nips nips2013 nips2013-326 knowledge-graph by maker-knowledge-mining

326 nips-2013-The Power of Asymmetry in Binary Hashing

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Author: Behnam Neyshabur, Nati Srebro, Ruslan Salakhutdinov, Yury Makarychev, Payman Yadollahpour

Abstract: When approximating binary similarity using the hamming distance between short binary hashes, we show that even if the similarity is symmetric, we can have shorter and more accurate hashes by using two distinct code maps. I.e. by approximating the similarity between x and x as the hamming distance between f (x) and g(x ), for two distinct binary codes f, g, rather than as the hamming distance between f (x) and f (x ). 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract When approximating binary similarity using the hamming distance between short binary hashes, we show that even if the similarity is symmetric, we can have shorter and more accurate hashes by using two distinct code maps. [sent-5, score-1.058]

2 by approximating the similarity between x and x as the hamming distance between f (x) and g(x ), for two distinct binary codes f, g, rather than as the hamming distance between f (x) and f (x ). [sent-8, score-0.831]

3 1 Introduction Encoding high-dimensional objects using short binary hashes can be useful for fast approximate similarity computations and nearest neighbor searches. [sent-9, score-0.569]

4 Calculating the hamming distance between two short binary strings is an extremely cheap computational operation, and the communication cost of sending such hash strings for lookup on a server (e. [sent-10, score-0.757]

5 sending hashes of all features or patches in an image taken on a mobile device) is low. [sent-12, score-0.236]

6 Furthermore, it is also possible to quickly look up nearby hash strings in populated hash tables. [sent-13, score-0.657]

7 Moreover, compact binary codes are remarkably storage efﬁcient, and allow one to store massive datasets in memory. [sent-15, score-0.411]

8 It is therefore desirable to ﬁnd short binary hashes that correspond well to some target notion of similarity. [sent-16, score-0.331]

9 Pioneering work on Locality Sensitive Hashing used random linear thresholds for obtaining bits of the hash [1]. [sent-17, score-0.47]

10 Later work suggested learning hash functions attuned to the distribution of the data [15, 11, 5, 7, 3]. [sent-18, score-0.308]

11 More recent work focuses on learning hash functions so as to optimize agreement with the target similarity measure on speciﬁc datasets [14, 8, 9, 6] . [sent-19, score-0.467]

12 It is important to obtain accurate and short hashes—the computational and communication costs scale linearly with the length of the hash, and more importantly, the memory cost of the hash table can scale exponentially with the length. [sent-20, score-0.412]

13 In all the above-mentioned approaches, similarity S(x, x ) between two objects is approximated by the hamming distance between the outputs of the same hash function, i. [sent-21, score-0.712]

14 The emphasis here is that the same hash function is applied to both x and x (in methods like LSH multiple hashes might be used to boost accuracy, but the comparison is still between outputs of the same function). [sent-24, score-0.505]

15 The only exception we are aware of is where a single mapping of objects to fractional vectors ˜ ˜ f (x) ∈ [−1, 1]k is used, its thresholding f (x) = sign f (x) ∈ {±1}k is used in the database, ˜ and similarity between x and x is approximated using f (x), f (x ) . [sent-25, score-0.424]

16 In this paper, we propose using two distinct mappings f (x), g(x) ∈ {±1}k and approximating the similarity S(x, x ) by the hamming distance between f (x) and g(x ). [sent-28, score-0.41]

17 Our main result is that even if the target similarity function is symmetric and “well behaved” (e. [sent-30, score-0.222]

18 , even if it is based on Euclidean distances between objects), using asymmetric binary hashes can be much more powerful, and allow better approximation of the target similarity with shorter code lengths. [sent-32, score-0.872]

19 Although actual data is not as extreme, our experimental results on real data sets demonstrate signiﬁcant beneﬁts from using asymmetric binary hashes. [sent-34, score-0.39]

20 Asymmetric hashes can be used in almost all places where symmetric hashes are typically used, usually without any additional storage or computational cost. [sent-35, score-0.499]

21 Consider the typical application of storing hash vectors for all objects in a database, and then calculating similarities to queries by computing the hash of the query and its hamming distance to the stored database hashes. [sent-36, score-1.161]

22 Using an asymmetric hash means using different hash functions for the database and for the query. [sent-37, score-1.04]

23 This neither increases the size of the database representation, nor the computational or communication cost of populating the database or performing a query, as the exact same operations are required. [sent-38, score-0.274]

24 In fact, when hashing the entire database, asymmetric hashes provide even more opportunity for improvement. [sent-39, score-0.668]

25 We argue that using two different hash functions to encode database objects and queries allows for much more ﬂexibility in choosing the database hash. [sent-40, score-0.697]

26 Unlike the query hash, which has to be stored compactly and efﬁciently evaluated on queries as they appear, if the database is ﬁxed, an arbitrary mapping of database objects to bit strings may be used. [sent-41, score-0.564]

27 We demonstrate that this can indeed increase similarity accuracy while reducing the bit length required. [sent-42, score-0.206]

28 2 Minimum Code Lengths and the Power of Asymmetry Let S : X × X → {±1} be a binary similarity function over a set of objects X , where we can interpret S(x, x ) to mean that x and x are “similar” or “dissimilar”, or to indicate whether they are “neighbors”. [sent-43, score-0.329]

29 A symmetric binary coding of X is a mapping f : X → {±1}k , where k is the bitlength of the code. [sent-44, score-0.239]

30 We are interested in constructing codes such that the hamming distance between f (x) and f (x ) corresponds to the similarity S(x, x ). [sent-45, score-0.525]

31 Although discussing the hamming distance, it is more convenient for us to work with the inner product u, v , which is equivalent to the hamming distance dh (u, v) since u, v = (k − 2dh (u, v)) for u, v ∈ {±1}k . [sent-47, score-0.287]

32 In this section, we will consider the problem of capturing a given similarity using an arbitrary binary code. [sent-48, score-0.227]

