nips nips2006 nips2006-77 knowledge-graph by maker-knowledge-mining

77 nips-2006-Fast Computation of Graph Kernels

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Author: Karsten M. Borgwardt, Nicol N. Schraudolph, S.v.n. Vishwanathan

Abstract: Using extensions of linear algebra concepts to Reproducing Kernel Hilbert Spaces (RKHS), we deﬁne a unifying framework for random walk kernels on graphs. Reduction to a Sylvester equation allows us to compute many of these kernels in O(n3 ) worst-case time. This includes kernels whose previous worst-case time complexity was O(n6 ), such as the geometric kernels of G¨ rtner et al. [1] and a the marginal graph kernels of Kashima et al. [2]. Our algebra in RKHS allow us to exploit sparsity in directed and undirected graphs more effectively than previous methods, yielding sub-cubic computational complexity when combined with conjugate gradient solvers or ﬁxed-point iterations. Experiments on graphs from bioinformatics and other application domains show that our algorithms are often more than 1000 times faster than existing approaches. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Reduction to a Sylvester equation allows us to compute many of these kernels in O(n3 ) worst-case time. [sent-17, score-0.177]

2 This includes kernels whose previous worst-case time complexity was O(n6 ), such as the geometric kernels of G¨ rtner et al. [sent-18, score-0.434]

3 [1] and a the marginal graph kernels of Kashima et al. [sent-19, score-0.36]

4 Our algebra in RKHS allow us to exploit sparsity in directed and undirected graphs more effectively than previous methods, yielding sub-cubic computational complexity when combined with conjugate gradient solvers or ﬁxed-point iterations. [sent-21, score-0.459]

5 Experiments on graphs from bioinformatics and other application domains show that our algorithms are often more than 1000 times faster than existing approaches. [sent-22, score-0.223]

6 Simple ways of comparing graphs which are based on pairwise comparison of nodes or edges, are possible in quadratic time, yet may neglect information represented by the structure of the graph. [sent-26, score-0.208]

7 Graph kernels, as originally proposed by G¨ rtner et al. [sent-27, score-0.18]

8 [3], a take the structure of the graph into account. [sent-30, score-0.186]

9 Even though the number of common random walks could potentially be exponential, polynomial time algorithms exist for computing these kernels. [sent-32, score-0.151]

10 Unfortunately for the practitioner, these kernels are still prohibitively expensive since their computation scales as O(n6 ), where n is the number of vertices in the input graphs. [sent-33, score-0.232]

11 In this paper, we extend common concepts from linear algebra to Reproducing Kernel Hilbert Spaces (RKHS), and use these extensions to deﬁne a unifying framework for random walk kernels. [sent-35, score-0.364]

12 We show that computing many random walk graph kernels including those of G¨ rtner et al. [sent-36, score-0.762]

13 We can now borrow concepts from tensor calculus to extend certain linear algebra operations to H: Deﬁnition 1 Let A ∈ X n×m, B ∈ X m×p, and C ∈ Rm×p. [sent-41, score-0.163]

14 [Φ(A) C]ik := j j Given A ∈ Rn×m and B ∈ Rp×q the Kronecker product A ⊗ B ∈ Rnp×mq and vec operator are     deﬁned as A∗1 A11 B A12 B . [sent-43, score-0.54]

15 (4) If p = q = n = m, direct computation of the right hand side of (4) requires O(n4 ) kernel evaluations. [sent-69, score-0.145]

16 3 Random Walk Kernels Random walk kernels on graphs are based on a simple idea: Given a pair of graphs perform a random walk on both of them and count the number of matching walks [1, 2, 3]. [sent-73, score-1.063]

17 These kernels mainly differ in the way the similarity between random walks is computed. [sent-74, score-0.284]

18 [1] count a the number of nodes in the random walk which have the same label. [sent-76, score-0.266]

19 [3] on the other hand, use a kernel deﬁned on nodes and edges in order to compute similarity between random walks, and deﬁne an initial probability distribution over nodes in order to ensure convergence. [sent-80, score-0.261]

20 In this section we present a unifying framework which includes the above mentioned kernels as special cases. [sent-81, score-0.155]

21 , a vector of all zeros with the i-th entry set to one), e to denote a vector with all entries set to one, 0 to denote the vector of all zeros, and I to denote the identity matrix. [sent-85, score-0.206]

22 A graph G ∈ G consists of an ordered and ﬁnite set of n vertices V denoted by {v1 , v2 , . [sent-87, score-0.291]

23 A vertex vi is said to be a neighbor of another vertex vj if they are connected by an edge. [sent-91, score-0.575]

24 G is said to be undirected if (vi , vj ) ∈ E ⇐⇒ (vj , vi ) ∈ E for all edges. [sent-92, score-0.363]

25 The unnormalized adjacency matrix of G is an n×n real matrix P with Pij = 1 if (vi , vj ) ∈ E, and 0 otherwise. [sent-93, score-0.387]

26 , Pij ∈ (0, ∞) if (vi , vj ) ∈ E and zero otherwise. [sent-96, score-0.169]

27 The matrix A := P D−1 is then called the normalized adjacency matrix, or simply adjacency matrix. [sent-98, score-0.271]

