nips nips2004 nips2004-18 knowledge-graph by maker-knowledge-mining

18 nips-2004-Algebraic Set Kernels with Application to Inference Over Local Image Representations

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Author: Amnon Shashua, Tamir Hazan

Abstract: This paper presents a general family of algebraic positive deﬁnite similarity functions over spaces of matrices with varying column rank. The columns can represent local regions in an image (whereby images have varying number of local parts), images of an image sequence, motion trajectories in a multibody motion, and so forth. The family of set kernels we derive is based on a group invariant tensor product lifting with parameters that can be naturally tuned to provide a cook-book of sorts covering the possible ”wish lists” from similarity measures over sets of varying cardinality. We highlight the strengths of our approach by demonstrating the set kernels for visual recognition of pedestrians using local parts representations. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Algebraic Set Kernels with Application to Inference Over Local Image Representations Amnon Shashua and Tamir Hazan ∗ Abstract This paper presents a general family of algebraic positive deﬁnite similarity functions over spaces of matrices with varying column rank. [sent-1, score-0.505]

2 The columns can represent local regions in an image (whereby images have varying number of local parts), images of an image sequence, motion trajectories in a multibody motion, and so forth. [sent-2, score-0.623]

3 The family of set kernels we derive is based on a group invariant tensor product lifting with parameters that can be naturally tuned to provide a cook-book of sorts covering the possible ”wish lists” from similarity measures over sets of varying cardinality. [sent-3, score-0.738]

4 We highlight the strengths of our approach by demonstrating the set kernels for visual recognition of pedestrians using local parts representations. [sent-4, score-0.344]

5 This dichotomy is probably most emphasized in the area of computer vision, where image understanding from observations involve data instances of images or image sequences containing huge amounts of data. [sent-9, score-0.163]

6 The ”holistic” representations suffer also from sensitivity to occlusions, invariance to local and global transformations, non-rigidity of local parts of the object, and so forth. [sent-11, score-0.523]

7 Practitioners in the area of data representations have long noticed that a collection of local representations (part-based representations) can be most effective to ameliorate changes of appearance [5, 10, 6]. [sent-12, score-0.3]

8 The local data representations vary in their sophistication, but share the same principle where an image corresponds to a collection of points each in a relatively small dimensional space — instead of a single point in high-dimensional space induced by holistic representations. [sent-13, score-0.4]

9 In general, the number of points (local parts) per image may vary and the dimension of each point may vary as well. [sent-14, score-0.128]

10 The key for unifying local and holistic representations for inference engines is to design positive deﬁnite similarity functions (a. [sent-16, score-0.458]

11 A set kernel would be useful also to other types of inference engines such as kernel versions of PCA, LDA, CCA, ridge regression and any algorithm which can be mapped onto inner-products between pairs of data instances (see [8] for details on kernel methods). [sent-21, score-0.369]

12 Formally, we consider an instance being represented by a collection of vectors, which for the sake of convenience, form the columns of a matrix. [sent-22, score-0.167]

13 We would like to ﬁnd an algebraic family of similarity functions sim(A, B) over matrices A, B which satisfy the following requirements: (i) sim(A, B) is an inner product, i. [sent-23, score-0.367]

14 The design of a kernel over sets of vectors has been recently attracting much attention in the computer vision and machine learning literature. [sent-27, score-0.158]

15 A possible approach is to ﬁt a distribution to the set of vectors and deﬁne the kernel as a distribution matching measure [9, 12, 4]. [sent-28, score-0.132]

16 This has the advantage that the number of local parts can vary but at the expense of ﬁtting a distribution to the variation over parts. [sent-29, score-0.278]

17 The variation could be quite complex at times, unlikely to ﬁt into a known family of distributions in many situations of interest, and in practice the sample size (number of columns of A) is not sufﬁciently large to reliably ﬁt a distribution. [sent-30, score-0.215]

18 The alternative, which is the approach taken in this paper, is to create a kernel over sets of vectors in a direct manner. [sent-31, score-0.132]

19 It is important to note that although we chose SVM over local representations as the application to demonstrate the use of set kernels, the need for adequately working with instances made out of sets of various cardinalities spans many other application domains. [sent-33, score-0.215]

20 2 The General Family of Inner-Products over Matrices We wish to derive the general family of positive deﬁnite similarity measures sim(A, B) over matrices A, B which have the same number of rows but possibly different column rank (in particular, different number of columns). [sent-36, score-0.433]

21 Let ai , bj be the column vectors of matrices A, B and let k(ai , bj ) be the local kernel function. [sent-39, score-0.8]