33 That is, we are given the entire similarity mapping S, e. [sent-49, score-0.178]

34 We ask for an encoding ui = f (xi ) ∈ {±1}k of each object xi ∈ X , and a threshold θ, such that Sij = sign( ui , uj − θ), or at least such that equality holds for as many pairs (i, j) as possible. [sent-55, score-0.18]

35 We now ask: Can allowing an asymmetric coding enable approximating a symmetric similarity matrix S with a shorter code length? [sent-57, score-0.735]

36 Denoting U ∈ {±1}n×k for the matrix whose columns contain the codewords ui , the minimal binary code length that allows exactly representing S is then given by the following matrix factorization problem: ks (S) = min k k,U,θ s. [sent-58, score-0.454]

37 2 θ∈R (1) We begin demonstrating the power of asymmetry by considering an asymmetric variant of the above problem. [sent-60, score-0.428]

38 That is, even if S is symmetric, we allow associating with each object xi two distinct binary codewords, ui ∈ {±1}k and vi ∈ {±1}k (we can think of this as having two arbitrary mappings ui = f (xi ) and vi = g(xi )), such that Sij = sign( ui , vj − θ). [sent-61, score-0.416]

39 The minimal asymmetric binary code length is then given by: ka (S) = min k s. [sent-62, score-0.582]

40 This is captured in the following Theorem, which establishes that there could be an exponential gap between the minimal asymmetry binary code length and the minimal symmetric code length, even if the matrix S is symmetric and very well behaved: Theorem 1. [sent-64, score-0.654]

41 For any r, there exists a set of n = 2r points in Euclidean space, with similarity matrix 1 if xi − xj ≤ 1 Sij = , such that ka (S) ≤ 2r but ks (S) ≥ 2r /2 −1 if xi − xj > 1 Proof. [sent-65, score-0.511]

42 if xi − xj > 1 2 Note that if i = j then Sij = 1; if i = j and (i, j) ∈ I1 × I1 ∪ I2 × I2 then xi − xj = Gii +Gjj −2Gij = 1+1/n > 1 and therefore Sij = −1. [sent-81, score-0.198]

43 Note that Yij ∈ [−ks , ks ] and thus θ ∈ [−ks + 1, ks − 1]. [sent-91, score-0.176]

44 The construction of Theorem 1 shows that there exists data sets for which an asymmetric binary hash might be much shorter then a symmetric hash. [sent-103, score-0.848]

45 This is an important observation as it demonstrates that asymmetric hashes could be much more powerful, and should prompt us to consider them instead of symmetric hashes. [sent-104, score-0.582]

46 3 10-D Uniform 35 60 25 50 20 bits bits LabelMe 70 30 15 Symmetric Asymmetric 10 30 Symmetric Asymmetric 20 5 0 0. [sent-106, score-0.324]

47 96 Average Precision Figure 1: Number of bits required for approximating two similarity matrices (as a function of average precision). [sent-121, score-0.343]

48 Left: uniform data in the 10-dimensional hypercube, similarity represents a thresholded Euclidean distance, set such that 30% of the similarities are positive. [sent-122, score-0.206]

49 Right: Semantic similarity of a subset of LabelMe images, thresholded such that 5% of the similarities are positive. [sent-123, score-0.206]

50 3 Approximate Binary Codes As we turn to real data sets, we also need to depart from seeking a binary coding that exactly captures the similarity matrix. [sent-124, score-0.247]

51 Rather, we are usually satisﬁed with merely approximating S, and for any ﬁxed code length k seek the (symmetric or asymmetric) k-bit code that “best captures” the similarity matrix S. [sent-125, score-0.388]

52 In our experiments, we replace the zero-one loss (·) with a continuous loss and perform local search by greedily updating single bits so as to improve this objective. [sent-131, score-0.162]

53 Before moving on to out-of-sample generalizations, we brieﬂy report on the number of bits needed empirically to ﬁnd good approximations of actual similarity matrices with symmetric and asymmetric codes. [sent-134, score-0.683]

54 We experimented with several data sets, attempting to ﬁt them with both symmetric and asymmetric codes, and then calculating average precision by varying the threshold θ (while keeping U and V ﬁxed). [sent-135, score-0.575]

55 Results for two similarity matrices, one based on Euclidean distances between points uniformly distributed in a hypoercube, and the other based on semantic similarity between images, are shown in Figure 1. [sent-136, score-0.35]

56 4 Out of Sample Generalization: Learning a Mapping So far we focused on learning binary codes over a ﬁxed set of objects by associating an arbitrary code word with each object and completely ignoring the input representation of the objects xi . [sent-137, score-0.683]

57 We discussed only how well binary hashing can approximate the similarity, but did not consider generalizing to additional new objects. [sent-138, score-0.263]

58 That is, we would like to learn a mapping f : X → {±1}k over an inﬁnite domain X using only a ﬁnite training set of objects, and then apply the mapping to obtain binary codes f (x) for future objects to be encountered, such that S(x, x ) ≈ sign( f (x), f (x ) − θ). [sent-140, score-0.5]

59 We already saw that asymmetric binary codes can allow for better approximations using shorter codes, so it is natural to seek asymmetric codes here as well. [sent-144, score-1.199]

60 This has the potential of allowing for better approximating the similarity, and thus better overall accuracy with shorter codes (despite possibly slightly harder generalization due to the increase in the number of parameters). [sent-146, score-0.314]

61 In fact, in a typical application where a database of objects is hashed for similarity search over future queries, asymmetry allows us to go even further. [sent-147, score-0.467]

62 Our goal is to hash these objects using short binary codes which would allow us to quickly compute approximate similarities between these objects (the “database”) and future objects x (the “query”). [sent-152, score-1.008]

63 That is, we would like to generate and store compact binary codes for objects in a database. [sent-153, score-0.471]

64 Then, given a new query object, we would like to efﬁciently compute a compact binary code for a given query and retrieve similar items in the database very fast by ﬁnding binary codes in the database that are within small hamming distance from the query binary code. [sent-154, score-1.153]

65 Recall that it is important to ensure that the bit length of the hashes are small, as short codes allow for very fast hamming distance calculations and low communication costs if the codes need to be sent remotely. [sent-155, score-0.946]