28 A walk w on G is a sequence of indices w1 , w2 , . [sent-99, score-0.234]

29 The length of a walk is equal to the number of edges encountered during the walk (here: t). [sent-103, score-0.518]

30 A graph is said to be connected if any two pairs of vertices can be connected by a walk; here we always work with connected graphs. [sent-104, score-0.406]

31 In other words, [At ]ij denotes the probability of a transition from vertex vi to vertex vj via a walk of length t. [sent-112, score-0.75]

32 We use this intuition to deﬁne random walk kernels on graphs. [sent-113, score-0.361]

33 Every edge labeled graph G is associated with a label matrix L ∈ X n×n , such that Lij = iff (vi , vj ) ∈ E, in other words only those edges / which are present in the graph get a non- label. [sent-115, score-0.811]

34 Let H be the RKHS endowed with the kernel κ : X × X → R, and let φ : X → H denote the corresponding feature map which maps to the zero element of H. [sent-116, score-0.184]

35 Henceforth we use the term labeled graph to denote an edge-labeled graph. [sent-119, score-0.25]

36 2 Product Graphs Given two graphs G(V, E) and G (V , E ), the product graph G× (V× , E× ) is a graph with nn vertices, each representing a pair of vertices from G and G , respectively. [sent-121, score-0.803]

37 An edge exists in E× iff the corresponding vertices are adjacent in both G and G . [sent-122, score-0.238]

38 Thus V× = {(vi , vi ) : vi ∈ V ∧ vi ∈ V }, (5) E× = {((vi ,vi ), (vj ,vj )) : (vi , vj ) ∈ E ∧ (vi , vj ) ∈ E }. [sent-123, score-0.827]

39 (6) If A and A are the adjacency matrices of G and G , respectively, the adjacency matrix of the product graph G× is A× = A ⊗ A . [sent-124, score-0.572]

40 An edge exists in the product graph iff an edge exits in both G and G , therefore performing a simultaneous random walk on G and G is equivalent to performing a random walk on the product graph [4]. [sent-125, score-1.323]

41 Then the initial probability distribution p× of the product graph is p× := p ⊗ p . [sent-127, score-0.301]

42 , the probability that a random walk ends at a given vertex), the stopping probability q× of the product graph is q× := q ⊗ q . [sent-130, score-0.572]

43 If G and G are edge-labeled, we can associate a weight matrix W× ∈ Rnn ×nn with G× , using our Kronecker product in RKHS (Deﬁnition 2): W× = Φ(L) ⊗ Φ(L ). [sent-131, score-0.17]

44 As a consequence of the deﬁnition of Φ(L) and Φ(L ), the entries of W× are non-zero only if the corresponding edge exists in the product graph. [sent-132, score-0.239]

45 The weight matrix is closely related to the adjacency matrix: assume that H = R endowed with the usual dot product, and φ(Lij ) = 1 if (vi , vj ) ∈ E or zero otherwise. [sent-133, score-0.378]

46 , the weight matrix is identical to the adjacency matrix of the product graph. [sent-136, score-0.333]

47 To extend the above discussion, assume that H = Rd endowed with the usual dot product, and that there are d distinct edge labels {1, 2, . [sent-137, score-0.158]

48 For each edge (vi , vj ) ∈ E we have φ(Lij ) = el if the edge (vi , vj ) is labeled l. [sent-141, score-0.532]

49 , its value between any two edges is one iff the labels on the edges match, and zero otherwise. [sent-145, score-0.183]

50 The weight matrix W× has a non-zero entry iff an edge exists in the product graph and the corresponding l edges in G and G have the same label. [sent-146, score-0.573]

51 Let A denote the adjacency matrix of the graph ﬁltered by l the label l, i. [sent-147, score-0.381]

52 Some simple algebra (omitted for the sake of brevity) shows that the weight matrix of the product graph can be written as d l l A⊗A . [sent-150, score-0.429]

53 3 (7) l=1 Kernel Deﬁnition Performing a random walk on the product graph G× is equivalent to performing a simultaneous random walk on the graphs G and G [4]. [sent-152, score-0.984]

54 Therefore, the (in + j, i n + j )-th entry of Ak represents × the probability of simultaneous k length random walks on G (starting from vertex vi and ending in vertex vj ) and G (starting from vertex vi and ending in vertex vj ). [sent-153, score-1.281]

55 The (in + j, i n + j )-th entry of W× represents the similarity between simultaneous k length random walks on G and G measured via the kernel function κ. [sent-155, score-0.336]

56 Given the weight matrix W× , initial and stopping probability distributions p× and q× , and an appropriately chosen discrete measure µ, we can deﬁne a random walk kernel on G and G as ∞ k µ(k) q× W× p× . [sent-156, score-0.432]

57 The lemma follows since a convex combination of kernels is itself a valid kernel. [sent-166, score-0.229]

58 [2] use marginalization and probabilities of random walks to deﬁne kernels on graphs. [sent-171, score-0.243]

59 Given transition probability matrices P and P associated with graphs G and G respectively, their kernel can be written as (see Eq. [sent-172, score-0.282]

