nips nips2009 nips2009-92 knowledge-graph by maker-knowledge-mining

92 nips-2009-Fast Graph Laplacian Regularized Kernel Learning via Semidefinite–Quadratic–Linear Programming

Source: pdf

Author: Xiao-ming Wu, Anthony M. So, Zhenguo Li, Shuo-yen R. Li

Abstract: Kernel learning is a powerful framework for nonlinear data modeling. Using the kernel trick, a number of problems have been formulated as semideﬁnite programs (SDPs). These include Maximum Variance Unfolding (MVU) (Weinberger et al., 2004) in nonlinear dimensionality reduction, and Pairwise Constraint Propagation (PCP) (Li et al., 2008) in constrained clustering. Although in theory SDPs can be efﬁciently solved, the high computational complexity incurred in numerically processing the huge linear matrix inequality constraints has rendered the SDP approach unscalable. In this paper, we show that a large class of kernel learning problems can be reformulated as semideﬁnite-quadratic-linear programs (SQLPs), which only contain a simple positive semideﬁnite constraint, a second-order cone constraint and a number of linear constraints. These constraints are much easier to process numerically, and the gain in speedup over previous approaches is at least of the order m2.5 , where m is the matrix dimension. Experimental results are also presented to show the superb computational efﬁciency of our approach.

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Using the kernel trick, a number of problems have been formulated as semideﬁnite programs (SDPs). [sent-18, score-0.227]

2 These include Maximum Variance Unfolding (MVU) (Weinberger et al. [sent-19, score-0.12]

3 , 2004) in nonlinear dimensionality reduction, and Pairwise Constraint Propagation (PCP) (Li et al. [sent-20, score-0.21]

4 Although in theory SDPs can be efﬁciently solved, the high computational complexity incurred in numerically processing the huge linear matrix inequality constraints has rendered the SDP approach unscalable. [sent-22, score-0.109]

5 In this paper, we show that a large class of kernel learning problems can be reformulated as semideﬁnite-quadratic-linear programs (SQLPs), which only contain a simple positive semideﬁnite constraint, a second-order cone constraint and a number of linear constraints. [sent-23, score-0.409]

6 These constraints are much easier to process numerically, and the gain in speedup over previous approaches is at least of the order m2. [sent-24, score-0.091]

7 Some well-known kernel methods include support vector machines (SVMs) (Vapnik, 2000), kernel principal component analysis (kernel PCA) (Sch¨ lkopf et al. [sent-28, score-0.482]

8 , 1998), and kernel o k-means (Shawe-Taylor & Cristianini, 2004). [sent-29, score-0.167]

9 Naturally, an important issue in kernel methods is kernel design. [sent-30, score-0.334]

10 Indeed, the performance of a kernel method depends crucially on the kernel used, where different choices of kernels often lead to quite different results. [sent-31, score-0.361]

11 For instance, in (Chapelle & Vapnik, 2000), parametric kernel functions are proposed, where the focus is on model selection (Chapelle & Vapnik, 2000). [sent-33, score-0.167]

12 For instance, in cases where only inner products of the input data are involved, kernel learning is equivalent to the learning of a kernel matrix. [sent-36, score-0.334]

13 Currently, many different kernel learning frameworks have been proposed. [sent-38, score-0.167]

14 These include spectral kernel learning (Li & Liu, 2009), multiple kernel learning (Lanckriet et al. [sent-39, score-0.454]

15 , 2004), and the Breg1 man divergence-based kernel learning (Kulis et al. [sent-40, score-0.287]

16 Typically, a kernel learning framework consists of two main components: the problem formulation in terms of the kernel matrix, and an optimization procedure for ﬁnding the kernel matrix that has certain desirable properties. [sent-42, score-0.611]

17 , the Maximum Variance Unfolding (MVU) method (Weinberger et al. [sent-45, score-0.12]

18 , 2004) for nonlinear dimensionality reduction (see (So, 2007) for related discussion) and Pairwise Constraint Propagation (PCP) (Li et al. [sent-46, score-0.238]

19 , 2008) for constrained clustering, a nice feature of such a framework is that the problem formulation often becomes straightforward. [sent-47, score-0.104]

20 Thus, the major challenge in optimizationbased kernel learning lies in the second component, where the key is to ﬁnd an efﬁcient procedure to obtain a positive semideﬁnite kernel matrix that satisﬁes certain properties. [sent-48, score-0.375]

21 Using the kernel trick, most kernel learning problems (Graepel, 2002; Weinberger et al. [sent-49, score-0.454]

22 , 2008) can naturally be formulated as semideﬁnite programs (SDPs). [sent-52, score-0.06]

23 An effective and widely used heuristic for speedup is to perform low-rank kernel approximation and matrix factorization (Weinberger et al. [sent-54, score-0.378]

24 In this paper, we investigate the possibility of further speedup by studying a class of convex quadratic semideﬁnite programs (QSDPs). [sent-58, score-0.151]

25 These QSDPs arise in many contexts, such as graph Laplacian regularized low-rank kernel learning in nonlinear dimensionality reduction (Sha & Saul, 2005; Weinberger et al. [sent-59, score-0.461]

26 , 2008; Singer, 2008) and constrained clustering (Li et al. [sent-61, score-0.185]

27 In this paper, we propose a novel reformulation that exploits the structure of the QSDP and leads to a semideﬁnite-quadratic-linear program (SQLP) that can be solved by the standard software SDPT3 (T¨ t¨ nc¨ et al. [sent-65, score-0.211]

28 Such a uu u reformulation has the advantage that it only has one positive semideﬁnite constraint on a matrix of size m × m, one second-order cone constraint of size O(m2 ) and a number of linear constraints on O(m2 ) variables. [sent-67, score-0.465]

29 Experimental results show that our formulation is indeed far more efﬁcient than previous ones. [sent-71, score-0.069]

30 We review related kernel learning problems in Section 2 and present our formulation in Section 3. [sent-73, score-0.236]

31 2 The Problems In this section, we brieﬂy review some kernel learning problems that arise in dimensionality reduction and constrained clustering. [sent-76, score-0.272]

32 , 2008), (Singer, 2008), Pairwise Semideﬁnite Embedding (PSDE) (Globerson & Roweis, 2007), and PCP (Li et al. [sent-79, score-0.12]

33 MVU maximizes the variance of the embedding while preserving local distances of the input data. [sent-81, score-0.06]

34 PSDE derives an embedding that strictly respects known similarities, in the sense that objects known to be similar are always closer in the embedding than those known to be dissimilar. [sent-83, score-0.163]

35 PCP is designed for constrained clustering, which embeds the data on the unit hypersphere such that two objects that are known to be from the same cluster are embedded to the same point, while two objects that are known to be from different clusters are embedded orthogonally. [sent-84, score-0.175]

36 In particular, PCP seeks the smoothest mapping for such an embedding, thereby propagating pairwise constraints. [sent-85, score-0.093]

37 Initially, each of the above problems is formulated as an SDP, whose kernel matrix K is of size n×n, where n denotes the number of objects. [sent-86, score-0.239]

38 In other words, one takes K = QY QT , where Q ∈ Rn×m consists of the smoothest m eigenvectors of the graph Laplacian (m n), and Y is of size m × m (Sha & Saul, 2005; Weinberger et al. [sent-88, score-0.255]

39 , 2008; Globerson & Roweis, 2007; Singer, 2008; Li et al. [sent-90, score-0.12]

40 The learning of K is then reduced to the learning of Y , leading to a quadratic semideﬁnite program (QSDP) that is similar to a standard quadratic program (QP), except that the feasible set of a QSDP resides in the positive semideﬁnite cone as well. [sent-92, score-0.298]

41 The intuition behind this low-rank kernel approximation is that a kernel matrix of the 2 form K = QY QT actually, to some degree, preserves the proximity of objects in the feature space. [sent-93, score-0.418]

42 Next, we use MVU and PCP as representatives to demonstrate how the SDP formulations emerge from nonlinear dimensionality reduction and constrained clustering. [sent-95, score-0.153]

43 The constraint in (2) centers the embedding at the origin, thus removing the translation freedom. [sent-101, score-0.126]

44 The constraint K 0 in (4) speciﬁes that K must be symmetric and positive semideﬁnite, which is necessary since K is taken as the inner product matrix of the embedding. [sent-103, score-0.107]