66 More importantly, if we would like to store the database in a hash table allowing immediate lookup, the size of the hash table is exponential in the code length. [sent-156, score-0.854]

67 The asymmetric approach described above would be to ﬁnd two parametric mappings f : X → {±1}k and g : X → {±1}k such that S(x, xi ) ≈ sign( f (x), g(xi ) − θ), and then calculate and store g(xi ). [sent-160, score-0.477]

68 Hence, we can consider allowing the database hash function g(·) to be an arbitrary mapping. [sent-163, score-0.433]

69 , xn in the database, such that S(x, xi ) ≈ sign( f (x), vi − θ) for future queries x and for the objects xi , . [sent-170, score-0.313]

70 This form of asymmetry can allow us for greater approximation power, and thus better accuracy with shorter codes, at no additional computational or storage cost. [sent-174, score-0.187]

71 In Section 6 we evaluate empirically both of the above asymmetric strategies and demonstrate their beneﬁts. [sent-175, score-0.299]

72 But before doing so, in the next Section, we discuss a local-search approach for ﬁnding the mappings f, g, or the mapping f and the codes v1 , . [sent-176, score-0.324]

73 5 Optimization We focus on x ∈ X ⊂ Rd and linear threshold hash maps of the form f (x) = sign(W x), where W ∈ Rk×d . [sent-180, score-0.353]

74 6 Empirical Evaluation In order to empirically evaluate the beneﬁts of asymmetry in hashing, we replicate the experiments of [8], which were in turn based on [5], on six datasets using learned (symmetric) linear threshold codes. [sent-215, score-0.172]

75 These datasets include: LabelMe and Peekaboom are collections of images, represented as 512D GIST features [13], Photo-tourism is a database of image patches, represented as 128 SIFT features [12], MNIST is a collection of 785D greyscale handwritten images, and Nursery contains 8D features. [sent-216, score-0.189]

76 , xn , represented as vectors in X = Rd , and the binary similarities S(xi , xj ) between the points, with +1 corresponding to xi and xj being neighbors and -1 otherwise. [sent-224, score-0.298]

77 Based on these n training points, [8] present a sophisticated optimization approach for learning a thresholded linear hash function of the form f (x) = sign(W x), where W ∈ Rk×d . [sent-225, score-0.341]

78 This hash function is then applied and f (x1 ), . [sent-226, score-0.308]

79 [8] evaluate the quality of the hash by considering an independent set of test points and comparing S(x, xi ) to sign( f (x), f (xi ) − θ) on the test points x and the database objects (i. [sent-230, score-0.658]

80 In our experiments, we followed the same protocol, but with the two asymmetric variations L IN :L IN and L IN :V, using the optimization method discussed in Sec. [sent-233, score-0.299]

81 In order to obtain different balances between precision and recall, we should vary β in (3), obtaining different codes for each value of 6 10-D Uniform 1 LIN:V LIN:LIN MLH KSH BRE LSH 0. [sent-235, score-0.331]

82 2 64 8 12 16 20 Number of Bits 24 28 32 36 40 44 48 52 56 60 64 Number of Bits Figure 2: Average Precision (AP) of points retrieved using Hamming distance as a function of code length for six datasets. [sent-259, score-0.238]

83 1 In our ﬁrst set of experiments, we test performance of the asymmetric hash codes as a function of the bit length. [sent-287, score-0.863]

84 Figure 2 displays Average Precision (AP) of data points retrieved using Hamming distance as a function of code length. [sent-288, score-0.201]

85 Observe that for all six datasets both variants of our method, asymmetric L IN :L IN and asymmetric L IN :V , consistently outperform all other methods for different binary code length. [sent-290, score-0.797]

86 For example, for the LabelMe dataset, MLH and KSH with 16 bits achieve AP of 0. [sent-292, score-0.162]

87 These results clearly show that an asymmetric binary hash can be much more compact than a symmetric hash. [sent-298, score-0.811]

88 More speciﬁcally, we set the number of points for each hash function in BRE to 50 and the number of anchors in KSH to 300 (the default values). [sent-301, score-0.34]

89 Results on previous 6 datasets show that asymmetric binary codes can signiﬁcantly outperform other state-of-the-art methods on relatively small scale datasets. [sent-337, score-0.636]

90 The dataset also provides a semantic similarity S(x, x ) between two images based on semantic content (object labels overlap in two images). [sent-340, score-0.257]

91 As argued by [8], hash functions learned using semantic labels should be more useful for content-based image retrieval compared to Euclidean distances. [sent-341, score-0.373]

92 Figure 5 shows that L IN :V with 64 bits substantially outperforms MLH and KSH with 64 bits. [sent-342, score-0.162]

93 7 Summary The main point we would like to make is that when considering binary hashes in order to approximate similarity, even if the similarity measure is entirely symmetric and “well behaved”, much power can be gained by considering asymmetric codes. [sent-343, score-0.834]

94 We substantiate this claim by both a theoretical analysis of the possible power of asymmetric codes, and by showing, in a fairly direct experimental replication, that asymmetric codes outperform state-of-the-art results obtained for symmetric codes. [sent-344, score-0.932]

95 However, even using this crude approach, we could ﬁnd asymmetric codes that outperformed well-optimized symmetric codes. [sent-346, score-0.608]

96 Although we demonstrated our results in a speciﬁc setting using linear threshold codes, we believe the power of asymmetry is far more widely applicable in binary hashing, and view the experiments here as merely a demonstration of this power. [sent-348, score-0.265]

97 Using asymmetric codes instead of symmetric codes could be much more powerful, and allow for shorter and more accurate codes, and is usually straightforward and does not require any additional computational, communication or signiﬁcant additional memory resources when using the code. [sent-349, score-0.919]

98 We would therefore encourage the use of such asymmetric codes (with two distinct hash mappings) wherever binary hashing is used to approximate similarity. [sent-350, score-1.115]

99 Asymmetric distance estimation with sketches for similarity search in high-dimensional spaces. [sent-364, score-0.181]

100 Iterative quantization: A procrustean approach to learning binary codes for large-scale image retrieval. [sent-371, score-0.333]

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same-paper 1 0.9999997 326 nips-2013-The Power of Asymmetry in Binary Hashing