60 The edge kernel κ(Lij , Li j ) := Pij Pi j κ(Lij , Li,j ) with λ = 1 recovers (9). [sent-176, score-0.187]

61 [1] use the adjacency matrix of the product graph to deﬁne the so-called geometric a kernel ∞ n n λk [Ak ]ij . [sent-178, score-0.57]

62 (11) k(G, G ) = × i=1 j=1 k=0 To recover their kernel in our framework, assume an uniform distribution over the vertices of G and G , i. [sent-179, score-0.211]

63 We now show that if the weight matrix W× can be written as (7) then the problem of computing the graph kernel (9) can be reduced to the problem of solving the following Sylvester equation: i i A λ X A + X0 , X= (14) i where vec(X0 ) = p× . [sent-192, score-0.382]

64 (17) (18) The right-hand side of (18) is the graph kernel (9). [sent-196, score-0.292]

65 Given the solution X of the Sylvester equation (14), the graph kernel can be obtained as q× vec(X) in O(n2 ) time. [sent-197, score-0.342]

66 Since solving the generalized Sylvester equation takes O(dn3 ) time, computing the graph kernel in this fashion is signiﬁcantly faster than the O(n6 ) time required by the direct approach. [sent-198, score-0.416]

67 Finding the nearest Kronecker product approximating a matrix such as W× is a well-studied problem in numerical linear algebra and efﬁcient algorithms which exploit sparsity of W× are readily available [6]. [sent-201, score-0.338]

68 2 Conjugate Gradient Methods Given a matrix M and a vector b, conjugate gradient (CG) methods solve the system of equations M x = b efﬁciently [7]. [sent-203, score-0.213]

69 Furthermore, if computing matrix-vector products is cheap — because M is sparse, for instance — the CG solver can be sped up signiﬁcantly [7]. [sent-208, score-0.196]

70 Speciﬁcally, if computing M v for an arbitrary vector v requires O(k) time, and the effective rank of the matrix is m, then a CG solver requires only O(mk) time to solve M x = b. [sent-209, score-0.241]

71 The graph kernel (9) can be computed by a two-step procedure: First we solve the linear system (I −λW× ) x = p× , (19) for x, then we compute q×x. [sent-210, score-0.335]

72 Directly computing the matrix-vector product W× r, requires O(n4 ) time. [sent-213, score-0.15]

73 Key to our speed-ups is the ability to exploit Lemma 1 to compute this matrix-vector product more efﬁciently: Recall that W× = Φ(L) ⊗ Φ(L ). [sent-214, score-0.174]

74 Letting xt denote the value of x at iteration t, we set x0 := p× , then compute xt+1 = p× + λW× xt (22) repeatedly until ||xt+1 − xt || < ε, where || · || denotes the Euclidean norm and ε some pre-deﬁned tolerance. [sent-221, score-0.197]

75 Since each iteration of (22) involves computation of the matrix-vector product W× xt , all speed-ups for computing the matrix-vector product discussed in Section 4. [sent-224, score-0.344]

76 In particular, we exploit the fact that W× is a sum of Kronecker products to reduce the worst-case time complexity to O(n3 ) in our experiments, in contrast to Kashima et al. [sent-226, score-0.152]

77 5 Experiments To assess the practical impact of our algorithmic improvements, we compared our techniques from Section 4 with G¨ rtner et al. [sent-228, score-0.18]

78 We tested the practical feasibility of the presented techniques on four real-world datasets whose size mandates fast graph kernel computation; two datasets of molecular compounds (MUTAG and PTC), and two datasets with hundreds of graphs describing protein tertiary structure (Protein and Enzyme). [sent-239, score-0.655]

79 Graph kernels provide useful measures of similarity for all these graphs; please refer to the addendum for more details on these datasets, and applications for graph kernels on them. [sent-240, score-0.481]

80 Figure 1: Time (in seconds on a log-scale) to compute 100×100 kernel matrix for unlabeled (left) resp. [sent-241, score-0.197]

81 Compare the conventional direct method (black) to our fast Sylvester equation, conjugate gradient (CG), and ﬁxed-point iteration (FP) approaches. [sent-243, score-0.219]

82 1 Unlabeled Graphs In a ﬁrst series of experiments, we compared graph topology only on our 4 datasets, i. [sent-245, score-0.186]

83 We report the time taken to compute the full graph kernel matrix for various sizes (number of graphs) in Table 1 and show the results for computing a 100 × 100 sub-matrix in Figure 1 (left). [sent-248, score-0.382]

84 On unlabeled graphs, conjugate gradient and ﬁxed-point iteration — sped up via our Lemma 1 — are consistently about two orders of magnitude faster than the conventional direct method. [sent-249, score-0.321]

85 The Sylvester approach is very competitive on smaller graphs (outperforming CG on MUTAG) but slows down with increasing number of nodes per graph; this is because we were unable to incorporate Lemma 1 into Matlab’s black-box dlyap solver. [sent-250, score-0.295]

86 2 Labeled Graphs In a second series of experiments, we compared graphs with node and edge labels. [sent-253, score-0.257]

87 On our two protein datasets we employed a linear kernel to measure similarity between edge labels representing ˚ distances (in angstr¨ ms) between secondary structure elements. [sent-254, score-0.36]

88 On our two chemical datasets we o used a delta kernel to compare edge labels reﬂecting types of bonds in molecules. [sent-255, score-0.305]