45 Note that given the constraint in (2), the variance of the embedding 1 is characterized by V(K) = 2n i,j (kii + kjj − 2kij ) = tr(K) (Weinberger et al. [sent-104, score-0.277]

46 Thus, the SDP in (1-4) maximizes the variance of the embedding while keeping local distances unchanged. [sent-106, score-0.06]

47 After K is obtained, kernel PCA is applied to K to compute the low-dimensional embedding. [sent-107, score-0.167]

48 , n, kij = 1, ∀(i, j) ∈ M, kij = 0, ∀(i, j) ∈ C, K 0, (6) (7) (8) (9) ¯ where L denotes the normalized graph Laplacian, M denotes the set of object pairs that are known to be from the same cluster, and C denotes those that are known to be from different clusters. [sent-115, score-0.251]

49 The constraints in (6) map the objects to the unit hypersphere. [sent-116, score-0.084]

50 The constraints in (7) map two objects that are known to be from the same cluster to the same vector. [sent-117, score-0.084]

51 The constraints in (8) map two objects that are known to be from different clusters to vectors that are orthogonal. [sent-118, score-0.084]

52 Then, the degree of smoothness of φ on the data graph can be captured by (Zhou et al. [sent-120, score-0.176]

53 , 2004): n S(φ) = 1 φ(xi ) φ(xj ) − wij √ 2 i,j=1 dii djj n j=1 2 ¯ = L • K, (10) F where wij is the similarity of xi and xj , dii = wij , and · F is the distance metric in F. [sent-121, score-0.18]

54 Thus, the SDP in (5-9) seeks the smoothest feature map that embeds the data on the unit hypersphere and at the same time respects the pairwise constraints. [sent-123, score-0.147]

55 After K is solved, kernel k-means is then applied to K to obtain the clusters. [sent-124, score-0.167]

56 3 Low-Rank Approximation: from SDP to QSDP The SDPs in MVU and PCP are difﬁcult to solve efﬁciently because their computational complexity scales at least cubically in the size of the matrix variable and the number of constraints (Borchers, 1999). [sent-126, score-0.082]

57 A useful heuristic is to use low-rank kernel approximation, which is motivated by the observation that the degree of freedom in the data is often much smaller than the number of parameters in a fully nonparametric kernel matrix K. [sent-127, score-0.375]

58 Another more speciﬁc observation is that it is often desirable to have nearby objects mapped to nearby points, as is done in MVU or PCP. [sent-129, score-0.111]

59 The form K = QY QT , where Q and Y are of sizes n × m and m × m, respectively, where m matrix Q should be smooth in the sense that nearby objects in the input space are mapped to nearby points (the i-th row of Q is taken as a new representation of xi ). [sent-131, score-0.152]

60 In this way, the learning of a kernel matrix K is reduced to the learning of a much smaller Y , subject to the constraint that Y 0. [sent-133, score-0.274]

61 , 2009) to speed up MVU and PCP, respectively, and is also adopted in Colored MVU (Song et al. [sent-136, score-0.12]

62 , vm+1 ), where {vi } are the eigenvectors of the graph Laplacian. [sent-143, score-0.081]

63 , um ), where {ui } are the eigenvectors of the normalized graph Laplacian. [sent-148, score-0.081]

64 Since such Q’s are obtained from graph Laplacians, we call the learning of K of the form K = QY QT the Graph Laplacian Regularized Kernel Learning problem, and denote the methods in (Weinberger et al. [sent-149, score-0.176]

65 Y 0, (13) (14) 2 where y = vec(Y ) ∈ Rm denotes the vector obtained by concatenating all the columns of Y , and A 0 (Weinberger et al. [sent-155, score-0.175]

66 Problem (13-14) is an instance of the so-called convex quadratic semideﬁnite program (QSDP), where the objective is a quadratic function in the matrix variable Y . [sent-159, score-0.173]

67 4 Previous Approach: from QSDP to SDP Currently, a typical approach for tackling a QSDP is to use the Schur complement (Boyd & Vandenberghe, 2004) to rewrite it as an SDP (Sha & Saul, 2005; Weinberger et al. [sent-163, score-0.173]

68 , 2008; Globerson & Roweis, 2007; Singer, 2008; Graepel, 2002), and then solve it using an SDP solver such as CSDP1 (Borchers, 1999) or SDPT32 (Toh et al. [sent-166, score-0.12]

69 Y 0 and Im2 1 (A 2 y)T 1 A2 y ν 0, (16) 1 where A 2 is the matrix square root of A, Im2 is the identity matrix of size m2 × m2 , and ν is a slack variable serving as an upper bound of yT Ay. [sent-178, score-0.082]

70 The second semideﬁnite cone constraint is equivalent 1 1 to (A 2 y)T (A 2 y) ≤ ν by the Schur complement. [sent-179, score-0.182]

71 Although the SDP in (15-16) has only m(m + 1)/2 + 1 variables, it has two semideﬁnite cone constraints, of sizes m×m and (m2 +1)×(m2 +1), respectively. [sent-180, score-0.116]

72 Thus, it is not surprising that m is set to a very small value in the related methods—for example, m = 10 in (Weinberger et al. [sent-184, score-0.12]

73 However, as noted by the authors in (Weinberger et al. [sent-187, score-0.12]

74 Is this formulation from QSDP to SDP the best we can have? [sent-191, score-0.069]

75 In the next section, we present a novel formulation that leads to a semideﬁnite-quadratic-linear program (SQLP), which is much more efﬁcient and scalable than the one above. [sent-193, score-0.119]

76 With Cholesky factorization, we can obtain an r × m2 matrix B such that A = B T B, as A is symmetric positive semideﬁnite and of rank r (Golub & Loan, 1996). [sent-199, score-0.078]

77 (19) (20) Next, we show that the constraint in (19) is equivalent to a second-order cone constraint. [sent-204, score-0.182]

78 Let Kn be the second-order cone of dimension n, i. [sent-205, score-0.116]

79 1, the problem in (17-20) 5 swiss roll sample (n=2000) (a) (b) Figure 1: Swiss Roll. [sent-225, score-0.254]

80 (e1 + e2 )T u = 1, By − Cu = 0, u ∈ Kr+2 , Y 0, (22) (23) (24) (25) y,u which is an instance of the SQLP problem (T¨ t¨ nc¨ et al. [sent-230, score-0.12]

81 Note that in this formulation, uu u we have traded the semideﬁnite cone constraint of size (m2 + 1) × (m2 + 1) in (16) with one second-order cone constraint of size r + 2 and r + 1 linear constraints. [sent-232, score-0.458]

82 As it turns out, such a formulation is much easier to process numerically and can be solved much more efﬁciently. [sent-233, score-0.069]

83 This compares very favorably with the m9 arithmetic complexity uu u of the SCSDP approach, and our experimental results indicate that the speedup in computation is quite substantial. [sent-237, score-0.182]

84 Moreover, in contrast with the SCSDP formulation, which does not take advantage of the low rank structure of A, our formulation does take advantage of such a structure. [sent-238, score-0.106]

85 4 Experimental Results In this section, we perform several experiments to demonstrate the viability of our SQLP formulation and its superior computational performance. [sent-239, score-0.069]

86 Since both the SQLP formulation and the previous SCSDP formulation can be solved by standard softwares to a satisfying gap tolerance, the focus in this comparison is not on the accuracy aspect but on the computational efﬁciency aspect. [sent-240, score-0.138]

87 Swiss Roll (Figure 1(a)) is a standard manifold model used for manifold learning and nonlinear dimensionality reduction. [sent-255, score-0.09]

88 , wij = 1 if xi is within the 4NN of xj or vice versa, and wij = 0 otherwise. [sent-341, score-0.12]

89 For RegPCP, we construct the graph following the approach suggested in (Li et al. [sent-343, score-0.176]

90 Speciﬁcally, we have wij = exp(−d2 /(2σ 2 )) if xi is within 20NN of xj or vice versa, and wij = 0 otherwise. [sent-345, score-0.12]

91 For the pairwise constraints used in RegPCP, we randomly generate 20 similarity constraints for each class, and 20 dissimilarity constraints for every two classes, yielding a total of 1100 constraints. [sent-347, score-0.162]

92 This is not very surprising, as the SQLP formulation takes advantage of the low rank structure of the data matrix A. [sent-374, score-0.147]