Author: Behnam Neyshabur, Nati Srebro, Ruslan Salakhutdinov, Yury Makarychev, Payman Yadollahpour

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Abstract: We go beyond the notion of pairwise similarity and look into search problems with k-way similarity functions. In this paper, we focus on problems related to 3-way Jaccard similarity: R3way = |S1 ∩S2 ∩S3 | , S1 , S2 , S3 ∈ C, where C is a |S1 ∪S2 ∪S3 | size n collection of sets (or binary vectors). We show that approximate R3way similarity search problems admit fast algorithms with provable guarantees, analogous to the pairwise case. Our analysis and speedup guarantees naturally extend to k-way resemblance. In the process, we extend traditional framework of locality sensitive hashing (LSH) to handle higher-order similarities, which could be of independent theoretical interest. The applicability of R3way search is shown on the “Google Sets” application. 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This paper fulﬁlls this requirement for k-way resemblance and its derived similarities. In particular, we show fast algorithms with provable query time guarantees for approximate k-way resemblance search. Our algorithms and analysis naturally provide a framework to extend classical LSH framework [14, 13] to handle higher-order similarities, which could be of independent theoretical interest. Organization In Section 2, we review approximate near neighbor search and classical Locality Sensitive Hashing (LSH). In Section 3, we formulate the 3-way similarity search problems. Sections 4, 5, and 6 describe provable fast algorithms for several search problems. Section 7 demonstrates the applicability of 3-way resemblance search in real applications. 2 Classical c-NN and Locality Sensitive Hashing (LSH) Initial attempts of ﬁnding efﬁcient (sub-linear time) algorithms for exact near neighbor search, based on space partitioning, turned out to be a disappointment with the massive dimensionality of current datasets [11, 28]. Approximate versions of the problem were proposed [14, 13] to break the linear query time bottleneck. One widely adopted formalism is the c-approximate near neighbor (c-NN). Deﬁnition 1 (c-Approximate Near Neighbor or c-NN). Consider a set of points, denoted by P, in a D-dimensional space RD , and parameters R0 > 0, δ > 0. The task is to construct a data structure which, given any query point q, if there exist an R0 -near neighbor of q in P, it reports some cR0 -near neighbor of q in P with probability 1 − δ. The usual notion of c-NN is for distance. Since we deal with similarities, we deﬁne R0 -near neighbor of point q as a point p with Sim(q, p) ≥ R0 , where Sim is the similarity function of interest. Locality sensitive hashing (LSH) [14, 13] is a popular framework for c-NN problems. LSH is a family of functions, with the property that similar input objects in the domain of these functions have a higher probability of colliding in the range space than non-similar ones. In formal terms, consider H a family of hash functions mapping RD to some set S Deﬁnition 2 (Locality Sensitive Hashing (LSH)). A family H is called (R0 , cR0 , p1 , p2 )-sensitive if for any two points x, y ∈ RD and h chosen uniformly from H satisﬁes the following: • if Sim(x, y) ≥ R0 then P rH (h(x) = h(y)) ≥ p1 • if Sim(x, y) ≤ cR0 then P rH (h(x) = h(y)) ≤ p2 For approximate nearest neighbor search typically, p1 > p2 and c < 1 is needed. Note, c < 1 as we are deﬁning neighbors in terms of similarity. Basically, LSH trades off query time with extra preprocessing time and space which can be accomplished off-line. 2 Fact 1 Given a family of (R0 , cR0 , p1 , p2 ) -sensitive hash functions, one can construct a data structure for c-NN with O(nρ log1/p2 n) query time and space O(n1+ρ ), where ρ = log 1/p1 . log 1/p2 Minwise Hashing for Pairwise Resemblance One popular choice of LSH family of functions associated with resemblance similarity is, Minwise Hashing family [5, 6, 13]. Minwise Hashing family applies an independent random permutation π : Ω → Ω, on the given set S ⊆ Ω, and looks at the minimum element under π, i.e. min(π(S)). Given two sets S1 , S2 ⊆ Ω = {0, 1, 2, ..., D − 1}, it can be shown by elementary probability argument that P r (min(π(S1 )) = min(π(S2 ))) = |S1 ∩ S2 | = R2way . |S1 ∪ S2 | (1) The recent work on b-bit minwise hashing [20, 23] provides an improvement by storing only the lowest b bits of the hashed values: min(π(S1 )), min(π(S2 )). [26] implemented the idea of building hash tables for near neighbor search, by directly using the bits from b-bit minwise hashing. 3 3-way Similarity Search Formulation Our focus will remain on binary vectors which can also be viewed as sets. We illustrate our method |S1 ∩S2 ∩S3 | using 3-way resemblance similarity function Sim(S1 , S2 , S3 ) = |S1 ∪S2 ∪S3 | . The algorithm and guarantees naturally extend to k-way resemblance. Given a size n collection C ⊆ 2Ω of sets (or binary vectors), we are particularly interested in the following three problems: 1. Given two query sets S1 and S2 , ﬁnd S3 ∈ C that maximizes Sim(S1 , S2 , S3 ). 2. Given a query set S1 , ﬁnd two sets S2 , S3 ∈ C maximizing Sim(S1 , S2 , S3 ). 3. Find three sets S1 , S2 , S3 ∈ C maximizing Sim(S1 , S2 , S3 ). The brute force way of enumerating all possibilities leads to the worst case query time of O(n), O(n2 ) and O(n3 ) for problem 1, 2 and 3, respectively. In a hope to break this barrier, just like the case of pairwise near neighbor search, we deﬁne the c-approximate (c < 1) versions of the above three problems. As in the case of c-NN, we are given two parameters R0 > 0 and δ > 0. For each of the following three problems, the guarantee is with probability at least 1 − δ: 1. (3-way c-Near Neighbor or 3-way c-NN) Given two query sets S1 and S2 , if there ′ exists S3 ∈ C with Sim(S1 , S2 , S3 ) ≥ R0 , then we report some S3 ∈ C so that ′ Sim(S1 , S2 , S3 ) ≥ cR0 . 