89 In the experiments with the delta kernel, conjugate gradient and ﬁxedpoint iteration are still at least two orders of magnitude faster. [sent-258, score-0.223]

90 Since we did not have access to a generalized Sylvester equation (13) solver, we had to use a Kronecker product approximation [6] which dramatically slowed down the Sylvester equation approach. [sent-259, score-0.215]

91 Table 1: Time to compute kernel matrix for given number of unlabeled graphs from various datasets. [sent-260, score-0.373]

92 Table 2: Time to compute kernel matrix for given number of labeled graphs from various datasets. [sent-284, score-0.369]

93 kernel dataset #graphs Direct Sylvester Conjugate Fixed Point delta MUTAG 100 230 7. [sent-285, score-0.15]

94 6 Outlook and Discussion We have shown that computing random walk graph kernels is essentially equivalent to solving a large linear system. [sent-299, score-0.582]

95 Even though the Sylvester equation method has a worst-case complexity of O(n3 ), the conjugate gradient and ﬁxed-point methods tend to be faster on all our datasets. [sent-303, score-0.165]

96 This is because computing matrix-vector products via Lemma 1 is quite efﬁcient when the graphs are sparse, so that the feature matrices Φ(L) and Φ(L ) contain only O(n) non- entries. [sent-304, score-0.257]

97 Matlab’s black-box dlyap solver is unable to exploit this sparsity; we are working on more capable alternatives. [sent-305, score-0.223]

98 We have shown that sparsity, low effective rank, and Kronecker product structure can be exploited to greatly reduce the computational cost of graph kernels; taking advantage of other forms of structure in W× remains a challenge. [sent-310, score-0.327]

99 Now that the computation of random walk graph kernels is viable for practical problem sizes, it will open the doors for their application in hitherto unexplored domains. [sent-311, score-0.547]

100 The algorithmic challenge now is how to integrate higher-order structures, such as spanning trees, in graph comparisons. [sent-312, score-0.186]