93 5 Conclusions We have studied a class of convex optimization programs called convex Quadratic Semideﬁnite Programs (QSDPs), which arise naturally from graph Laplacian regularized kernel learning (Sha & Saul, 2005; Weinberger et al. [sent-375, score-0.403]

94 A QSDP is similar to a QP, except that it is subject to a semideﬁnite cone constraint as well. [sent-379, score-0.182]

95 To tackle the QSDP, one typically uses the Schur complement to rewrite it as an SDP (SCSDP), thus resulting in a large linear matrix inequality constraint. [sent-380, score-0.067]

96 In this paper, we argue that this formulation is not computationally optimal and have proposed a novel formulation that leads to a semideﬁnite-quadratic-linear program (SQLP). [sent-381, score-0.188]

97 Our formulation introduces one positive semidefinite constraint, one second-order cone constraint and a set of linear constraints. [sent-382, score-0.251]

98 This should be contrasted with the two large semideﬁnite cone constraints in the SCSDP. [sent-383, score-0.157]

99 Our complexity analysis and experimental results have shown that the proposed SQLP formulation scales far better than the SCSDP formulation. [sent-384, score-0.069]

100 On the Implementation and Usage of SDPT3 — A MATLAB uu u Software Package for Semideﬁnite–Quadratic–Linear Programming, Version 4. [sent-534, score-0.094]

similar papers computed by tfidf model

tfidf for this paper:

wordName wordTfidf (topN-words)

[('qsdp', 0.341), ('sqlp', 0.305), ('mvu', 0.287), ('semide', 0.263), ('sdp', 0.257), ('scsdp', 0.233), ('weinberger', 0.213), ('pcp', 0.179), ('kernel', 0.167), ('iter', 0.157), ('li', 0.141), ('qy', 0.141), ('swiss', 0.128), ('roll', 0.126), ('regpcp', 0.126), ('et', 0.12), ('cone', 0.116), ('regmvu', 0.108), ('uu', 0.094), ('sha', 0.093), ('qsdps', 0.09), ('sdps', 0.09), ('qt', 0.085), ('saul', 0.083), ('song', 0.075), ('globerson', 0.072), ('toh', 0.072), ('borchers', 0.072), ('psde', 0.072), ('formulation', 0.069), ('constraint', 0.066), ('zt', 0.064), ('vapnik', 0.064), ('tij', 0.063), ('wij', 0.06), ('embedding', 0.06), ('programs', 0.06), ('roweis', 0.059), ('schur', 0.058), ('nemirovski', 0.058), ('nite', 0.056), ('laplacian', 0.056), ('graph', 0.056), ('kong', 0.054), ('smoothest', 0.054), ('usps', 0.053), ('singer', 0.051), ('hong', 0.051), ('kij', 0.051), ('nc', 0.051), ('program', 0.05), ('speedup', 0.05), ('kr', 0.049), ('nonlinear', 0.048), ('kii', 0.047), ('ie', 0.047), ('graepel', 0.043), ('objects', 0.043), ('colored', 0.043), ('dimensionality', 0.042), ('quadratic', 0.041), ('matrix', 0.041), ('constraints', 0.041), ('reformulation', 0.041), ('pairwise', 0.039), ('arithmetic', 0.038), ('bt', 0.038), ('rank', 0.037), ('massachusetts', 0.037), ('loan', 0.036), ('sqlps', 0.036), ('constrained', 0.035), ('nearby', 0.034), ('chinese', 0.034), ('kjj', 0.031), ('csdp', 0.031), ('cuhk', 0.031), ('unfolding', 0.031), ('denotes', 0.031), ('clustering', 0.03), ('cristianini', 0.03), ('tolerance', 0.03), ('smola', 0.03), ('qp', 0.029), ('todd', 0.029), ('reduction', 0.028), ('lkopf', 0.028), ('day', 0.027), ('kernels', 0.027), ('tackling', 0.027), ('embeds', 0.027), ('rendered', 0.027), ('vandenberghe', 0.027), ('hypersphere', 0.027), ('complement', 0.026), ('sch', 0.026), ('liu', 0.026), ('eigenvectors', 0.025), ('programming', 0.025), ('concatenating', 0.024)]

similar papers list:

simIndex simValue paperId paperTitle

same-paper 1 0.99999994 92 nips-2009-Fast Graph Laplacian Regularized Kernel Learning via Semidefinite–Quadratic–Linear Programming

Author: Xiao-ming Wu, Anthony M. So, Zhenguo Li, Shuo-yen R. Li

2 0.093935624 61 nips-2009-Convex Relaxation of Mixture Regression with Efficient Algorithms

Author: Novi Quadrianto, John Lim, Dale Schuurmans, Tibério S. Caetano

Abstract: We develop a convex relaxation of maximum a posteriori estimation of a mixture of regression models. Although our relaxation involves a semideﬁnite matrix variable, we reformulate the problem to eliminate the need for general semideﬁnite programming. In particular, we provide two reformulations that admit fast algorithms. The ﬁrst is a max-min spectral reformulation exploiting quasi-Newton descent. The second is a min-min reformulation consisting of fast alternating steps of closed-form updates. We evaluate the methods against Expectation-Maximization in a real problem of motion segmentation from video data. 1

3 0.08727017 128 nips-2009-Learning Non-Linear Combinations of Kernels

Author: Corinna Cortes, Mehryar Mohri, Afshin Rostamizadeh

Abstract: This paper studies the general problem of learning kernels based on a polynomial combination of base kernels. We analyze this problem in the case of regression and the kernel ridge regression algorithm. We examine the corresponding learning kernel optimization problem, show how that minimax problem can be reduced to a simpler minimization problem, and prove that the global solution of this problem always lies on the boundary. We give a projection-based gradient descent algorithm for solving the optimization problem, shown empirically to converge in few iterations. Finally, we report the results of extensive experiments with this algorithm using several publicly available datasets demonstrating the effectiveness of our technique.

4 0.083528504 120 nips-2009-Kernels and learning curves for Gaussian process regression on random graphs

