.-r. Fang, and Y. Saad. Fast approximate knn graph construc- tion for high dimensi...	2401.09350#270	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	271	H. Edelsbrunner and N. R. Shah. Incremental topological flipping works for In Proceedings of the 8th Annual Symposium on regular triangulations. Computational Geometry, pages 43â52, 1992. S. Fortune. Voronoi Diagrams and Delaunay Triangulations, pages 377â388. CRC Press, Inc., 1997. C. Fu and D. Cai. Efanna : An ex...	2401.09350#271	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	272	S. Gollapudi, N. Karia, V. Sivashankar, R. Krishnaswamy, N. Begwani, S. Raz, Y. Lin, Y. Zhang, N. Mahapatro, P. Srinivasan, A. Singh, and H. V. Simhadri. Filtered-diskann: Graph algorithms for approximate near- est neighbor search with filters. In Proceedings of the ACM Web Conference 2023, pages 3406â3416, 2023. L. ...	2401.09350#272	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	273	B. Harwood and T. Drummond. Fanng: Fast approximate nearest neigh- bour graphs. In 2016 IEEE Conference on Computer Vision and Pattern Recognition, pages 5713â5722, 2016. P. Indyk and H. Xu. Worst-case performance of popular approximate nearest neighbor search implementations: Guarantees and limitations. In Proceed- ...	2401.09350#273	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	274	J. Kleinberg. The small-world phenomenon: An algorithmic perspective. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pages 163â170, 2000. W. Li, Y. Zhang, Y. Sun, W. Wang, M. Li, W. Zhang, and X. Lin. Approx- imate nearest neighbor search on high dimensional data â experiments, analyses, an...	2401.09350#274	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	275	S. Milgram. The Small-World Problem. Psychology Today, 1(1):61â67, 1967. S. Morozov and A. Babenko. Non-metric similarity graphs for maximum inner product search. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, pages 4726â4735, 2018. G. Navarro. Searching in metric spac...	2401.09350#275	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	276	M. Wang, X. Xu, Q. Yue, and Y. Wang. A comprehensive survey and exper- imental comparison of graph-based approximate nearest neighbor search. Proceedings of the VLDB Endowment, 14(11):1964â1978, jul 2021. Z. Zhou, S. Tan, Z. Xu, and P. Li. MÂ¨obius transformation for fast inner product search on graph. In Advances in...	2401.09350#276	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	277	# 7.1 Algorithm As usual, we begin by indexing a collection of m data points X â Rd. Except in this paradigm, that involves invoking a clustering function, Î¶ : Rd â [C], that is appropriate for the distance function Î´(Â·, Â·), to map every data point to one of C clusters, where C is an arbitrary parameter. A typi...	2401.09350#277	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	279	Fig. 7.1: Illustration of the clustering-based retrieval method. The collection of points (left) is first partitioned into clusters (regions enclosed by dashed boundary on the right). When processing a query q using Equation (7.1), we compute Î´(q, Â·) for the centroid (solid circles) of every cluster and conduct our s...	2401.09350#279	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	280	This simple protocolâwith some variant of KMeans as Î¶ and Ï as in Equa- tion (7.1)âworks well in practice [Auvolat et al., 2015, JÂ´egou et al., 2011, Bruch et al., 2023b, Babenko and Lempitsky, 2012, Chierichetti et al., 2007]. We present the results of our own experiments on various real-world datasets in Figur...	2401.09350#280	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	281	7.2 Closing Remarks 107 âCURACY 0.6) a ) âpeerin a Feven-MiNLM B04 -@-QUORA-MINILM > Feven-TasB -@-NQ-MiniLM- MS Manco Passace-MiniLM + NO-TasB 1% 3% 1% 2% ah ah me AS PERCENT OF C 1.0 a ) aan Deerisâ4-Gisr =O-GL0VE-200 AMS Tense srr 1% 5% 1% 3% cu; 3% AS PERCENT OF C (a) MIPS (b) NN Fig. 7.2: Performance of th...	2401.09350#281	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	282	functionâincluding learnt functions tailored to a query distributionâthat uses higher-order statistics from the cluster distributions. In spite of these shortcomings, the algorithmic framework above contains a fascinating insight that is actually useful for a rather different end-goal: vector compression, or more p...	2401.09350#282	