22 For example, in the context where the column vectors represent local parts of an image, then the matching function k(·, ·) between pairs of local parts provides the building blocks of the overall similarity function. [sent-40, score-0.746]

23 The local kernel is some positive deﬁnite function k(x, y) = φ(x) φ(y) which is the inner-product between the ”feature”-mapped vectors x, y for some feature map φ(·). [sent-41, score-0.321]

24 The local kernels can be combined in a linear or non-linear manner. [sent-43, score-0.253]

25 When the combination is linear the similarity becomes the analogue of the inner-product between vectors extended to matrices. [sent-44, score-0.209]

26 We will refer to the linear family as sim(A, B) =< A, B > and that will be the focus of this section. [sent-45, score-0.106]

27 In the next section we will derive the general (algebraic) nonlinear family which is based on ”lifting” the input matrices A, B onto higher dimensional spaces and feeding the result onto the < ·, · > machinery developed in this section, i. [sent-46, score-0.337]

28 We will start by embedding A, B onto m × m matrices by zero padding as follows. [sent-49, score-0.178]

29 The the embedding is represented by linear combinations of tensor products: n k A→ n aij ei ⊗ ej , q B→ blt el ⊗ et . [sent-55, score-0.568]

30 Let S be a positive p semi deﬁnite m2 × m2 matrix represented by S = r=1 Gr ⊗ Fr where Gr , Fr are m × m 1 ˆ ˆ matrices . [sent-57, score-0.293]

31 The matrices Gr , Fr 1 Any S can be represented as a sum over tensor products: given column-wise ordering, the matrix G ⊗ F is composed of n × n blocks of the form fij G. [sent-67, score-0.514]

32 Therefore, take Gr to be the n × n blocks of S and Fr to be the elemental matrices which have ”1” in coordinate r = (i, j) and zero everywhere else. [sent-68, score-0.175]

33 p must be selected such that r=1 Gr ⊗ Fr is positive semi deﬁnite. [sent-69, score-0.099]

34 The problem of deciding on the the necessary conditions on Fr and Gr such that the sum over tensor products is p. [sent-70, score-0.221]

35 The sufﬁcient conditions are easy — choosing Gr , Fr to be positive p semi deﬁnite would make r=1 Gr ⊗ Fr positive semi deﬁnite as well. [sent-74, score-0.198]

36 In this context (of separable S) we need one more constraint in order to work with non-linear local kerˆ ˜ ˜ nels k(x, y) = φ(x) φ(y): the matrices Gr = Mr Mr must ”distribute with the kernel”, namely there exist Mr such that ˜ ˜ ˆ k(Mr x, Mr y) = φ(Mr x) φ(Mr y) = φ(x) Mr Mr φ(y) = φ(x) Gr φ(y). [sent-75, score-0.258]

37 ˆ The role of the matrices Gr is to perform global coordinate changes of Rn before application of the kernel k() on the columns of A, B. [sent-77, score-0.35]

38 These global transformations include projections (say onto prototypical ”parts”) that may be given or ”learned” from a training set. [sent-78, score-0.078]

39 ˆ The matrices Fr determine the range of interaction between columns of A and columns of ˆ ˆ ˆ B. [sent-79, score-0.44]

40 The various choices of F determine the type of invariances one could obtain from the similarity measure. [sent-85, score-0.198]

41 For example, when F = I the similarity is simply the sum (average) of the local kernels k(ai , bi ) thereby assuming we have a strict alignment between the local parts represented by A and the local parts represented by B. [sent-86, score-0.876]

42 On the other end of the invariance spectrum, when F = 11 (all entries are ”1”) the similarity measure averages over all interactions of local parts k(ai , bj ) thereby achieving an invariance to the order of the parts. [sent-87, score-0.695]

43 A decaying weighted interaction such as fij = σ −|i−j| would provide a middle ground between the assumption of strict alignment and the assumption of complete lack of alignment. [sent-88, score-0.122]

44 3 Lifting Matrices onto Higher Dimensions The family of sim(A, B) =< A, B > forms a weighted linear superposition of the local kernel k(ai , bj ). [sent-91, score-0.505]

45 Non-linear combinations of local kernels emerge using mappings ψ(A) from the input matrices onto other higher-dimensional matrices, thus forming sim(A, B) =< ψ(A), ψ(B) >. [sent-92, score-0.434]

46 Additional invariance properties and parameters controlling the perfromance of sim(A, B) emerge with the introduction of non-linear combinations of local kernels, and those will be discussed later on in this section. [sent-93, score-0.22]