Author: Peter Sollich, Matthew Urry, Camille Coti

Abstract: We investigate how well Gaussian process regression can learn functions deﬁned on graphs, using large regular random graphs as a paradigmatic example. Random-walk based kernels are shown to have some non-trivial properties: within the standard approximation of a locally tree-like graph structure, the kernel does not become constant, i.e. neighbouring function values do not become fully correlated, when the lengthscale σ of the kernel is made large. Instead the kernel attains a non-trivial limiting form, which we calculate. The fully correlated limit is reached only once loops become relevant, and we estimate where the crossover to this regime occurs. Our main subject are learning curves of Bayes error versus training set size. We show that these are qualitatively well predicted by a simple approximation using only the spectrum of a large tree as input, and generically scale with n/V , the number of training examples per vertex. We also explore how this behaviour changes for kernel lengthscales that are large enough for loops to become important. 1 Motivation and Outline Gaussian processes (GPs) have become a standard part of the machine learning toolbox [1]. Learning curves are a convenient way of characterizing their capabilities: they give the generalization error as a function of the number of training examples n, averaged over all datasets of size n under appropriate assumptions about the process generating the data. We focus here on the case of GP regression, where a real-valued output function f (x) is to be learned. The general behaviour of GP learning curves is then relatively well understood for the scenario where the inputs x come from a continuous space, typically Rn [2, 3, 4, 5, 6, 7, 8, 9, 10]. For large n, the learning curves then typically decay as a power law ∝ n−α with an exponent α ≤ 1 that depends on the dimensionality n of the space as well as the smoothness properties of the function f (x) as encoded in the covariance function. But there are many interesting application domains that involve discrete input spaces, where x could be a string, an amino acid sequence (with f (x) some measure of secondary structure or biological function), a research paper (with f (x) related to impact), a web page (with f (x) giving a score used to rank pages), etc. In many such situations, similarity between different inputs – which will govern our prior beliefs about how closely related the corresponding function values are – can be represented by edges in a graph. One would then like to know how well GP regression can work in such problem domains; see also [11] for a related online regression algorithm. We study this 1 problem here theoretically by focussing on the paradigmatic example of random regular graphs, where every node has the same connectivity. Sec. 2 discusses the properties of random-walk inspired kernels [12] on such random graphs. These are analogous to the standard radial basis function kernels exp[−(x − x )2 /(2σ 2 )], but we ﬁnd that they have surprising properties on large graphs. In particular, while loops in large random graphs are long and can be neglected for many purposes, by approximating the graph structure as locally tree-like, here this leads to a non-trivial limiting form of the kernel for σ → ∞ that is not constant. The fully correlated limit, where the kernel is constant, is obtained only because of the presence of loops, and we estimate when the crossover to this regime takes place. In Sec. 3 we move on to the learning curves themselves. A simple approximation based on the graph eigenvalues, using only the known spectrum of a large tree as input, works well qualitatively and predicts the exact asymptotics for large numbers of training examples. When the kernel lengthscale is not too large, below the crossover discussed in Sec. 2 for the covariance kernel, the learning curves depend on the number of examples per vertex. We also explore how this behaviour changes as the kernel lengthscale is made larger. Sec. 4 summarizes the results and discusses some open questions. 2 Kernels on graphs and trees We assume that we are trying to learn a function deﬁned on the vertices of a graph. Vertices are labelled by i = 1 . . . V , instead of the generic input label x we used in the introduction, and the associated function values are denoted fi ∈ R. By taking the prior P (f ) over these functions f = (f1 , . . . , fV ) as a (zero mean) Gaussian process we are saying that P (f ) ∝ exp(− 1 f T C −1 f ). 2 The covariance function or kernel C is then, in our graph setting, just a positive deﬁnite V × V matrix. The graph structure is characterized by a V × V adjacency matrix, with Aij = 1 if nodes i and j are connected by an edge, and 0 otherwise. All links are assumed to be undirected, so that Aij = Aji , V and there are no self-loops (Aii = 0). The degree of each node is then deﬁned as di = j=1 Aij . The covariance kernels we discuss in this paper are the natural generalizations of the squaredexponential kernel in Euclidean space [12]. They can be expressed in terms of the normalized graph Laplacian, deﬁned as L = 1 − D −1/2 AD −1/2 , where D is a diagonal matrix with entries d1 , . . . , dV and 1 is the V × V identity matrix. An advantage of L over the unnormalized Laplacian D − A, which was used in the earlier paper [13], is that the eigenvalues of L (again a V × V matrix) lie in the interval [0,2] (see e.g. [14]). From the graph Laplacian, the covariance kernels we consider here are constructed as follows. The p-step random walk kernel is (for a ≥ 2) C ∝ (1 − a−1 L)p = 1 − a−1 1 + a−1 D −1/2 AD −1/2 p (1) while the diffusion kernel is given by 1 C ∝ exp − 2 σ 2 L ∝ exp 1 2 −1/2 AD −1/2 2σ D (2) We will always normalize these so that (1/V ) i Cii = 1, which corresponds to setting the average (over vertices) prior variance of the function to be learned to unity. To see the connection of the above kernels to random walks, assume we have a walker on the graph who at each time step selects randomly one of the neighbouring vertices and moves to it. The probability for a move from vertex j to i is then Aij /dj . The transition matrix after s steps follows as (AD −1 )s : its ij-element gives the probability of being on vertex i, having started at j. We can now compare this with the p-step kernel by expanding the p-th power in (1): p p ( p )a−s (1−a−1 )p−s (D −1/2 AD −1/2 )s = D −1/2 s C∝ s=0 ( p )a−s (1−a−1 )p−s (AD −1 )s D 1/2 s s=0 (3) Thus C is essentially a random walk transition matrix, averaged over the number of steps s with s ∼ Binomial(p, 1/a) 2 (4) a=2, d=3 K1 1 1 Cl,p 0.9 p=1 p=2 p=3 p=4 p=5 p=10 p=20 p=50 p=100 p=200 p=500 p=infty 0.8 0.6 0.4 d=3 0.8 0.7 0.6 a=2, V=infty a=2, V=500 a=4, V=infty a=4, V=500 0.5 0.4 0.3 0.2 0.2 ln V / ln(d-1) 0.1 0 0 5 10 l 0 15 1 10 p/a 100 1000 Figure 1: (Left) Random walk kernel C ,p plotted vs distance along graph, for increasing number of steps p and a = 2, d = 3. Note the convergence to a limiting shape for large p that is not the naive fully correlated limit C ,p→∞ = 1. (Right) Numerical results for average covariance K1 between neighbouring nodes, averaged over neighbours and over randomly generated regular graphs. This shows that 1/a can be interpreted as the probability of actually taking a step at each of p “attempts”. To obtain the actual C the resulting averaged transition matrix is premultiplied by D −1/2 and postmultiplied by D 1/2 , which ensures that the kernel C is symmetric. For the diffusion kernel, one ﬁnds an analogous result but the number of random walk steps is now distributed as s ∼ Poisson(σ 2 /2). This implies in particular that the diffusion kernel is the limit of the p-step kernel for p, a → ∞ at constant p/a = σ 2 /2. Accordingly, we discuss mainly the p-step kernel in this paper because results for the diffusion kernel can be retrieved as limiting cases. In the limit of a large number of steps s, the random walk on a graph will reach its stationary distribution p∞ ∝ De where e = (1, . . . , 1). (This form of p∞ can be veriﬁed by checking that it remains unchanged after multiplication with the transition matrix AD −1 .) The s-step transition matrix for large s is then p∞ eT = DeeT because we converge from any starting vertex to the stationary distribution. It follows that for large p or σ 2 the covariance kernel becomes C ∝ D 1/2 eeT D 1/2 , i.e. Cij ∝ (di dj )1/2 . This is consistent with the interpretation of σ or (p/a)1/2 as a lengthscale over which the random walk can diffuse along the graph: once this lengthscale becomes large, the covariance kernel Cij is essentially independent of the distance (along the graph) between the vertices i and j, and the function f becomes fully correlated across the graph. (Explicitly f = vD 1/2 e under the prior, with v a single Gaussian random variable.) As we next show, however, the approach to this fully correlated limit as p or σ are increased is non-trivial. We focus in this paper on kernels on random regular graphs. This means we consider adjacency matrices A which are regular in the sense that they give for each vertex the same degree, di = d. A uniform probability distribution is then taken across all A that obey this constraint [15]. What will the above kernels look like on typical samples drawn from this distribution? Such random regular graphs will have long loops, of length of order ln(V ) or larger if V is large. Their local structure is then that of a regular tree of degree d, which suggests that it should be possible to calculate the kernel accurately within a tree approximation. In a regular tree all nodes are equivalent, so the kernel can only depend on the distance between two nodes i and j. Denoting this kernel value C ,p for a p-step random walk kernel, one has then C ,p=0 = δ ,0 and γp+1 C0,p+1 γp+1 C ,p+1 = = 1− 1 ad C 1 a C0,p + −1,p 1 a + 1− C1,p 1 a C (5) ,p + d−1 ad C +1,p for ≥1 (6) where γp is chosen to achieve the desired normalization C0,p = 1 of the prior variance for every p. Fig. 1(left) shows results obtained by iterating this recursion numerically, for a regular graph (in the tree approximation) with degree d = 3, and a = 2. As expected the kernel becomes more longranged initially as p increases, but eventually it is seen to approach a non-trivial limiting form. This can be calculated as C ,p→∞ = [1 + (d − 1)/d](d − 1)− /2 (7) 3 and is also plotted in the ﬁgure, showing good agreement with the numerical iteration. There are (at least) two ways of obtaining the result (7). One is to take the limit σ → ∞ of the integral representation of the diffusion kernel on regular trees given in [16] (which is also quoted in [13] but with a typographical error that effectively removes the factor (d − 1)− /2 ). Another route is to ﬁnd the steady state of the recursion for C ,p . This is easy to do but requires as input the unknown steady state value of γp . To determine this, one can map from C ,p to the total random walk probability S ,p in each “shell” of vertices at distance from the starting vertex, changing variables to S0,p = C0,p and S ,p = d(d − 1) −1 C ,p ( ≥ 1). Omitting the factors γp , this results in a recursion for S ,p that simply describes a biased random walk on = 0, 1, 2, . . ., with a probability of 1 − 1/a of remaining at the current , probability 1/(ad) of moving to the left and probability (d − 1)/(ad) of moving to the right. The point = 0 is a reﬂecting barrier where only moves to the right are allowed, with probability 1/a. The time evolution of this random walk starting from = 0 can now be analysed as in [17]. As expected from the balance of moves to the left and right, S ,p for large p is peaked around the average position of the walk, = p(d − 2)/(ad). For smaller than this S ,p has a tail behaving as ∝ (d − 1) /2 , and converting back to C ,p gives the large- scaling of C ,p→∞ ∝ (d − 1)− /2 ; this in turn ﬁxes the value of γp→∞ and so eventually gives (7). The above analysis shows that for large p the random walk kernel, calculated in the absence of loops, does not approach the expected fully correlated limit; given that all vertices have the same degree, the latter would correspond to C ,p→∞ = 1. This implies, conversely, that the fully correlated limit is reached only because of the presence of loops in the graph. It is then interesting to ask at what point, as p is increased, the tree approximation for the kernel breaks down. To estimate this, we note that a regular tree of depth has V = 1 + d(d − 1) −1 nodes. So a regular graph can be tree-like at most out to ≈ ln(V )/ ln(d − 1). Comparing with the typical number of steps our random walk takes, which is p/a from (4), we then expect loop effects to appear in the covariance kernel when p/a ≈ ln(V )/ ln(d − 1) (8) To check this prediction, we measure the analogue of C1,p on randomly generated [15] regular graphs. Because of the presence of loops, the local kernel values are not all identical, so the appropriate estimate of what would be C1,p on a tree is K1 = Cij / Cii Cjj for neighbouring nodes i and j. Averaging over all pairs of such neighbours, and then over a number of randomly generated graphs we ﬁnd the results in Fig. 1(right). The results for K1 (symbols) accurately track the tree predictions (lines) for small p/a, and start to deviate just around the values of p/a expected from (8), as marked by the arrow. The deviations manifest themselves in larger values of K1 , which eventually – now that p/a is large enough for the kernel to “notice” the loops - approach the fully correlated limit K1 = 1. 3 Learning curves We now turn to the analysis of learning curves for GP regression on random regular graphs. We assume that the target function f ∗ is drawn from a GP prior with a p-step random walk covariance kernel C. Training examples are input-output pairs (iµ , fi∗ + ξµ ) where ξµ is i.i.d. Gaussian noise µ of variance σ 2 ; the distribution of training inputs iµ is taken to be uniform across vertices. Inference from a data set D of n such examples µ = 1, . . . , n takes place using the prior deﬁned by C and a Gaussian likelihood with noise variance σ 2 . We thus assume an inference model that is matched to the data generating process. This is obviously an over-simpliﬁcation but is appropriate for the present ﬁrst exploration of learning curves on random graphs. We emphasize that as n is increased we see more and more function values from the same graph, which is ﬁxed by the problem domain; the graph does not grow. ˆ The generalization error is the squared difference between the estimated function fi and the target fi∗ , averaged across the (uniform) input distribution, the posterior distribution of f ∗ given D, the distribution of datasets D, and ﬁnally – in our non-Euclidean setting – the random graph ensemble. Given the assumption of a matched inference model, this is just the average Bayes error, or the average posterior variance, which can be expressed explicitly as [1] (n) = V −1 Cii − k(i)T Kk−1 (i) i 4 D,graphs (9) where the average is over data sets and over graphs, K is an n × n matrix with elements Kµµ = Ciµ ,iµ + σ 2 δµµ and k(i) is a vector with entries kµ (i) = Ci,iµ . The resulting learning curve depends, in addition to n, on the graph structure as determined by V and d, and the kernel and noise level as speciﬁed by p, a and σ 2 . We ﬁx d = 3 throughout to avoid having too many parameters to vary, although similar results are obtained for larger d. Exact prediction of learning curves by analytical calculation is very difﬁcult due to the complicated way in which the random selection of training inputs enters the matrix K and vector k in (9). However, by ﬁrst expressing these quantities in terms of kernel eigenvalues (see below) and then approximating the average over datasets, one can derive the approximation [3, 6] =g n + σ2 V , g(h) = (λ−1 + h)−1 α (10) α=1 This equation for has to be solved self-consistently because also appears on the r.h.s. In the Euclidean case the resulting predictions approximate the true learning curves quite reliably. The derivation of (10) for inputs on a ﬁxed graph is unchanged from [3], provided the kernel eigenvalues λα appearing in the function g(h) are deﬁned appropriately, by the eigenfunction condition Cij φj = λφi ; the average here is over the input distribution, i.e. . . . = V −1 j . . . From the deﬁnition (1) of the p-step kernel, we see that then λα = κV −1 (1 − λL /a)p in terms of the corα responding eigenvalue of the graph Laplacian L. The constant κ has to be chosen to enforce our normalization convention α λα = Cjj = 1. Fortunately, for large V the spectrum of the Laplacian of a random regular graph can be approximated by that of the corresponding large regular tree, which has spectral density [14] L ρ(λ ) = 4(d−1) − (λL − 1)2 d2 2πdλL (2 − λL ) (11) in the range λL ∈ [λL , λL ], λL = 1 + 2d−1 (d − 1)1/2 , where the term under the square root is ± + − positive. (There are also two isolated eigenvalues λL = 0, 2 but these have weight 1/V each and so can be ignored for large V .) Rewriting (10) as = V −1 α [(V λα )−1 + (n/V )( + σ 2 )−1 ]−1 and then replacing the average over kernel eigenvalues by an integral over the spectral density leads to the following prediction for the learning curve: = dλL ρ(λL )[κ−1 (1 − λL /a)−p + ν/( + σ 2 )]−1 (12) with κ determined from κ dλL ρ(λL )(1 − λL /a)p = 1. A general consequence of the form of this result is that the learning curve depends on n and V only through the ratio ν = n/V , i.e. the number of training examples per vertex. The approximation (12) also predicts that the learning curve will have two regimes, one for small ν where σ 2 and the generalization error will be essentially 2 independent of σ ; and another for large ν where σ 2 so that can be neglected on the r.h.s. and one has a fully explicit expression for . We compare the above prediction in Fig. 2(left) to the results of numerical simulations of the learning curves, averaged over datasets and random regular graphs. The two regimes predicted by the approximation are clearly visible; the approximation works well inside each regime but less well in the crossover between the two. One striking observation is that the approximation seems to predict the asymptotic large-n behaviour exactly; this is distinct to the Euclidean case, where generally only the power-law of the n-dependence but not its prefactor come out accurately. To see why, we exploit that for large n (where σ 2 ) the approximation (9) effectively neglects ﬂuctuations in the training input “density” of a randomly drawn set of training inputs [3, 6]. This is justiﬁed in the graph case for large ν = n/V , because the number of training inputs each vertex receives, Binomial(n, 1/V ), has negligible relative ﬂuctuations away from its mean ν. In the Euclidean case there is no similar result, because all training inputs are different with probability one even for large n. Fig. 2(right) illustrates that for larger a the difference in the crossover region between the true (numerically simulated) learning curves and our approximation becomes larger. This is because the average number of steps p/a of the random walk kernel then decreases: we get closer to the limit of uncorrelated function values (a → ∞, Cij = δij ). In that limit and for low σ 2 and large V the 5 V=500 (filled) & 1000 (empty), d=3, a=2, p=10 V=500, d=3, a=4, p=10 0 0 10 10 ε ε -1 -1 10 10 -2 10 -2 10 2 σ = 0.1 2 σ = 0.1 2 -3 10 σ = 0.01 2 σ = 0.01 -3 10 2 σ = 0.001 2 σ = 0.001 2 -4 10 2 σ = 0.0001 σ = 0.0001 -4 10 2 σ =0 -5 2 σ =0 -5 10 0.1 1 ν=n/V 10 10 0.1 1 ν=n/V 10 Figure 2: (Left) Learning curves for GP regression on random regular graphs with degree d = 3 and V = 500 (small ﬁlled circles) and V = 1000 (empty circles) vertices. Plotting generalization error versus ν = n/V superimposes the results for both values of V , as expected from the approximation (12). The lines are the quantitative predictions of this approximation. Noise level as shown, kernel parameters a = 2, p = 10. (Right) As on the left but with V = 500 only and for larger a = 4. 2 V=500, d=3, a=2, p=20 0 0 V=500, d=3, a=2, p=200, σ =0.1 10 10 ε ε simulation -1 2 10 1/(1+n/σ ) theory (tree) theory (eigenv.) -1 10 -2 10 2 σ = 0.1 -3 10 -4 10 -2 10 2 σ = 0.01 2 σ = 0.001 2 σ = 0.0001 -3 10 2 σ =0 -5 10 -4 0.1 1 ν=n/V 10 10 1 10 100 n 1000 10000 Figure 3: (Left) Learning curves for GP regression on random regular graphs with degree d = 3 and V = 500, and kernel parameters a = 2, p = 20; noise level σ 2 as shown. Circles: numerical simulations; lines: approximation (12). (Right) As on the left but for much larger p = 200 and for a single random graph, with σ 2 = 0.1. Dotted line: naive estimate = 1/(1 + n/σ 2 ). Dashed line: approximation (10) using the tree spectrum and the large p-limit, see (17). Solid line: (10) with numerically determined graph eigenvalues λL as input. α true learning curve is = exp(−ν), reﬂecting the probability of a training input set not containing a particular vertex, while the approximation can be shown to predict = max{1 − ν, 0}, i.e. a decay of the error to zero at ν = 1. Plotting these two curves (not displayed here) indeed shows the same “shape” of disagreement as in Fig. 2(right), with the approximation underestimating the true generalization error. Increasing p has the effect of making the kernel longer ranged, giving an effect opposite to that of increasing a. In line with this, larger values of p improve the accuracy of the approximation (12): see Fig. 3(left). One may ask about the shape of the learning curves for large number of training examples (per vertex) ν. The roughly straight lines on the right of the log-log plots discussed so far suggest that ∝ 1/ν in this regime. This is correct in the mathematical limit ν → ∞ because the graph kernel has a nonzero minimal eigenvalue λ− = κV −1 (1−λL /a)p : for ν σ 2 /(V λ− ), the square bracket + 6 in (12) can then be approximated by ν/( +σ 2 ) and one gets (because also regime) ≈ σ 2 /ν. σ 2 in the asymptotic However, once p becomes reasonably large, V λ− can be shown – by analysing the scaling of κ, see Appendix – to be extremely (exponentially in p) small; for the parameter values in Fig. 3(left) it is around 4 × 10−30 . The “terminal” asymptotic regime ≈ σ 2 /ν is then essentially unreachable. A more detailed analysis of (12) for large p and large (but not exponentially large) ν, as sketched in the Appendix, yields ∝ (cσ 2 /ν) ln3/2 (ν/(cσ 2 )), c ∝ p−3/2 (13) This shows that there are logarithmic corrections to the naive σ 2 /ν scaling that would apply in the true terminal regime. More intriguing is the scaling of the coefﬁcient c with p, which implies that to reach a speciﬁed (low) generalization error one needs a number of training examples per vertex of order ν ∝ cσ 2 ∝ p−3/2 σ 2 . Even though the