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	283	108 7 Clustering That we have a method that is efficient in practice, but its efficiency and the conditions under which it is efficient are unexplained, constitutes a sub- stantial gap and thus presents multiple consequential open questions. These questions involve optimal clustering, routing, and bounds on retrieval a...	2401.09350#283	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	284	Even when we know what the right clustering algorithm is, there is still the issue of âbalanceâ that we must understand how to handle. It would, for example, be far from ideal if the clusters end up having very different sizes. Unfortunately, that happens quite naturally if the data points have highly variable norm...	2401.09350#284	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	285	We may answer these questions differently if we had some idea of what the query distribution looks like. Assuming access to a set of training queries, it may be possible to learn a more optimal sketch using supervised learning methods. Concepts from learning-to-rank [Bruch et al., 2023a] seem particu- larly relevant to...	2401.09350#285	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	286	Take any clustering algorithm, Î¶. If one could show formally that Î¶ behaves like an LSH family, then clustering-based top-k retrieval simply collapses to LSH. In that case, not only do the results from that literature apply, but the techniques developed for LSH (such as multi-probe LSH) too port over to clustering. S...	2401.09350#286	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	287	A. Babenko and V. Lempitsky. The inverted multi-index. In 2012 IEEE Con- ference on Computer Vision and Pattern Recognition, pages 3069â3076, 2012. S. Bruch. An alternative cross entropy loss for learning-to-rank. In Proceed- ings of the Web Conference 2021, page 118â126, 2021. S. Bruch, X. Wang, M. Bendersky, and ...	2401.09350#287	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	288	109 110 7 Clustering F. Chierichetti, A. Panconesi, P. Raghavan, M. Sozio, A. Tiberi, and E. Upfal. Finding near neighbors through cluster pruning. In Proceedings of the Twenty-Sixth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, pages 103â112, 2007. R. Guo, P. Sun, E. Lindgren, Q. Geng, D. Sim...	2401.09350#288	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	289	# Chapter 8 Sampling Algorithms Abstract Nearly all of the data structures and algorithms we reviewed in the previous chapters are designed specifically for either nearest neighbor search or maximum cosine similarity search. MIPS is typically an afterthought. It is often cast as NN or MCS through a rank-preserving tran...	2401.09350#289	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	290	Approximating the ranks or scores of data points uses some form of sam- pling: we either sample data points according to a distribution defined by inner products, or sample a dimension to compute partial inner products with and eliminate sub-optimal data points iteratively. In the former, the more frequently a data poi...	2401.09350#290	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	291	# 8.2 Approximating the Ranks We are interested in finding the top-k data points with the largest inner product with a query q â Rd, from a collection X â Rd of m points. Suppose that we had an efficient way of sampling a data point from X where the point u â X has probability proportional to â¨q, uâ© of being ...	2401.09350#291	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	292	The key to tackling that challenge is the linearity of inner product. Follow- ing a few simple derivations using Bayesâ theorem, we can break up the sam- pling procedure into two steps, each using marginal distributions only [Loren- zen and Pham, 2021, Ballard et al., 2015, Cohen and Lewis, 1997, Ding et al., 2019]. ...	2401.09350#292	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	293	Plt\|qlx Ss Us = Ss Uts (8.2) UuEx Ucrk and, Plu At \| q] ur ut x . Plt \| q] Ut ocx Ut ocx Ut Plu \| tq] (8.3) ut vâX vt vâX vt Ì¸= 0; if that sum is 0 we can Plt \| q] Ut In the above, we have assumed that },,<y simply discard the t-th dimension. In the above, we have assumed that },,<y vr 4 0; if that sum is 0 we can...	2401.09350#293	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	294	The distribution over dimensions given a query, P{t \| g], must be computed online using Equation (8.2), which requires O(d) operations, assuming we compute >> ,,Â¢y Ur offline for each t and store them in our index. Again, using he alias method, we can subsequently draw samples with O(1) operations. The procedure descr...	