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With some abuse of notation, we deﬁne E as the set of line segments in 2 corresponding to the edges in the graph. The regions of 2 \ E are deﬁned as the faces of the graph. The face which corresponds to an unbounded region is called the external face. Given a planar graph G, its dual graph G∗ is deﬁned in the following way: the vertices of G∗ correspond to faces of G, and there is an edge between two vertices in G∗ iff the two corresponding faces in G share an edge. If the graph G is weighted, the weight on an edge in G∗ is the weight on the edge shared by the corresponding faces in G. A plane triangulation of a planar graph G is obtained from G by adding edges such that all the faces of the resulting graph have exactly three vertices. Thus a plane triangulated graph has a dual where all vertices have degree three. It can be shown that every plane graph can be plane triangulated [4]. We shall also need the notion of a perfect matching on a graph. A perfect matching on a graph G is deﬁned as a set of edges H ⊆ E such that every vertex in G has exactly one edge in H incident on it. If the graph is weighted, the weight of the matching is deﬁned as the product of the weights of the edges in the matching. Finally, we recall the deﬁnition of a marginal polytope of a graph [12]. Consider an MRF over a graph G where fij are given by Equation 2. Denote the probability of the event I(xi = xj ) under p(x) by τij . The marginal polytope of G, denoted by M(G), is deﬁned as the set of values τij that can be obtained under some assignment to the parameters θij . For a general graph G the polytope M(G) cannot be described using a polynomial number of inequalities. However, for planar graphs, it turns out that a set of O(n3 ) constraints, commonly referred to as triangle inequalities, sufﬁce to describe M(G) (see [3] page 434). The triangle inequalities are deﬁned by 1 TRI(n) = {τij : τij + τjk − τik ≤ 1, τij + τjk + τik ≥ 1, ∀i, j, k ∈ {1, . . . , n}} (3) Note that the above inequalities actually contain variables τij which do not correspond to edges in the original graph G. Thus the equality M(G) = TRI(n) should be understood as referring only to the values of τij that correspond to edges in the graph. Importantly, the values of τij for edges not in the graph need not be valid marginals for any MRF. In other words M(G) is a projection of TRI(n) on the set of edges of G. It is well known that the marginal polytope for trees is described via pairwise constraints. It is thus interesting that for planar graphs, it is triplets, rather than pairwise 1 The deﬁnition here is slightly different from that in [3], since here we refer to agreement probabilities, whereas [3] refers to disagreement probabilities. This polytope is also referred to as the cut polytope. constraints, that characterize the polytope. In this sense, planar graphs and trees may be viewed as a hierarchy of polytope complexity classes. It remains an interesting problem to characterize other structures in this hierarchy and their related inference algorithms. 2 Exact calculation of partition function using perfect matching The seminal works of Kasteleyn [7] and Fisher [5] have shown how one can calculate the partition function for a binary MRF over a planar graph with pure interaction potentials. We brieﬂy review Fisher’s construction, which we will use in what follows. Our interpretation of the method differs somewhat from that of Fisher, but we believe it is more straightforward. The key idea in calculating the partition function is to convert the summation over values of x to the problem of calculating the sum of weights of all perfect matchings in a graph constructed from G, as shown below. In this section, we consider weighted graphs (graphs with numbers assigned to their edges). For the graph G associated with the pairwise MRF, we assign weights wij = e2θij to the edges. The ﬁrst step in the construction is to plane triangulate the graph G. Let us call the resulting graph GT . We deﬁne an MRF on GT by assigning a parameter θij = 0 to the edges that have been added to G, and the corresponding weight wij = 1. Thus GT essentially describes the same distribution as G, and therefore has the same partition function. We can thus restrict our attention to calculating the partition function for the MRF on GT . As a ﬁrst step in calculating a partition function over GT , we introduce the following deﬁnition: a ˆ set of edges E in GT is an agreement edge set (or AES) if for every triangle face F in GT one of the ˆ ˆ following holds: The edges in F are all in E, or exactly one of the edges in F is in E. The weight ˆ is deﬁned as the product of the weights of the edges in E. ˆ of a set E It can be shown that there exists a bijection between pairs of assignments {x, −x} and agreement edge sets. The mapping from x to an edge set is simply the set of edges such that xi = xj . It is easy to see that this is an agreement edge set. The reverse mapping is obtained by ﬁnding an assignment x such that xi = xj iff the corresponding edge is in the agreement edge set. The existence of this mapping can be shown by induction on the number of (triangle) faces. P The contribution of a given assignment x to the partition function is e ˆ sponds to an AES denoted by E it is easy to see that P e ij∈E θij I(xi =xj ) = e− P ij∈E θij P e ˆ ij∈E 2θij = ce P ˆ ij∈E ij∈E 2θij θij I(xi =xj ) =c wij . If x corre(4) ˆ ij∈E P where c = e− ij∈E θij . Deﬁne the superset Λ as the set of agreement edge sets. The above then implies that Z(θ) = 2c E∈Λ ij∈E wij , and is thus proportional to the sum of AES weights. ˆ ˆ To sum over agreement edge sets, we use the following elegant trick introduced by Fisher [5]. Construct a new graph GPM from the dual of GT by introducing new vertices and edges according to the following rule: Replace each original vertex with three vertices that are connected to each other, and assign a weight of one to the new edges. Next, consider the three neighbors of the original vertex 2 . Connect each of the three new vertices to one of these three neighbors, keeping the original weights on these edges. The transformation is illustrated in Figure 1. The new graph GPM has O(3n) vertices, and is also planar. It can be seen that there is a one to one correspondence between perfect matchings in GPM and agreement edge sets in GT . Deﬁne Ω to be the set of perfect matchings in GPM . Then Z(θ) = 2c M ∈Ω ij∈M wij where we have used the fact that all the new weights have a value of one. Thus, the partition function is a sum over the weights of perfect matchings in GPM . Finally, we need a way of summing over the weights of the set of perfect matchings in a graph. Kasteleyn [7] proved that for a planar graph GPM , this sum may be obtained using the following sequence of steps: • Direct the edges of the graph GPM such that for every face (except possibly the external face), the number of edges on its perimeter oriented in a clockwise manner is odd. Kasteleyn showed that such a so called Pfafﬁan orientation may be constructed in polynomial time for a planar graph (see also [8] page 322). 