covariance kernel C ,p – in the same tree approximation that also went into (12) – approaches a limiting form for large p as discussed in Sec. 2, generalization performance thus continues to improve with increasing p. The explanation for this must presumably be that C ,p converges to the limit (7) only at ﬁxed , while in the tail ∝ p, it continues to change. For ﬁnite graph sizes V we know of course that loops will eventually become important as p increases, around the crossover point estimated in (8). The approximation for the learning curve in (12) should then break down. The most naive estimate beyond this point would be to say that the kernel becomes nearly fully correlated, Cij ∝ (di dj )1/2 which in the regular case simpliﬁes to Cij = 1. With only one function value to learn, and correspondingly only one nonzero kernel eigenvalue λα=1 = 1, one would predict = 1/(1 + n/σ 2 ). Fig. 3(right) shows, however, that this signiﬁcantly underestimates the actual generalization error, even though for this graph λα=1 = 0.994 is very close to unity so that the other eigenvalues sum to no more than 0.006. An almost perfect prediction is obtained, on the other hand, from the approximation (10) with the numerically calculated values of the Laplacian – and hence kernel – eigenvalues. The presence of the small kernel eigenvalues is again seen to cause logarithmic corrections to the naive ∝ 1/n scaling. Using the tree spectrum as an approximation and exploiting the large-p limit, one ﬁnds indeed (see Appendix, Eq. (17)) that ∝ (c σ 2 /n) ln3/2 (n/c σ 2 ) where now n enters rather than ν = n/V , c being a constant dependent only on p and a: informally, the function to be learned only has a ﬁnite (rather than ∝ V ) number of degrees of freedom. The approximation (17) in fact provides a qualitatively accurate description of the data Fig. 3(right), as the dashed line in the ﬁgure shows. We thus have the somewhat unusual situation that the tree spectrum is enough to give a good description of the learning curves even when loops are important, while (see Sec. 2) this is not so as far as the evaluation of the covariance kernel itself is concerned. 4 Summary and Outlook We have studied theoretically the generalization performance of GP regression on graphs, focussing on the paradigmatic case of random regular graphs where every vertex has the same degree d. Our initial concern was with the behaviour of p-step random walk kernels on such graphs. If these are calculated within the usual approximation of a locally tree-like structure, then they converge to a non-trivial limiting form (7) when p – or the corresponding lengthscale σ in the closely related diffusion kernel – becomes large. The limit of full correlation between all function values on the graph is only reached because of the presence of loops, and we have estimated in (8) the values of p around which the crossover to this loop-dominated regime occurs; numerical data for correlations of function values on neighbouring vertices support this result. In the second part of the paper we concentrated on the learning curves themselves. We assumed that inference is performed with the correct parameters describing the data generating process; the generalization error is then just the Bayes error. The approximation (12) gives a good qualitative description of the learning curve using only the known spectrum of a large regular tree as input. It predicts in particular that the key parameter that determines the generalization error is ν = n/V , the number of training examples per vertex. We demonstrated also that the approximation is in fact more useful than in the Euclidean case because it gives exact asymptotics for the limit ν 1. Quantitatively, we found that the learning curves decay as ∝ σ 2 /ν with non-trivial logarithmic correction terms. Slower power laws ∝ ν −α with α < 1, as in the Euclidean case, do not appear. 7 We attribute this to the fact that on a graph there is no analogue of the local roughness of a target function because there is a minimum distance (one step along the graph) between different input points. Finally we looked at the learning curves for larger p, where loops become important. These can still be predicted quite accurately by using the tree eigenvalue spectrum as an approximation, if one keeps track of the zero graph Laplacian eigenvalue which we were able to ignore previously; the approximation shows that the generalization error scales as σ 2 /n with again logarithmic corrections. In future work we plan to extend our analysis to graphs that are not regular, including ones from application domains as well as artiﬁcial ones with power-law tails in the distribution of degrees d, where qualitatively new effects are to be expected. It would also be desirable to improve the predictions for the learning curve in the crossover region ≈ σ 2 , which should be achievable using iterative approaches based on belief propagation that have already been shown to give accurate approximations for graph eigenvalue spectra [18]. These tools could then be further extended to study e.g. the effects of model mismatch in GP regression on random graphs, and how these are mitigated by tuning appropriate hyperparameters. Appendix We sketch here how to derive (13) from (12) for large p. Eq. (12) writes = g(νV /( + σ 2 )) with λL + g(h) = dλL ρ(λL )[κ−1 (1 − λL /a)−p + hV −1 ]−1 (14) λL − and κ determined from the condition g(0) = 1. (This g(h) is the tree spectrum approximation to the g(h) of (10).) Turning ﬁrst to g(0), the factor (1 − λL /a)p decays quickly to zero as λL increases above λL . One can then approximate this factor according to (1 − λL /a)p [(a − λL )/(a − λL )]p ≈ − − − (1 − λL /a)p exp[−(λL − λL )p/(a − λL )]. In the regime near λL one can also approximate the − − − − spectral density (11) by its leading square-root increase, ρ(λL ) = r(λL − λL )1/2 , with r = (d − − 1)1/4 d5/2 /[π(d − 2)2 ]. Switching then to a new integration variable y = (λL − λL )p/(a − λL ) and − − extending the integration limit to ∞ gives ∞ √ 1 = g(0) = κr(1 − λL /a)p [p/(a − λL )]−3/2 dy y e−y (15) − − 0 and this ﬁxes κ. Proceeding similarly for h > 0 gives ∞ g(h) = κr(1−λL /a)p [p/(a−λL )]−3/2 F (hκV −1 (1−λL /a)p ), − − − F (z) = √ dy y (ey +z)−1 0 (16) Dividing by g(0) = 1 shows that simply g(h) = F (hV −1 c−1 )/F (0), where c = 1/[κ(1 − σ2 λL /a)p ] = rF (0)[p/(a − λL )]−3/2 which scales as p−3/2 . In the asymptotic regime − − 2 2 we then have = g(νV /σ ) = F (ν/(cσ ))/F (0) and the desired result (13) follows from the large-z behaviour of F (z) ≈ z −1 ln3/2 (z). One can proceed similarly for the regime where loops become important. Clearly the zero Laplacian eigenvalue with weight 1/V then has to be taken into account. If we assume that the remainder of the Laplacian spectrum can still be approximated by that of a tree [18], we get (V + hκ)−1 + r(1 − λL /a)p [p/(a − λL )]−3/2 F (hκV −1 (1 − λL /a)p ) − − − g(h) = (17) V −1 + r(1 − λL /a)p [p/(a − λL )]−3/2 F (0) − − The denominator here is κ−1 and the two terms are proportional respectively to the covariance kernel eigenvalue λ1 , corresponding to λL = 0 and the constant eigenfunction, and to 1−λ1 . Dropping the 1 ﬁrst terms in the numerator and denominator of (17) by taking V → ∞ leads back to the previous analysis as it should. For a situation as in Fig. 3(right), on the other hand, where λ1 is close to unity, we have κ ≈ V and so g(h) ≈ (1 + h)−1 + rV (1 − λL /a)p [p/(a − λL )]−3/2 F (h(1 − λL /a)p ) (18) − − − The second term, coming from the small kernel eigenvalues, is the more slowly decaying because it corresponds to ﬁne detail of the target function that needs many training examples to learn accurately. It will therefore dominate the asymptotic behaviour of the learning curve: = g(n/σ 2 ) ∝ F (n/(c σ 2 )) with c = (1 − λL /a)−p independent of V . The large-n tail of the learning curve in − Fig. 3(right) is consistent with this form. 8 References [1] C E Rasmussen and C K I Williams. Gaussian processes for regression. In D S Touretzky, M C Mozer, and M E Hasselmo, editors, Advances in Neural Information Processing Systems 8, pages 514–520, Cambridge, MA, 1996. MIT Press. [2] M Opper. Regression with Gaussian processes: Average case performance. In I K Kwok-Yee, M Wong, I King, and Dit-Yun Yeung, editors, Theoretical Aspects of Neural Computation: A Multidisciplinary Perspective, pages 17–23. Springer, 1997. [3] P Sollich. Learning curves for Gaussian processes. In M S Kearns, S A Solla, and D A Cohn, editors, Advances in Neural Information Processing Systems 11, pages 344–350, Cambridge, MA, 1999. MIT Press. [4] M Opper and F Vivarelli. General bounds on Bayes errors for regression with Gaussian processes. In M Kearns, S A Solla, and D Cohn, editors, Advances in Neural Information Processing Systems 11, pages 302–308, Cambridge, MA, 1999. MIT Press. [5] C K I Williams and F Vivarelli. Upper and lower bounds on the learning curve for Gaussian processes. Mach. Learn., 40(1):77–102, 2000. [6] D Malzahn and M Opper. Learning curves for Gaussian processes regression: A framework for good approximations. In T K Leen, T G Dietterich, and V Tresp, editors, Advances in Neural Information Processing Systems 13, pages 273–279, Cambridge, MA, 2001. MIT Press. [7] D Malzahn and M Opper. A variational approach to learning curves. In T G Dietterich, S Becker, and Z Ghahramani, editors, Advances in Neural Information Processing Systems 14, pages 463–469, Cambridge, MA, 2002. MIT Press. [8] P Sollich and A Halees. Learning curves for Gaussian process regression: approximations and bounds. Neural Comput., 14(6):1393–1428, 2002. [9] P Sollich. Gaussian process regression with mismatched models. In T G Dietterich, S Becker, and Z Ghahramani, editors, Advances in Neural Information Processing Systems 14, pages 519–526, Cambridge, MA, 2002. MIT Press. [10] P Sollich. Can Gaussian process regression be made robust against model mismatch? In Deterministic and Statistical Methods in Machine Learning, volume 3635 of Lecture Notes in Artiﬁcial Intelligence, pages 199–210. 2005. [11] M Herbster, M Pontil, and L Wainer. Online learning over graphs. In ICML ’05: Proceedings of the 22nd international conference on Machine learning, pages 305–312, New York, NY, USA, 2005. ACM. [12] A J Smola and R Kondor. Kernels and regularization on graphs. In M Warmuth and B Sch¨ lkopf, o editors, Proc. Conference on Learning Theory (COLT), Lect. Notes Comp. Sci., pages 144–158. Springer, Heidelberg, 2003. [13] R I Kondor and J D Lafferty. Diffusion kernels on graphs and other discrete input spaces. In ICML ’02: Proceedings of the Nineteenth International Conference on Machine Learning, pages 315–322, San Francisco, CA, USA, 2002. Morgan Kaufmann. [14] F R K Chung. Spectral graph theory. Number 92 in Regional Conference Series in Mathematics. Americal Mathematical Society, 1997. [15] A Steger and N C Wormald. Generating random regular graphs quickly. Combinator. Probab. Comput., 8(4):377–396, 1999. [16] F Chung and S-T Yau. Coverings, heat kernels and spanning trees. The Electronic Journal of Combinatorics, 6(1):R12, 1999. [17] C Monthus and C Texier. Random walk on the Bethe lattice and hyperbolic brownian motion. J. Phys. A, 29(10):2399–2409, 1996. [18] T Rogers, I Perez Castillo, R Kuehn, and K Takeda. Cavity approach to the spectral density of sparse symmetric random matrices. Phys. Rev. E, 78(3):031116, 2008. 9