2. (3-way c-Close Pair or 3-way c-CP) Given a query set S1 , if there exists a pair of ′ ′ set S2 , S3 ∈ C with Sim(S1 , S2 , S3 ) ≥ R0 , then we report sets S2 , S3 ∈ C so that ′ ′ Sim(S1 , S2 , S3 ) ≥ cR0 . 3. (3-way c-Best Cluster or 3-way c-BC) If there exist sets S1 , S2 , S3 ∈ C with ′ ′ ′ ′ ′ ′ Sim(S1 , S2 , S3 ) ≥ R0 , then we report sets S1 , S2 , S3 ∈ C so that Sim(S1 , S2 , S3 ) ≥ cR0 . 4 Sub-linear Algorithm for 3-way c-NN The basic philosophy behind sub-linear search is bucketing, which allows us to preprocess dataset in a fashion so that we can ﬁlter many bad candidates without scanning all of them. LSH-based techniques rely on randomized hash functions to create buckets that probabilistically ﬁlter bad candidates. This philosophy is not restricted for binary similarity functions and is much more general. Here, we ﬁrst focus on 3-way c-NN problem for binary data. Theorem 1 For R3way c-NN one can construct a data structure with O(nρ log1/cR0 n) query time and O(n1+ρ ) space, where ρ = 1 − log 1/c log 1/c+log 1/R0 . The argument for 2-way resemblance can be naturally extended to k-way resemblance. Speciﬁcally, given three sets S1 , S2 , S3 ⊆ Ω and an independent random permutation π : Ω → Ω, we have: P r (min(π(S1 )) = min(π(S2 )) = min(π(S3 ))) = R3way . (2) Eq.( 2) shows that minwise hashing, although it operates on sets individually, preserves all 3-way (in fact k-way) similarity structure of the data. The existence of such a hash function is the key requirement behind the existence of efﬁcient approximate search. For the pairwise case, the probability event was a simple hash collision, and the min-hash itself serves as the bucket index. In case 3 of 3-way (and higher) c-NN problem, we have to take care of a more complicated event to create an indexing scheme. In particular, during preprocessing we need to create buckets for each individual S3 , and while querying we need to associate the query sets S1 and S2 to the appropriate bucket. We need extra mechanisms to manipulate these minwise hashes to obtain a bucketing scheme. Proof of Theorem 1: We use two additional functions: f1 : Ω → N for manipulating min(π(S3 )) and f2 : Ω × Ω → N for manipulating both min(π(S1 )) and min(π(S2 )). Let a ∈ N+ such that |Ω| = D < 10a . We deﬁne f1 (x) = (10a + 1) × x and f2 (x, y) = 10a x + y. This choice ensures that given query S1 and S2 , for any S3 ∈ C, f1 (min(π(S3 ))) = f2 (min(π(S1 )), min(π(S2 ))) holds if and only if min(π(S1 )) = min(π(S2 )) = min(π(S2 )), and thus we get a bucketing scheme. To complete the proof, we introduce two integer parameters K and L. Deﬁne a new hash function by concatenating K events. To be more precise, while preprocessing, for every element S3 ∈ C create buckets g1 (S3 ) = [f1 (h1 (S3 )); ...; f1 (hK (S3 ))] where hi is chosen uniformly from minwise hashing family. For given query points S1 and S2 , retrieve only points in the bucket g2 (S1 , S2 ) = [f2 (h1 (S1 ), h1 (S2 )); ...; f2 (hK (S1 ), hK (S2 ))]. Repeat this process L times independently. For any K S3 ∈ C, with Sim(S1 , S2 , S3 ) ≥ R0 , is retrieved with probability at least 1 − (1 − R0 )L . Using log 1/c log K = ⌈ log n ⌉ and L = ⌈nρ log( 1 )⌉, where ρ = 1 − log 1/c+log 1/R0 , the proof can be obtained 1 δ cR0 using standard concentration arguments used to prove Fact 1, see [14, 13]. It is worth noting that the probability guarantee parameter δ gets absorbed in the constants as log( 1 ). Note, the process is δ stopped as soon as we ﬁnd some element with R3way ≥ cR0 . Theorem 1 can be easily extended to k-way resemblance with same query time and space guarantees. Note that k-way c-NN is at least as hard as k ∗ -way c-NN for any k ∗ ≤ k, because we can always choose (k −k ∗ +1) identical query sets in k-way c-NN, and it reduces to k ∗ -way c-NN problem. So, any improvements in R3way c-NN implies improvement in the classical min-hash LSH for Jaccard similarity. The proposed analysis is thus tight in this sense. The above observation makes it possible to also perform the traditional pairwise c-NN search using the same hash tables deployed for 3-way c-NN. In the query phase we have an option, if we have two different queries S1 , S2 , then we retrieve from bucket g2 (S1 , S2 ) and that is usual 3-way c-NN search. If we are just interested in pairwise near neighbor search given one query S1 , then we will look into bucket g2 (S1 , S1 ), and we know that the 3-way resemblance between S1 , S1 , S3 boils down to the pairwise resemblance between S1 and S3 . So, the same hash tables can be used for both the purposes. This property generalizes, and hash tables created for k-way c-NN can be used for any k ∗ -way similarity search so long as k ∗ ≤ k. The approximation guarantees still holds. This ﬂexibility makes k-way c-NN bucketing scheme more advantageous over the pairwise scheme. ρ 1 One of the peculiarity of LSH based techniques is that the query complexity exponent ρ < 1 is dependent on the choice R0=0.01 0.8 of the threshold R0 we are interested in and the value of c 0.05 0.1 0.3 0.6 which is the approximation ratio that we will tolerate. Figure 1 0.2 0.4 0.8 log 1/c 0.5 plots ρ = 1− log 1/c+log 1/R0 with respect to c, for selected R0 0.4 0.6 0.9 0.7 values from 0.01 to 0.99. For instance, if we are interested in 0.2 0.95 highly similar pairs, i.e. R0 ≈ 1, then we are looking at near R =0.99 0 O(log n) query complexity for c-NN problem as ρ ≈ 0. On 0 0 0.2 0.4 0.6 0.8 1 the other hand, for very lower threshold R0 , there is no much c log 1/c of hope of time-saving because ρ is close to 1. Figure 1: ρ = 1 − log 1/c+log 1/R0 . 5 Other Efﬁcient k-way Similarities We refer to the k-way similarities for which there exist sub-linear algorithms for c-NN search with query and space complexity exactly as given in Theorem 1 as efﬁcient . We have demonstrated existence of one such example of efﬁcient similarities, which is the k-way resemblance. This leads to a natural question: “Are there more of them?”