2401.09350#294	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	295	First, we must ensure that the marginal distributions are valid. That is easy to do: In Equations (8.2) and (8.3), we replace each term with its absolute value. So, P[t \| g] becomes proportional to }>,<y\|qeus\|, and Plu \| toa] & lusl/ Cyexltel We then use the resulting distributions to sample data points as before, but ...	2401.09350#295	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	296	Taking expectation over the dimension t yields: E[Z] =E(E(Z\| 4] = Ya Plt \| 4] vex \|dUe\| d UU Dover lave ,] a t=1 Vvea lure Viet Voex lari (q,u) ~Sd a), Viet Vocal urel ââ 8.2 Approximating the Ranks # 8.2.3 Sample Complexity We have formalized an efficient way to sample data points according to the distribution of ...	2401.09350#296	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	297	If S$ is the number of samples to be drawn, for a vector u, denote by Z,,i a random variable that is 0 if u was not sampled in round 7, and otherwise SIGN(qu,) if t is the sampled dimension. Once the sampling has concluded, the final value for point wu is simply Z, = >; Zu,:. Note that, from Lemma 8.1, we have that E[Z...	2401.09350#297	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	298	S â¥ max uÌ¸=uâ (1 + âu)2 uh( âu(1+âu) Ï2 Ï2 u ) log m Î´ , where h(x) = (1 + x) log(1 + x) â x, then P[Zuâ > Zu â u Ì¸= uâ] â¥ 1 â Î´. Before proving the theorem above, let us make a quick observation. Clearly Ï2 u â¤ O(dâu) and (1 + âu) â 1. Because h(Â·) is monotone increasing in its ar...	2401.09350#298	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	299	Plugging this into Theorem 8.1 gives us S â¤ O( d # â log m Î´ ). Theorem 8.1 tells us that, if we draw O( d Î´ ) samples, we can iden- tify the top-1 solution to MIPS with high probability. Observe that, â is a measure of the difficulty of the query: When inner products are close to each other, â becomes smalle...	2401.09350#299	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	300	Setting the right-hand-side to Î´ m , we arrive at: So (1+ Ay)Au 6 vol a+ Teh 2 ) < m u Aull + Au) m = S(1+ Au) ?o2h( > log 5: o It is easy to see that for x > 0, h(x) > 0. Observing that âu(1 + âu)/Ï2 u is positive, that implies that h(âu(1 + âu)/Ï2 u) > 0, and therefore we can re-arrange the expression abov...	2401.09350#300	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	301	where we have used the union bound to obtain the inequality. # 8.3 Approximating the Scores The method we have just presented avoids the computation of inner products altogether but estimates the rank of each data point with respect to a query using a sampling procedure. In this section, we introduce another sampling m...	2401.09350#301	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	302	That is the core idea in this section. For each data point, we sample a few dimensions without replacement, and compute its partial inner product with the query along the chosen dimensions. Based on the scores so far, we can eliminate data points whose full inner product is projected, with high confidence, to be too sm...	2401.09350#302	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	303	117 ââ 118 8 Sampling Algorithms # Algorithm 4: The BoundedME algorithm for MIPS. Input: Query point q â¬ RÂ¢; k > 1 for top-k retrieval; confidence parameters Â¢,6 â¬ (0,1); and data points Y Cc R* Result: (1 â 5)-confident e-approximate top-k set to MIPS with respect to q. Li 1 2: Xi << X ; > Initialize the s...	2401.09350#303	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	304	their algorithm, we find it makes for a clearer presentation if we avoided the Bandit terminology. # 8.3.1 The BoundedME Algorithm The top-k retrieval algorithm developed by Liu et al. [2019] is presented in Algorithm 4. It is important to note that, for the algorithm to be correctâas we will explain laterâeach par...	2401.09350#304	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	305	The number of dimensions to sample is adaptive and changes from it- eration