5 0.074173242 33 nips-2009-Analysis of SVM with Indefinite Kernels

Author: Yiming Ying, Colin Campbell, Mark Girolami

Abstract: The recent introduction of indeﬁnite SVM by Luss and d’Aspremont [15] has effectively demonstrated SVM classiﬁcation with a non-positive semi-deﬁnite kernel (indeﬁnite kernel). This paper studies the properties of the objective function introduced there. In particular, we show that the objective function is continuously differentiable and its gradient can be explicitly computed. Indeed, we further show that its gradient is Lipschitz continuous. The main idea behind our analysis is that the objective function is smoothed by the penalty term, in its saddle (min-max) representation, measuring the distance between the indeﬁnite kernel matrix and the proxy positive semi-deﬁnite one. Our elementary result greatly facilitates the application of gradient-based algorithms. Based on our analysis, we further develop Nesterov’s smooth optimization approach [17, 18] for indeﬁnite SVM which has an optimal convergence rate for smooth problems. Experiments on various benchmark datasets validate our analysis and demonstrate the efﬁciency of our proposed algorithms.

6 0.070190303 95 nips-2009-Fast subtree kernels on graphs

7 0.067952789 118 nips-2009-Kernel Choice and Classifiability for RKHS Embeddings of Probability Distributions

8 0.067899361 191 nips-2009-Positive Semidefinite Metric Learning with Boosting

9 0.067732833 79 nips-2009-Efficient Recovery of Jointly Sparse Vectors

10 0.065838486 202 nips-2009-Regularized Distance Metric Learning:Theory and Algorithm

11 0.065210685 77 nips-2009-Efficient Match Kernel between Sets of Features for Visual Recognition

12 0.065103605 257 nips-2009-White Functionals for Anomaly Detection in Dynamical Systems

13 0.06466642 119 nips-2009-Kernel Methods for Deep Learning

14 0.063186586 229 nips-2009-Statistical Analysis of Semi-Supervised Learning: The Limit of Infinite Unlabelled Data

15 0.06068778 64 nips-2009-Data-driven calibration of linear estimators with minimal penalties

16 0.059484139 208 nips-2009-Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization

17 0.058716778 142 nips-2009-Locality-sensitive binary codes from shift-invariant kernels

18 0.053442094 22 nips-2009-Accelerated Gradient Methods for Stochastic Optimization and Online Learning

19 0.053338073 139 nips-2009-Linear-time Algorithms for Pairwise Statistical Problems