47 Consider the general d-fold lifting ψ(A) = A⊗d which can be viewed as a nd × k d matrix. [sent-94, score-0.112]

48 The key for computational simpliﬁcation lay in the fact that choices of Fr determine not only local interactions (as in the linear case) but also group invariances. [sent-111, score-0.251]

49 The group invariances are a result of applying symmetric operators on the tensor product space — we will consider two of those operators here, known as the the d-fold alternating tensor A∧d = A ∧ . [sent-112, score-0.611]

50 These lifting operations introduce the determinant and permanent operations on ˆ submatrices of A Gr B, as described below. [sent-120, score-0.208]

51 The alternating tensor is a multilinear map of Rn , (A ∧ . [sent-121, score-0.266]

52 < id ≤ n form a basis of the alternating d − f old tensor product of Rn , denoted d n n×k n as Λ R . [sent-149, score-0.357]

53 If A ∈ R is a linear map on R sending points to Rk , then A∧d is a d n linear map on Λ R sending x1 ∧ . [sent-150, score-0.198]

54 , jd ]) where the determinant is of the d × d block constructed by choosing the rows i1 , . [sent-171, score-0.175]

55 In other words, Cd (A) has n rows and k columns d d (instead of nd × k d necessary for A⊗d ) whose entries are equal to the d × d minors of A. [sent-178, score-0.231]

56 Taken together, the ”d-fold alternating kernel” Λd (A, B) is deﬁned by: ˆ ˆ trace Cd (A Gr B)Fr , (2) Λd (A, B) =< A∧d , B ∧d >=< Cd (A), Cd (B) >= r ˆ where Fr is the × upper-left submatrix of the p. [sent-181, score-0.141]

57 Note d d ˆ ˆ that the local kernel plugs in as the entries of (A Gr B)ij = k(Mr ai , Mr bj ) where Gr = Mr Mr . [sent-184, score-0.494]

58 q d k d Another symmetric operator on the tensor product space is via the d-fold symmetric tensor space Symd Rn whose points are: 1 xσ(1) ⊗ . [sent-185, score-0.495]

59 , jd ]) and which stands for the map A·d (A · · · A)(x1 · · · xd ) = Ax1 · · · Axd . [sent-203, score-0.171]

60 In other words, Rd (A) has n+d−1 rows and k+d−1 columns whose entries are equal to d d the d×d permanents of A. [sent-204, score-0.231]

61 The ensuing kernel similarity function, referred to as the ”d-fold symmetric kernel” is: ˆ ˆ Symd (A, B) =< A·d , B ·d >=< Rd (A), Rd (B) >= trace Rd (A Gr B)Fr (3) r ˆ where Fr is the q+d−1 × k+d−1 upper-left submatrix of the positive deﬁnite m+d−1 × d d d n+d−1 matrix Fr . [sent-206, score-0.372]

62 1 Practical Considerations To recap, the family of similarity functions sim(A, B) comprise of the linear version < A, B > (eqn. [sent-209, score-0.196]

63 2,3) which are group projections of the general kernel < A⊗d , B ⊗d >. [sent-211, score-0.116]

64 These different similarity functions are controlled by the choice of three items: Gr , Fr and the parameter d representing the degree of the tensor product operator. [sent-212, score-0.326]

65 The role of Gr is fairly interesting as it can be viewed as a projection operator from ”parts” to prototypical parts that can be learned from a training set but we leave this to the full length article that will appear later. [sent-214, score-0.144]

66 Practically, to compute Λd (A, B) one needs to run over all d × d blocks of the k × q matrix A B (whose entries are k(ai , bj )) and for each block compute the determinant. [sent-215, score-0.312]

67 The similarity function is a weighted sum of all those determinants weighted by fij . [sent-216, score-0.213]

68 These determinants have an interesting geometric interpretation if those are computed over unitary matrices — as described next. [sent-218, score-0.215]

69 , QA has orthonormal columns which span the column space of A, then it has been recently shown −1 [14] that RA can be computed from A using only operations over k(ai , aj ). [sent-221, score-0.237]

70 Therefore, −T −1 the product QA QB , which is equal to RA A BRB , can be computed using only local −1 kernel applications. [sent-222, score-0.259]

71 In other words, for each A compute RA (can be done using only inner-products over columns of A), then when it comes to compute A B compute in−T −1 stead RA A BRB which is equivalent to computing QA QB . [sent-223, score-0.134]

72 Now, Λd (QA , QB ) for unitary matrices is the sum over the product of the cosine principal angles between d-dim subspaces spanned by columns of A and B. [sent-225, score-0.591]

73 The value of each determinant of the d × d blocks of QA QB is equal to the product of the cosine principal angles between the respective d-dim subspaces determined by corresponding selection of d columns from A and d columns from B. [sent-226, score-0.613]