. [9] analyzed all the transformations on similarities that preserve existence of efﬁcient LSH search. In particular, they showed that if S is a similarity for which there exists an LSH family, then there also exists an LSH family for any similarity which is a probability generating function (PGF) transfor∑∞ mation on S. PGF transformation on S is deﬁned as P GF (S) = i=1 pi S i , where S ∈ [0, 1] and ∑∞ pi ≥ 0 satisﬁes i=1 pi = 1. Similar theorem can also be shown in the case of 3-way resemblance. 4 Theorem 2 Any PGF transformation on 3-way resemblance R3way is efﬁcient. Recall in the proof of Theorem 1, we created hash assignments f1 (min(π(S3 ))) and f2 (min(π(S1 )), min(π(S2 ))), which lead to a bucketing scheme for the 3-way resemblance search, where the collision event E = {f1 (min(π(S3 )) = f2 (min(π(S1 )), min(π(S2 )))} happens with probability P r(E) = R3way . To prove the above Theorem 2, we will need to create hash events ∑∞ i having probability P GF (R3way ) = i=1 pi (R3way ) . Note that 0 ≤ P GF (R3way ) ≤ 1. We will make use of the following simple lemma. Lemma 1 (R3way )n is efﬁcient for all n ∈ N. n n Proof: Deﬁne new hash assignments g1 (S3 ) = [f1 (h1 (S3 )); ...; f1 (hn (S3 ))] and g2 (S1 , S2 ) = n n [f2 (h1 (S1 ), h1 (S2 )); ...; f2 (hn (S1 ), hn (S2 ))]. The collision event g1 (S3 ) = g2 (S1 , S2 ) has n n probability (R3way )n . We now use the pair < g1 , g2 > instead of < f1 , f2 > and obtain same 3way n guarantees, as in Theorem 1, for (R ) as well. i i Proof of Theorem 2: From Lemma 1, let < g1 , g2 > be the hash pair corresponding to (R3way )i i i as used in above lemma. We sample one hash pair from the set {< g1 , g2 >: i ∈ N}, where i i the probability of sampling < g1 , g2 > is proportional to pi . Note that pi ≥ 0, and satisﬁes ∑∞ is i=1 pi = 1, and so the above sampling ∑ valid. It is not difﬁcult to see that the collision of the ∞ sampled hash pair has probability exactly i=1 pi (R3way )i . Theorem 2 can be naturally extended to k-way similarity for any k ≥ 2. Thus, we now have inﬁnitely many k-way similarity functions admitting efﬁcient sub-linear search. One, that might be interesting, because of its radial basis kernel like nature, is shown in the following corollary. Corollary 1 eR k−way −1 is efﬁcient. Proof: Use the expansion of eR k−way normalized by e to see that eR k−way −1 is a PGF on Rk−way . 6 Fast Algorithms for 3-way c-CP and 3-way c-BC Problems For 3-way c-CP and 3-way c-BC problems, using bucketing scheme with minwise hashing family will save even more computations. Theorem 3 For R3way c-Close Pair Problem (or c-CP) one can construct a data structure with log 1/c O(n2ρ log1/cR0 n) query time and O(n1+2ρ ) space, where ρ = 1 − log 1/c+log 1/R0 . Note that we can switch the role of f1 and f2 in the proof of Theorem 1. We are thus left with a c-NN problem with search space O(n2 ) (all pairs) instead of n. A bit of analysis, similar to Theorem 1, will show that this procedure achieves the required query time O(n2ρ log1/cR0 n), but uses a lot more space, O(n2(1+ρ )), than shown in the above theorem. It turns out that there is a better way of doing c-CP that saves us space. Proof of Theorem 3: We again start with constructing hash tables. For every element Sc ∈ C, we create a hash-table and store Sc in bucket B(Sc ) = [h1 (Sc ); h2 (Sc ); ...; hK (Sc )], where hi is chosen uniformly from minwise independent family of hash functions H. We create L such hash-tables. For a query element Sq we look for all pairs in bucket B(Sq ) = [h1 (Sq ); h2 (Sq ); ...; hK (Sq )] and repeat this for each of the L tables. Note, we do not form pairs of elements retrieved from different tables as they do not satisfy Eq. (2). If there exists a pair S1 , S2 ∈ C with Sim(Sq , S1 , S2 ) ≥ R0 , using K Eq. (2), we can see that we will ﬁnd that pair in bucket B(Sq ) with probability 1 − (1 − R0 )L . Here, we cannot use traditional choice of K and L, similar to what we did in Theorem 1, as there 2 log are O(n2 ) instead of O(n) possible pairs. We instead use K = ⌈ log 1n ⌉ and L = ⌈n2ρ log( 1 )⌉, δ cR0 log 1/c with ρ = 1 − log 1/c+log 1/R0 . With this choice of K and L, the result follows. Note, the process is stopped as soon as we ﬁnd pairs S1 and S2 with Sim(Sq , S1 , S2 ) ≥ cR0 . The key argument that saves space from O(n2(1+ρ) ) to O(n1+2ρ ) is that we hash n points individually. Eq. (2) makes it clear that hashing all possible pairs is not needed when every point can be processed individually, and pairs formed within each bucket itself ﬁlter out most of the unnecessary combinations. 5 Theorem 4 For R3way c-Best Cluster Problem (or c-BC) there exist an algorithm with running time log 1/c O(n1+2ρ log1/cR0 n), where ρ = 1 − log 1/c+log 1/R0 . The argument similar to one used in proof of Theorem 3 leads to the running time of O(n1+3ρ log1/cR0 n) as we need L = O(n3ρ ), and we have to processes all points at least once. Proof of Theorem 4: Repeat c-CP problem n times for every element in collection C acting as query once. We use the same set of hash tables and hash functions every time. The preprocessing time is O(n1+2ρ log1/cR0 n) evaluations of hash functions and the total querying time is O(n × n2ρ log1/cR0 n), which makes the total running time O(n1+2ρ log1/cR0 n). For k-way c-BC Problem, we can achieve O(n1+(k−1)ρ log1/cR0 n) running time. If we are interested in very high similarity cluster, with R0 ≈ 1, then ρ ≈ 0, and the running time is around O(n log n). This is a huge saving over the brute force O(nk ). In most practical cases, specially in big data regime where we have enormous amount of data, we can expect the k-way similarity of good clusters to be high and ﬁnding them should be efﬁcient. We can see that with increasing k, hashing techniques save more computations. 7 Experiments In this section, we demonstrate the usability of 3-way and higher-order similarity search using (i) Google Sets, and (ii) Improving retrieval quality. 