to iteration. It is determined using the quantity on Line 7 of the 8.3 Approximating the Scores algorithm, where the function h(Â·) is defined as follows: (8.5) lta etal, A(z) = min { I+a/dâ1+a/d At the end of iteration i with the remaining d...	2401.09350#305	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	306	As for the time complexity of Algorithm 4, it can be shown that it re- quires o(@4 flog(1/5)) operations. That is simply due to the fact that in each iteration, the number of data points is cut in half, combined with the inequality h(x) < O(Vdz) for x > 0. Theorem 8.2 The time complexity of Algorithm 4 is O( m â o(2Â...	2401.09350#306	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	307	â l+a x+a/d _ 1 Jal Fay + t/a) ho < fe ipa x(1 + 2)(1+ 1/d) â Of) \| dx ~ 1+2/d = OTF) < O(Vaz). Note that, in the i-th iteration there are at most m/2i data points to examine. Moreover, for each data point that is eliminated in round i, we will have computed at most ti partial inner products (see Line 7 of Algorith...	2401.09350#307	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	308	The proof of Theorem 8.3 requires the concentration inequality due to Bar- denet and Maillard [2015], repeated below for completeness. Lemma 8.3 Let J â [0, 1] be a finite set of size d with mean Âµ. Let {J1, J2, . . . , Jn} be n < d samples from J without replacement. Then for any n â¤ d and any Î´ â [0, 1] it ho...	2401.09350#308	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	309	l+2 etal > mi ; n> min{ ap 1l+a/d where x = log(1/Î´)/2Ïµ2, then the following holds: PE a-n<d >1-6. t=1 ââ 8.3 Approximating the Scores Proof. By Lemma 8.3 we can see that: so long as: log 1 Î´ â¤ Ïµ =â n Ïn â¥ 1 2Ïµ2 log 1 Î´ . There are two cases to consider. First, if Ïn = 1 â (n â 1)/d, then: n 1 1 n...	2401.09350#309	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	310	20. )n â x â¥ 0. To make the closed-form solution more manageable, Liu et al. [2019] relax the problem above and solve n in the following problem instead. Note that, any solution to the problem below is a valid solution to the problem above. (1+ yn? -(@=â)n-2-130 = [a+ 5)n-2-1][n +1] 20 â 1 + & â ne l+a/d By c...	2401.09350#310	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	311	P [<i â Gini < Â«\| 21-46, (8.6) then the theorem immediately follows: P [61 â Gogm SÂ¢] 21-6, because: logm log m 5 5 Y= La sd% and, ala Kee in Me alo a Kio Na L \| logm log m # Equation So we focus on proving Equation (8.6). Suppose we are in the i-th iteration. Collect in ZÏµi every data point in u â Xi such tha...	2401.09350#311	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	312	What is the probability that a data point u in Â¥; \ Z., has a higher partial inner product than any data point in Z,,? Assuming that u* is the data point that achieves Â¢;, we can write: PA, >A, Vue Z.) < P[A. >A, \| PA, >A, Vue Z.) < P[A. >A, \| <P[A, > (gu) +S V Au SG 3] <P[A. > (qu) + S$] +P [Aw <6 - $]. We can apply...	2401.09350#312	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	313	That completes the proof of Equation (8.6) and, therefore, the theorem. ââ # 8.4 Closing Remarks The algorithms in this chapter were unique in two ways. First, they directly took on the challenging problem of MIPS. This is in contrast to earlier chap- ters where MIPS was only an afterthought. Second, there is littl...	2401.09350#313	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	314	Another area that would benefit from further research is the sampling strategy itself. In particular, in the BoundedME algorithm, the dimensions that are sampled next are drawn randomly. While that simplifies analysisâ which follows the analysis of popular Bandit algorithmsâit is not hard to argue that the strategy...	2401.09350#314	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]
2401.09350	315	Q. Ding, H.-F. Yu, and C.-J. Hsieh. A fast sampling algorithm for maxi- mum inner product search. In K. Chaudhuri and M. Sugiyama, editors, Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics, volume 89 of Proceedings of Machine Learning Research, pages 3004â3012, 4 2019. T. Lat...	2401.09350#315	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 rele...	http://arxiv.org/pdf/2401.09350	Sebastian Bruch	cs.DS, cs.IR	null	null	cs.DS	20240117	20240117	[]

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