2 Note that in the dual of GT all vertices have degree three, since GT is plane triangulated. 1.2 0.7 0.6 1 1 1 0.8 0.6 0.8 1.5 1.4 1.5 1 1 1.2 1 1 1 1 0.7 1.4 1 1 1 Figure 1: Illustration of the graph transformations in Section 2 for a complete graph with four vertices. Left panel shows the original weighted graph (dotted edges and grey vertices) and its dual (solid edges and black vertices). Right panel shows the dual graph with each vertex replaced by a triangle (the graph GPM in the text). Weights for dual graph edges correspond to the weights on the original graph. • Deﬁne the matrix P (GPM ) to be a skew symmetric matrix such that Pij = 0 if ij is not an edge, Pij = wij if the arrow on edge ij runs from i to j and Pij = −wij otherwise. • The sum over weighted matchings can then be shown to equal |P (GPM )|. The partition function is thus given by Z(θ) = 2c |P (GPM )|. To conclude this section we reiterate the following two key points: the partition function of a binary MRF over a planar graph with interaction potentials as in Equation 2 may be calculated in polynomial time by calculating the determinant of a matrix of size O(3n). An important outcome of this result is that the functional relation between Z(θ) and the parameters θij is known, a fact we shall use in what follows. 3 Partition function bounds via planar decomposition Given a non-planar graph G over binary variables with a vector of interaction potentials θ, we wish to use the exact planar computation to obtain a bound on the partition function of the MRF on G. We assume for simplicity that the potentials on the MRF for G are given in the form of Equation 2. Thus, G violates the assumptions of the previous section only in its non-planarity. Deﬁne G(r) as a set of spanning planar subgraphs of G, i.e., each graph G(r) is planar and contains all the vertices of G and some its edges. Denote by m the number of such graphs. Introduce the following deﬁnitions: (r) • θ (r) is a set of parameters on the edges of G(r) , and θij is an element in this set. Z(θ (r) ) is the partition function of the MRF on G(r) with parameters θ (r) . ˆ (r) ˆ(r) • θ is a set of parameters on the edges of G such that if edge (ij) is in G(r) then θij = (r) ˆ(r) θ , and otherwise θ = 0. ij ij Given a distribution ρ(r) on the graphs G(r) (i.e., ρ(r) ≥ 0 for r = 1, . . . , m and assume that the parameters for G(r) are such that ˆ ρ(r)θ θ= (r) r ρ(r) = 1), (5) r Then, by the convexity of the log partition function, as a function of the model parameters, we have ρ(r) log Z(θ (r) ) ≡ f (θ, ρ, θ (r) ) log Z(θ) ≤ (6) r Since by assumption the graphs G(r) are planar, this bound can be calculated in polynomial time. Since this bound is true for any set of parameters θ (r) which satisﬁes the condition in Equation 5 and for any distribution ρ(r), we may optimize over these two variables to obtain the tightest bound possible. Deﬁne the optimal bound for a ﬁxed value of ρ(r) by g(ρ, θ) (optimization is w.r.t. θ (r) ) g(ρ, θ) = f (θ, ρ, θ (r) ) min θ (r) : P ˆ ρ(r)θ (r) =θ (7) Also, deﬁne the optimum of the above w.r.t. ρ by h(θ). h(θ) = min g(θ, ρ) ρ(r) ≥ 0, ρ(r) = 1 (8) Thus, h(θ) is the optimal upper bound for the given parameter vector θ. In the following section we argue that we can in fact ﬁnd the global optimum of the above problem. 4 Globally Optimal Bound Optimization First consider calculating g(ρ, θ) from Equation 7. Note that since log Z(θ (r) ) is a convex function of θ (r) , and the constraints are linear, the overall optimization is convex and can be solved efﬁciently. In the current implementation, we use a projected gradient algorithm [2]. The gradient of f (θ, ρ, θ (r) ) w.r.t. θ (r) is given by ∂f (θ, ρ, θ (r) ) (r) ∂θij (r) = ρ(r) 1 + eθij (r) P −1 (GPM ) (r) k(i,j) Sign(Pk(i,j) (GPM )) (9) where k(i, j) returns the row and column indices of the element in the upper triangular matrix of (r) (r) P (GPM ), which contains the element e2θij . Since the optimization in Equation 7 is convex, it has an equivalent convex dual. Although we do not use this dual for optimization (because of the difﬁculty of expressing the entropy of planar models solely in terms of triplet marginals), it nevertheless allows some insight into the structure of the problem. The dual in this case is closely linked to the notion of the marginal polytope deﬁned in Section 1. Using a derivation similar to [11], we arrive at the following characterization of the dual g(ρ, θ) = max τ ∈TRI(n) ρ(r)H(θ (r) (τ )) θ·τ + (10) r where θ (r) (τ ) denotes the parameters of an MRF on G(r) such that its marginals are given by the restriction of τ to the edges of G(r) , and H(θ (r) (τ )) denotes the entropy of the MRF over G(r) with parameters θ (r) (τ ). The maximized function in Equation 10 is linear in ρ and thus g(ρ, θ) is a pointwise maximum over (linear) convex functions in ρ and is thus convex in ρ. It therefore has no (r) local minima. Denote by θmin (ρ) the set of parameters that minimizes Equation 7 for a given value of ρ. Using a derivation similar to that in [11], the gradient of g(ρ, θ) can be shown to be ∂g(ρ, θ) (r) = H(θmin (ρ)) ∂ρ(r) (11) Since the partition function for G(r) can be calculated efﬁciently, so can the entropy. We can now summarize the algorithm for calculating h(θ) • Initialize ρ0 . Iterate: – For ρt , ﬁnd θ (r) which solves the minimization in Equation 7. – Calculate the gradient