74 For example, the case k = q = d produces Λd (QA , QB ) = det(QA QB ) which is the product of the eigenvalues of the matrix QA QB . [sent-227, score-0.074]

75 Those eigenvalues are the cosine of the principal angles between the column space of A and the column space of B [2]. [sent-228, score-0.248]

76 Therefore, det(QA QB ) measures the ”angle” between the two subspaces spanned by the respective columns of the input matrices — in particular is invariant to the order of the columns. [sent-229, score-0.406]

77 For smaller values of d we obtain the sum over such products between subspaces spanned by subsets of d columns between A and B. [sent-230, score-0.304]

78 The advantage of smaller values of d is two fold: ﬁrst it enables to compute the similarity when k = q and second breaks down the similarity between subspaces into smaller pieces. [sent-231, score-0.299]

79 The entries of the matrix F determine which subspaces are being considered and the interaction between subspaces in A and B. [sent-232, score-0.385]

80 A diagonal F compares corresponding subspaces (a) (b) Figure 1: (a) The conﬁguration of the nine sub-regions is displayed over the gradient image. [sent-233, score-0.119]

81 between A and B whereas off-diagonal entries would enable comparisons between different choices of subspaces in A and in B. [sent-235, score-0.223]

82 For example, we may want to consider choices of d columns arranged in a ”sliding” fashion, i. [sent-236, score-0.168]

83 This selection is associated with a sparse diagonal F where the non-vanishing entries along the diagonal have the value of ”1” and correspond to the sliding window selections. [sent-247, score-0.125]

84 To conclude, in the linear version < A, B > the role of F is to determine the range of interaction between columns of A and columns of B, whereas with the non-linear version it is the interaction between d-dim subspaces rather than individual columns. [sent-248, score-0.5]

85 We could select all possible interactions (exponential number) or any reduced interaction set such as the sliding window rule (linear number of choices) as described above. [sent-249, score-0.133]

86 4 Experiments We examined the performance of sim(A, B) on part-based representations for pedestrian detection using SVM for the inference engine. [sent-250, score-0.118]

87 We compared our results to the conventional down-sampled holistic representation where the raw images were down-sampled to size 20 × 20 and 32 × 32. [sent-255, score-0.158]

88 Our tests also included simulation of occlusions (in the test images) in order to examine the sensitivity of our sim(A, B) family to occlusions. [sent-256, score-0.136]

89 For the local part representation, the input image was divided into 9 ﬁxed regions where for each image local orientation statistics were were generated following [5, 7] with a total of 22 numbers per region (see Fig 1a), thereby making a 22 × 9 matrix representation to be fed into sim(A, B). [sent-257, score-0.421]

90 The table below summarizes the accuracy results for the raw-pixel (holistic) representation over three trials: (i) images down-sampled to 20 × 20, (ii) images down-sampled to 32 × 32, and (iii) test images were partially occluded (32 × 32 version). [sent-259, score-0.133]

91 5% The table below displays sim(A, B) with linear and RBF local kernels. [sent-263, score-0.155]

92 2% < A, B >, F = 11 85% 85% < A, B >, fij = 2−|i−j| Λ2 (A, B) 90. [sent-266, score-0.078]

93 4% 88% 90% One can see that the local part representation provides a sharp increase in accuracy compared to the raw pixel holistic representation. [sent-268, score-0.253]

94 The added power of invariance to order of parts induced by < A, B >, F = 11 is not required since the parts are aligned and therefore the accuracy is the highest for the linear combination of local RBF < A, B >, F = I. [sent-269, score-0.449]

95 The same applies for the non-linear version Λd (A, B) — the additional invariances that come with a non-linear combination of local parts are apparently not required. [sent-270, score-0.32]

96 The power of non-linearity associated with the combination of local parts comes to bear when the test images have occluded parts, i. [sent-271, score-0.309]

97 Although the experiments above are still preliminary they show the power and potential of the sim(A, B) family of kernels deﬁned over local kernels. [sent-274, score-0.309]

98 With the principles laid down in Section 3 one can construct a large number (we touched only a few) of algebraic kernels which combine the local kernels in non-linear ways thus creating invariances to order and increased performance against occlusion. [sent-275, score-0.468]

99 Further research is required for sifting through the various possibilities with this new family of kernels and extracting their properties, their invariances and behavior under changing parameters (Fr , Gr , d). [sent-276, score-0.253]

100 The kullback-leibler kernel as a framework for discriminant and localized representations for visual recognition. [sent-346, score-0.173]

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