20 0.052952018 212 nips-2009-Semi-Supervised Learning in Gigantic Image Collections

similar papers computed by lsi model

lsi for this paper:

topicId topicWeight

[(0, -0.157), (1, 0.092), (2, -0.046), (3, 0.06), (4, -0.05), (5, -0.015), (6, 0.002), (7, 0.102), (8, -0.032), (9, -0.037), (10, 0.057), (11, -0.031), (12, 0.065), (13, 0.043), (14, -0.13), (15, -0.083), (16, -0.021), (17, -0.004), (18, -0.093), (19, -0.007), (20, -0.026), (21, -0.038), (22, -0.003), (23, -0.02), (24, -0.017), (25, 0.054), (26, 0.034), (27, -0.063), (28, 0.016), (29, 0.025), (30, 0.027), (31, -0.05), (32, 0.021), (33, -0.052), (34, 0.017), (35, 0.02), (36, 0.078), (37, 0.113), (38, 0.066), (39, 0.065), (40, 0.011), (41, -0.022), (42, -0.074), (43, -0.063), (44, 0.005), (45, -0.031), (46, -0.099), (47, -0.024), (48, 0.005), (49, 0.013)]

similar papers list:

simIndex simValue paperId paperTitle

same-paper 1 0.93539411 92 nips-2009-Fast Graph Laplacian Regularized Kernel Learning via Semidefinite–Quadratic–Linear Programming

Author: Xiao-ming Wu, Anthony M. So, Zhenguo Li, Shuo-yen R. Li

2 0.66205084 33 nips-2009-Analysis of SVM with Indefinite Kernels

Author: Yiming Ying, Colin Campbell, Mark Girolami

3 0.62462652 128 nips-2009-Learning Non-Linear Combinations of Kernels

Author: Corinna Cortes, Mehryar Mohri, Afshin Rostamizadeh

4 0.56687629 120 nips-2009-Kernels and learning curves for Gaussian process regression on random graphs

Author: Peter Sollich, Matthew Urry, Camille Coti

5 0.56065756 95 nips-2009-Fast subtree kernels on graphs

Author: Nino Shervashidze, Karsten M. Borgwardt

Abstract: In this article, we propose fast subtree kernels on graphs. On graphs with n nodes and m edges and maximum degree d, these kernels comparing subtrees of height h can be computed in O(mh), whereas the classic subtree kernel by Ramon & G¨ rtner scales as O(n2 4d h). Key to this efﬁciency is the observation that the a Weisfeiler-Lehman test of isomorphism from graph theory elegantly computes a subtree kernel as a byproduct. Our fast subtree kernels can deal with labeled graphs, scale up easily to large graphs and outperform state-of-the-art graph kernels on several classiﬁcation benchmark datasets in terms of accuracy and runtime. 1

6 0.5446806 77 nips-2009-Efficient Match Kernel between Sets of Features for Visual Recognition

7 0.53716028 191 nips-2009-Positive Semidefinite Metric Learning with Boosting

8 0.52394515 227 nips-2009-Speaker Comparison with Inner Product Discriminant Functions

9 0.52320772 81 nips-2009-Ensemble Nystrom Method

10 0.51712543 79 nips-2009-Efficient Recovery of Jointly Sparse Vectors

11 0.49946648 182 nips-2009-Optimal Scoring for Unsupervised Learning

12 0.48579815 8 nips-2009-A Fast, Consistent Kernel Two-Sample Test

13 0.48164177 61 nips-2009-Convex Relaxation of Mixture Regression with Efficient Algorithms

14 0.47605559 118 nips-2009-Kernel Choice and Classifiability for RKHS Embeddings of Probability Distributions

15 0.47561151 106 nips-2009-Heavy-Tailed Symmetric Stochastic Neighbor Embedding

16 0.46312606 223 nips-2009-Sparse Metric Learning via Smooth Optimization

17 0.4580498 119 nips-2009-Kernel Methods for Deep Learning

18 0.4556284 124 nips-2009-Lattice Regression

19 0.45510516 30 nips-2009-An Integer Projected Fixed Point Method for Graph Matching and MAP Inference

20 0.44261834 64 nips-2009-Data-driven calibration of linear estimators with minimal penalties

similar papers computed by lda model

lda for this paper:

topicId topicWeight

[(24, 0.019), (25, 0.03), (35, 0.038), (36, 0.071), (39, 0.024), (58, 0.064), (71, 0.023), (86, 0.613)]

similar papers list:

simIndex simValue paperId paperTitle

1 0.98179084 120 nips-2009-Kernels and learning curves for Gaussian process regression on random graphs

Author: Peter Sollich, Matthew Urry, Camille Coti

2 0.97296035 6 nips-2009-A Biologically Plausible Model for Rapid Natural Scene Identification

Author: Sennay Ghebreab, Steven Scholte, Victor Lamme, Arnold Smeulders

Abstract: Contrast statistics of the majority of natural images conform to a Weibull distribution. This property of natural images may facilitate efficient and very rapid extraction of a scene's visual gist. Here we investigated whether a neural response model based on the Wei bull contrast distribution captures visual information that humans use to rapidly identify natural scenes. In a learning phase, we measured EEG activity of 32 subjects viewing brief flashes of 700 natural scenes. From these neural measurements and the contrast statistics of the natural image stimuli, we derived an across subject Wei bull response model. We used this model to predict the EEG responses to 100 new natural scenes and estimated which scene the subject viewed by finding the best match between the model predictions and the observed EEG responses. In almost 90 percent of the cases our model accurately predicted the observed scene. Moreover, in most failed cases, the scene mistaken for the observed scene was visually similar to the observed scene itself. Similar results were obtained in a separate experiment in which 16 other subjects where presented with artificial occlusion models of natural images. Together, these results suggest that Weibull contrast statistics of natural images contain a considerable amount of visual gist information to warrant rapid image identification.

same-paper 3 0.96566582 92 nips-2009-Fast Graph Laplacian Regularized Kernel Learning via Semidefinite–Quadratic–Linear Programming

Author: Xiao-ming Wu, Anthony M. So, Zhenguo Li, Shuo-yen R. Li

4 0.95703304 190 nips-2009-Polynomial Semantic Indexing

Author: Bing Bai, Jason Weston, David Grangier, Ronan Collobert, Kunihiko Sadamasa, Yanjun Qi, Corinna Cortes, Mehryar Mohri

Abstract: We present a class of nonlinear (polynomial) models that are discriminatively trained to directly map from the word content in a query-document or documentdocument pair to a ranking score. Dealing with polynomial models on word features is computationally challenging. We propose a low-rank (but diagonal preserving) representation of our polynomial models to induce feasible memory and computation requirements. We provide an empirical study on retrieval tasks based on Wikipedia documents, where we obtain state-of-the-art performance while providing realistically scalable methods. 1

5 0.94428825 253 nips-2009-Unsupervised feature learning for audio classification using convolutional deep belief networks

Author: Honglak Lee, Peter Pham, Yan Largman, Andrew Y. Ng

Abstract: In recent years, deep learning approaches have gained signiﬁcant interest as a way of building hierarchical representations from unlabeled data. However, to our knowledge, these deep learning approaches have not been extensively studied for auditory data. In this paper, we apply convolutional deep belief networks to audio data and empirically evaluate them on various audio classiﬁcation tasks. In the case of speech data, we show that the learned features correspond to phones/phonemes. In addition, our feature representations learned from unlabeled audio data show very good performance for multiple audio classiﬁcation tasks. We hope that this paper will inspire more research on deep learning approaches applied to a wide range of audio recognition tasks. 1

6 0.92963713 176 nips-2009-On Invariance in Hierarchical Models

7 0.8024804 151 nips-2009-Measuring Invariances in Deep Networks

8 0.76145077 119 nips-2009-Kernel Methods for Deep Learning

9 0.72301215 32 nips-2009-An Online Algorithm for Large Scale Image Similarity Learning

10 0.70015901 137 nips-2009-Learning transport operators for image manifolds

11 0.69262975 142 nips-2009-Locality-sensitive binary codes from shift-invariant kernels

12 0.68849027 95 nips-2009-Fast subtree kernels on graphs

13 0.68346119 241 nips-2009-The 'tree-dependent components' of natural scenes are edge filters

14 0.67786968 196 nips-2009-Quantification and the language of thought

15 0.67502838 84 nips-2009-Evaluating multi-class learning strategies in a generative hierarchical framework for object detection

16 0.67322159 104 nips-2009-Group Sparse Coding

17 0.65828323 2 nips-2009-3D Object Recognition with Deep Belief Nets

18 0.65711474 210 nips-2009-STDP enables spiking neurons to detect hidden causes of their inputs

19 0.6512863 212 nips-2009-Semi-Supervised Learning in Gigantic Image Collections

20 0.64714956 87 nips-2009-Exponential Family Graph Matching and Ranking