of g(ρ, θ) at ρt using the expression in Equation 11 – Update ρt+1 = ρt + αv where v is a feasible search direction calculated from the gradient of g(ρ, θ) and the simplex constraints on ρ. The step size α is calculated via an Armijo line search. – Halt when the change in g(ρ, θ) is smaller than some threshold. Note that the minimization w.r.t. θ (r) is not very time consuming since we can initialize it with the minimum from the previous step, and thus only a few iterations are needed to ﬁnd the new optimum, provided the change in ρ is not too big. The above algorithm is guaranteed to converge to a global optimum of ρ [2], and thus we obtain the tightest possible upper bound on Z(θ) given our planar graph decomposition. The procedure described here is asymmetric w.r.t. ρ and θ (r) . In a symmetric formulation the minimizing gradient steps could be carried out jointly or in an alternating sequence. The symmetric ˆ (r) formulation can be obtained by decoupling ρ and θ (r) in the bi-linear constraint ρ(r)θ = θ. Field Figure 2: Illustration of planar subgraph construction for a rectangular lattice with external ﬁeld. Original graph is shown on the left. The ﬁeld vertex is connected to all vertices (edges not shown). The graph on the right results from isolating the 4th ,5th columns of the original graph (shown in grey), and connecting the ﬁeld vertex to the external vertices of the three disconnected components. Note that the resulting graph is planar. ˜ ˜ Speciﬁcally, we introduce θ (r) = θ (r) ρ(r) and perform the optimization w.r.t. ρ and θ (r) . It can be ˜(r) ) with the relevant (de-coupled) constraint is equivalent shown that a stationary point of f (θ, ρ, θ to the procedure described above. The advantage of this approach is that the exact minimization w.r.t θ (r) is not required before modifying ρ. Our experiments have shown, however, that the methods take comparable times to converge, although this may be a property of the implementation. 5 Estimating Marginals The optimization problem as deﬁned above minimizes an upper bound on the partition function. However, it may also be of interest to obtain estimates of the marginals of the MRF over G. To obtain marginal estimates, we follow the approach in [11]. We ﬁrst characterize the optimum of Equation 7 for a ﬁxed value of ρ. Deriving the Lagrangian of Equation 7 w.r.t. θ (r) we obtain the (r) following characterization of θmin (ρ): Marginal Optimality Criterion: For any two graphs G(r) , G(s) such that the edge (ij) is in both (r) (s) graphs, the optimal parameter vector satisﬁes τij (θmin (ρ)) = τij (θmin (ρ)). Thus, the optimal set of parameters for the graphs G(r) is such that every two graphs agree on the marginals of all the edges they share. This implies that at the optimum, there is a well deﬁned set of marginals over all the edges. We use this set as an approximation to the true marginals. A different method for estimating marginals uses the partition function bound directly. We ﬁrst P calculate partition function bounds on the sums: αi (1) = x:xi =1 e ij∈E fij (xi ,xj ) and αi (−1) = P αi (1) e ij∈E fij (xi ,xj ) and then normalize αi (1)+αi (−1) to obtain an estimate for p(xi = 1). This method has the advantage of being more numerically stable (since it does not depend on derivatives of log Z). However, it needs to be calculated separately for each variable, so that it may be time consuming if one is interested in marginals for a large set of variables. x:xi =−1 6 Experimental Evaluation We study the application of our Planar Decomposition (PDC) P method to a binary MRF on a square P lattice with an external ﬁeld. The MRF is given by p(x) ∝ e ij∈E θij xi xj + i∈V θi xi where V are the lattice vertices, and θi and θij are parameters. Note that this interaction does not satisfy the conditions for exact calculation of the partition function, even though the graph is planar. This problem is in fact NP hard [1]. However, it is possible to obtain the desired interaction form by introducing an additional variable xn+1 that is connected to all the original variables.P Denote the correspondP ij∈E θij xi xj + i∈V θi,n+1 xi xn+1 , where ing graph by Gf . Consider the distribution p(x, xn+1 ) ∝ e θi,n+1 = θi . It is easy to see that any property of p(x) (e.g., partition function, marginals) may be calculated from the corresponding property of p(x, xn+1 ). The advantage of the latter distribution is that it has the desired interaction form. We can thus apply PDC by choosing planar subgraphs of the non-planar graph Gf . 0.25 0.15 0.1 0.05 0.5 1 1.5 Interaction Strength 0.03 Singleton Marginal Error Z Bound Error Pairwise Marginals Error 0.08 PDC TRW 0.2 0.07 0.06 0.05 0.04 0.03 0.02 2 0.5 1 1.5 Interaction Strength 0.025 0.02 0.015 0.01 0.005 2 0.5 1 1.5 Interaction Strength 2 !3 x 10 0.025 0.02 0.015 0.5 1 Field Strength 1.5 2 Singleton Marginal Error Pairwise Marginals Error Z Bound Error 0.03 0.03 0.025 0.02 0.015 0.5 1 Field Strength 1.5 2 9 8 7 6 5 4 3 0.5 1 Field Strength 1.5 2 Figure 3: Comparison of the TRW and Planar Decomposition (PDC) algorithms on a 7×7 square lattice. TRW results shown in red squares, and PDC in blue circles. Left column shows the error in the log partition bound. Middle column is the mean error for pairwise marginals, and right column is the error for the singleton marginal of the variable at the lattice center. Results in upper row are for ﬁeld parameters drawn from U[−0.05, 0.05] and various interaction parameters. Results in the lower row are for interaction parameters drawn from U [−0.5, 0.5] and various ﬁeld parameters. Error bars are standard errors calculated from 40 random trials. There are clearly many ways to choose spanning planar subgraphs of Gf . Spanning subtrees are one option, and were used in [11]. Since our optimization is polynomial in the number of subgraphs, √ we preferred to use a number of subgraphs that is linear in n. The key idea in generating these planar subgraphs is to generate disconnected components of the lattice and connect xn+1 only to the external vertices of these components. Here we generate three disconnected components by isolating two neighboring columns (or rows) from the rest of the graph, resulting in three components. This is √ illustrated in Figure 2. To this set of 2 n graphs, we add the independent variables graph consisting only of edges from the ﬁeld node to all the other nodes. We compared the performance of the PDC and TRW methods 3 4 on a 7 × 7 lattice . Since the exact partition function and marginals can be calculated for this case, we could compare both algorithms to the true values. The MRF parameters were set according to the two following scenarios: 1) Varying Interaction - The ﬁeld parameters θi were drawn uniformly from U[−0.05, 0.05], and the interaction θij from U[−α, α] where α ∈ {0.2, 0.4, . . . , 2}. This is the setting tested in [11]. 