7.1 Google Sets: Generating Semantically Similar Words Here, the task is to retrieve words which are “semantically” similar to the given set of query words. We collected 1.2 million random documents from Wikipedia and created a standard term-doc binary vector representation of each term present in the collected documents after removing standard stop words and punctuation marks. More speciﬁcally, every word is represented as a 1.2 million dimension binary vector indicating its presence or absence in the corresponding document. The total number of terms (or words) was around 60,000 in this experiment. Since there is no standard benchmark available for this task, we show qualitative evaluations. For querying, we used the following four pairs of semantically related words: (i) “jaguar” and “tiger”; (ii) “artiﬁcial” and “intelligence”; (iii) “milky” and “way” ; (iv) “ﬁnger” and “lakes”. Given the query words w1 and w2 , we compare the results obtained by the following four methods. • Google Sets: We use Google’s algorithm and report 5 words from Google spreadsheets [1]. This is Google’s algorithm which uses its own data. • 3-way Resemblance (3-way): We use 3-way resemblance |w1 ∩w2 ∩w| to rank every word |w1 ∪w2 ∪w| w and report top 5 words based on this ranking. • Sum Resemblance (SR): Another intuitive method is to use the sum of pairwise resem|w2 ∩w| blance |w1 ∩w| + |w2 ∪w| and report top 5 words based on this ranking. |w1 ∪w| • Pairwise Intersection (PI): We ﬁrst retrieve top 100 words based on pairwise resemblance for each w1 and w2 independently. We then report the words common in both. If there is no word in common we do not report anything. The results in Table 1 demonstrate that using 3-way resemblance retrieves reasonable candidates for these four queries. An interesting query is “ﬁnger” and “lakes”. Finger Lakes is a region in upstate New York. Google could only relate it to New York, while 3-way resemblance could even retrieve the names of cities and lakes in the region. Also, for query “milky” and “way”, we can see some (perhaps) unrelated words like “dance” returned by Google. We do not see such random behavior with 3-way resemblance. Although we are not aware of the algorithm and the dataset used by Google, we can see that 3-way resemblance appears to be a right measure for this application. The above results also illustrate the problem with using the sum of pairwise similarity method. The similarity value with one of the words dominates the sum and hence we see for queries “artiﬁcial” and “intelligence” that all the retrieved words are mostly related to the word “intelligence”. Same is the case with query “ﬁnger” and “lakes” as well as “jaguar” and “tiger”. Note that “jaguar” is also a car brand. In addition, for all 4 queries, there was no common word in the top 100 words similar to the each query word individually and so PI method never returns anything. 6 Table 1: Top ﬁve words retrieved using various methods for different queries. “JAGUAR” AND “ TIGER” G OOGLE 3- WAY SR LION LEOPARD CHEETAH CAT DOG LEOPARD CHEETAH LION PANTHER CAT CAT LEOPARD LITRE BMW CHASIS “MILKY” AND “ WAY” G OOGLE 3- WAY SR DANCE STARS SPACE THE UNIVERSE GALAXY STARS EARTH LIGHT SPACE EVEN ANOTHER STILL BACK TIME PI — — — — — “ARTIFICIAL” AND “INTELLIGENCE” G OOGLE 3- WAY SR PI COMPUTER COMPUTER SECURITY — PROGRAMMING SCIENCE WEAPONS — INTELLIGENT SECRET — SCIENCE ROBOT HUMAN ATTACKS — ROBOTICS TECHNOLOGY HUMAN — PI — — — — — G OOGLE NEW YORK NY PARK CITY “FINGER” AND “LAKES” 3- WAY SR SENECA CAYUGA ERIE ROCHESTER IROQUOIS RIVERS FRESHWATER FISH STREAMS FORESTED PI — — — — — We should note the importance of the denominator term in 3-way resemblance, without which frequent words will be blindly favored. The exciting contribution of this paper is that 3-way resemblance similarity search admits provable sub-linear guarantees, making it an ideal choice. On the other hand, no such provable guarantees are known for SR and other heuristic based search methods. 7.2 Improving Retrieval Quality in Similarity Search We also demonstrate how the retrieval quality of traditional similarity search can be boosted by utilizing more query candidates instead of just one. For the evaluations we choose two public datasets: MNIST and WEBSPAM, which were used in a recent related paper [26] for near neighbor search with binary data using b-bit minwise hashing [20, 23]. The two datasets reﬂect diversity both in terms of task and scale that is encountered in practice. The MNIST dataset consists of handwritten digit samples. Each sample is an image of 28 × 28 pixel yielding a 784 dimension vector with the associated class label (digit 0 − 9). We binarize the data by settings all non zeros to be 1. We used the standard partition of MNIST, which consists of 10,000 samples in one set and 60,000 in the other. The WEBSPAM dataset, with 16,609,143 features, consists of sparse vector representation of emails labeled as spam or not. We randomly sample 70,000 data points and partitioned them into two independent sets of size 35,000 each. Table 2: Percentage of top candidates with the same labels as that of query retrieved using various similarity criteria. More indicates better retrieval quality (Best marked in bold). T OP Pairwise 3-way NNbor 4-way NNbor 1 94.20 96.90 97.70 MNIST 10 20 92.33 91.10 96.13 95.36 96.89 96.28 50 89.06 93.78 95.10 1 98.45 99.75 99.90 WEBSPAM 10 20 96.94 96.46 98.68 97.80 98.87 98.15 50 95.12 96.11 96.45 For evaluation, we need to generate potential similar search query candidates for k-way search. It makes no sense in trying to search for object simultaneously similar to two very different objects. To generate such query candidates, we took one independent set of the data and partition it according to the class labels. We then run a cheap k-mean clustering on each class, and randomly sample triplets < x1 , x2 , x3 > from each cluster for evaluating 2-way, 3-way and 4-way similarity search. For MNIST dataset, the standard 10,000 test set was partitioned according to the labels into 10 sets, each partition was then clustered into 10 clusters, and we choose 10 triplets randomly from each cluster. In all we had 100 such