2) Varying Field θi was drawn uniformly from U[−α, α], where α ∈ {0.2, 0.4, . . . , 2} and θij from U[−0.5, 0.5]. For each scenario, we calculated the following measures: 1) Normalized log partition error 1 1 alg − log Z true ). 2) Error in pairwise marginals |E| ij∈E |palg (xi = 1, xj = 1) − 49 (log Z ptrue (xi = 1, xj = 1)|. Pairwise marginals were calculated jointly using the marginal optimality criterion of Section 5. 3) Error in singleton marginals. We calculated the singleton marginals for the innermost node in the lattice (i.e., coordinate [3, 3]), which intuitively should be the most difﬁcult for the planar based algorithm. This marginal was calculated using two partition functions, as explained in Section 5 5 . The same method was used for TRW. The reported error measure is |palg (xi = 1) − ptrue (xi = 1)|. Results were averaged over 40 random trials. Results for the two scenarios and different evaluation measures are given in Figure 3. It can be seen that the partition function bound for PDC is signiﬁcantly better than TRW for almost all parameter settings, although the difference becomes smaller for large ﬁeld values. Error for the PDC pairwise 3 TRW and PDC bounds were optimized over both the subgraph parameters and the mixture parameters ρ. In terms of running time, PDC optimization for a ﬁxed value of ρ took about 30 seconds, which is still slower than the TRW message passing implementation. 5 Results using the marginal optimality criterion were worse for PDC, possibly due to its reduced numerical precision. 4 marginals are smaller than those of TRW for all parameter settings. For the singleton parameters, TRW slightly outperforms PDC. This is not surprising since the ﬁeld is modeled by every spanning tree in the TRW decomposition, whereas in PDC not all the structures model a given ﬁeld. 7 Discussion We have presented a method for using planar graphs as the basis for approximating non-planar graphs such as planar graphs with external ﬁelds. While the restriction to binary variables limits the applicability of our approach, it remains relevant in many important applications, such as coding theory and combinatorial optimization. Moreover, it is always possible to convert a non-binary graphical model to a binary one by introducing additional variables. The resulting graph will typically not be planar, even when the original graph over k−ary variables is. However, the planar decomposition method can then be applied to this non-planar graph. The optimization of the decomposition is carried out explicitly over the planar subgraphs, thus limiting the number of subgraphs that can be used in the approximation. In the TRW method this problem is circumvented since it is possible to implicitly optimize over all spanning trees. The reason this can be done for trees is that the entropy of an MRF over a tree may be written as a function of its marginal variables. We do not know of an equivalent result for planar graphs, and it remains a challenge to ﬁnd one. It is however possible to combine the planar and tree decompositions into one single bound, which is guaranteed to outperform the tree or planar approximations alone. The planar decomposition idea may in principle be applied to bounding the value of the MAP assignment. However, as in TRW, it can be shown that the solution is not dependent on the decomposition (as long as each edge appears in some structure), and the problem is equivalent to maximizing a linear function over the marginal polytope (which can be done in polynomial time for planar graphs). However, such a decomposition may suggest new message passing algorithms, as in [10]. Acknowledgments The authors acknowledge support from the Defense Advanced Research Projects Agency (Transfer Learning program). Amir Globerson is also supported by the Rothschild Yad-Hanadiv fellowship. The authors also wish to thank Martin Wainwright for providing his TRW code. References [1] F. Barahona. On the computational complexity of ising spin glass models. J. Phys. A., 15(10):3241–3253, 1982. [2] D. P. Bertsekas, editor. Nonlinear Programming. Athena Scientiﬁc, Belmont, MA, 1995. [3] M.M. Deza and M. Laurent. Geometry of Cuts and Metrics. Springe-Verlag, 1997. [4] R. Diestel. Graph Theory. Springer-Verlag, 1997. [5] M.E. Fisher. On the dimer solution of planar ising models. J. Math. Phys., 7:1776–1781, 1966. [6] M.I. Jordan, editor. Learning in graphical models. MIT press, Cambridge, MA, 1998. [7] P.W. Kasteleyn. Dimer statistics and phase transitions. Journal of Math. Physics, 4:287–293, 1963. [8] L. Lovasz and M.D. Plummer. Matching Theory, volume 29 of Annals of discrete mathematics. NorthHolland, New-York, 1986. [9] M. J. Wainwright, T. Jaakkola, and A. S. Willsky. Tree-based reparameterization framework for analysis of sum-product and related algorithms. IEEE Trans. on Information Theory, 49(5):1120–1146, 2003. [10] M. J. Wainwright, T. Jaakkola, and A. S. Willsky. Map estimation via agreement on trees: messagepassing and linear programming. IEEE Trans. on Information Theory, 51(11):1120–1146, 2005. [11] M. J. Wainwright, T. Jaakkola, and A. S. Willsky. A new class of upper bounds on the log partition function. IEEE Trans. on Information Theory, 51(7):2313–2335, 2005. [12] M.J. Wainwright and M.I. Jordan. Graphical models, exponential families, and variational inference. Technical report, UC Berkeley Dept. of Statistics, 2003. [13] J.S. Yedidia, W.T. W.T. Freeman, and Y. Weiss. Constructing free-energy approximations and generalized belief propagation algorithms. IEEE Trans. on Information Theory, 51(7):2282–2312, 2005.

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