triplets for each class, and thus 1000 overall query triplets. For WEBSPAM, which consists only of 2 classes, we choose one of the independent set and performed the same procedure. We selected 100 triplets from each cluster. We thus have 1000 triplets from each class making the total number of 2000 query candidates. The above procedures ensure that the elements in each triplets < x1 , x2 , x3 > are not very far from each other and are of the same class label. For each triplet < x1 , x2 , x3 >, we sort all the points x in the other independent set based on the following: • Pairwise: We only use the information in x1 and rank x based on resemblance 7 |x1 ∩x| |x1 ∪x| . • 3-way NN: We rank x based on 3-way resemblance • 4-way NN: We rank x based on 4-way resemblance |x1 ∩x2 ∩x| |x1 ∪x2 ∪x| . |x1 ∩x2 ∩x3 ∩x| |x1 ∪x2 ∪x3 ∪x| . We look at the top 1, 10, 20 and 50 points based on orderings described above. Since, all the query triplets are of the same label, The percentage of top retrieved candidates having same label as that of the query items is a natural metric to evaluate the retrieval quality. This percentage values accumulated over all the triplets are summarized in Table 2. We can see that top candidates retrieved by 3-way resemblance similarity, using 2 query points, are of better quality than vanilla pairwise similarity search. Also 4-way resemblance, with 3 query points, further improves the results compared to 3-way resemblance similarity search. This clearly demonstrates that multi-way resemblance similarity search is more desirable whenever we have more than one representative query in mind. Note that, for MNIST, which contains 10 classes, the boost compared to pairwise retrieval is substantial. The results follow a consistent trend. 8 Future Work While the work presented in this paper is promising for efﬁcient 3-way and k-way similarity search in binary high-dimensional data, there are numerous interesting and practical research problems we can study as future work. In this section, we mention a few such examples. One-permutation hashing. Traditionally, building hash tables for near neighbor search required many (e.g., 1000) independent hashes. This is both time- and energy-consuming, not only for building tables but also for processing un-seen queries which have not been processed. One permutation hashing [22] provides the hope of reducing many permutations to merely one. The version in [22], however, was not applicable to near neighbor search due to the existence of many empty bins (which offer no indexing capability). The most recent work [27] is able to ﬁll the empty bins and works well for pairwise near neighbor search. It will be interesting to extend [27] to k-way search. Non-binary sparse data. This paper focuses on minwise hashing for binary data. Various extensions to real-valued data are possible. For example, our results naturally apply to consistent weighted sampling [25, 15], which is one way to handle non-binary sparse data. The problem, however, is not solved if we are interested in similarities such as (normalized) k-way inner products, although the line of work on Conditional Random Sampling (CRS) [19, 18] may be promising. CRS works on non-binary sparse data by storing a bottom subset of nonzero entries after applying one permutation to (real-valued) sparse data matrix. CRS performs very well for certain applications but it does not work in our context because the bottom (nonzero) subsets are not properly aligned. Building hash tables by directly using bits from minwise hashing. This will be a different approach from the way how the hash tables are constructed in this paper. For example, [26] directly used the bits from b-bit minwise hashing [20, 23] to build hash tables and demonstrated the signiﬁcant advantages compared to sim-hash [8, 12] and spectral hashing [29]. It would be interesting to see the performance of this approach in k-way similarity search. k-Way sign random projections. It would be very useful to develop theory for k-way sign random projections. For usual (real-valued) random projections, it is known that the volume (which is related to the determinant) is approximately preserved [24, 17]. We speculate that the collision probability of k-way sign random projections might be also a (monotonic) function of the determinant. 9 Conclusions We formulate a new framework for k-way similarity search and obtain fast algorithms in the case of k-way resemblance with provable worst-case approximation guarantees. We show some applications of k-way resemblance search in practice and demonstrate the advantages over traditional search. Our analysis involves the idea of probabilistic hashing and extends the well-known LSH family beyond the pairwise case. We believe the idea of probabilistic hashing still has a long way to go. Acknowledgement The work is supported by NSF-III-1360971, NSF-Bigdata-1419210, ONR-N00014-13-1-0764, and AFOSR-FA9550-13-1-0137. 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Approximate nearest neighbor: Towards removing the curse of dimensionality. Theory of Computing, 8(14):321–350, 2012. [14] Piotr Indyk and Rajeev Motwani. Approximate nearest neighbors: Towards removing the curse of dimensionality. In STOC, pages 604–613, Dallas, TX, 1998. [15] Sergey Ioffe. Improved consistent sampling, weighted minhash and l1 sketching. In ICDM, 2010. [16] Yugang Jiang, Chongwah Ngo, and Jun Yang. Towards optimal bag-of-features for object categorization and semantic video retrieval. In CIVR, pages 494–501, Amsterdam, Netherlands, 2007. [17] Alex Kulesza and Ben Taskar. Determinantal point processes for machine learning. Technical report, arXiv:1207.6083, 2013. [18] Ping Li and Kenneth W. Church. A sketch algorithm for estimating two-way and multi-way associations. Computational Linguistics (Preliminary results appeared in HLT/EMNLP 2005), 33(3):305–354, 2007. [19] Ping Li, Kenneth W. Church, and Trevor J. Hastie. 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