Fast Jensen-Shannon Graph Kernel Bai Lu and Edwin Hancock Department of Computer Science University of York Supported by a Royal Society Wolfson Research Merit Award Structural Variations Protein-Protein Interaction Networks Manipulating graphs Is structure similar (graph isomorphism, inexact match)? Is complexity similar (are graphs from same class but different in detail)? Is complexity (type of structure) uniform? Goals Can we capture determine the similarity of structure using measures that capture their intrinsic complexity. Can graph entropies be used for this purpose. If they can then they lead naturally to information theoretic kernels and description length for learning over graph data. Outline Literature Review: State of the Art Graph Kernels Existing graph kernel methods：Graph kernels based on a) walks, b) paths or c) subgraph or subtree structures. Prior Work: Recently we have developed on information theoretic graph kernel based on Jensen-Shannon divergence probability distributions on graphs. Fast Jensen-Shannon Graph Kernel: Based on depth depth-based subgraph representation of a graph Based around graph centroid Experiments Conclusion Literature Review: Graph Kernels Existing Graph Kernels (i.e Graph Kernels from the Rconvolution [Haussler, 1999]) fall into three classes: Restricted subgraph or subtree kernels Weisfeiler-Lehman subtree kernel [Shevashidze et al., 2009, NIPS] Random walk kernels Product graph kernels [Gartner et al., 2003, ICML] Marginalized kernels on graphs [Kashima et al., 2003, ICML] Path based kernels Shortest path kernel [Borgwardt, 2005, ICDM] Motivation Limitations of existing graph kernel Can not scale up to substructures of large size (e.g. (sub)graphs with hundreds or even thousands vertices). Compromised to substructures of limited size and only roughly capture topological arrangement within a graph. Even for relatively small subgraphs, most graph kernels still require significant computational overheads. Aim: develop a novel subgraph kernel for efficient computation, even when a pair of fully sized subgraphs are compared. Approach Investigate how to kernelize depth-based graph representations by similarity for K-layer subgraphs using the Jensen-Shannon divergence. Commence by showing how to compute a fast JensenShannon diffusion kernel for a pair of (sub)graphs. Describe how to compute a fast depth-based graph representation., based on complexity of structure. Combine ideas to compute fast Jensen-Shannon subgraph kernel. Notation Notation Consider a graph , adjacency matrix has elements The vertex degree matrix of is given by Normalaised Laplacian and its spectrum ˆ ˆ ˆT Lˆ D 1/ 2 ( D A) D 1/ 2 The Jensen-Shannon Diffusion Kernel Jensen-Shannon diffusion kernel for graphs: For graphs Gp and Gq, the Jensen-Shannon divergence is where is entropy of composite structure formed from two (sub)graphs being compared (here we use the disjoint union). The Jensen-Shannon diffusion kernel for Gp and Gq is where entropy H(·) is either Shannon or the von Neumann. Composite Structure Composite entropy of disjoint union A disjoint union of a pair of graph of graphs Gp and Gq is Graphs Gp and Gq are the connected components of the disjoint union graph GDU. Let p = |V p|/|V Entropy (i.e. the composite entropy) of GDU is DU | and q = |V q|/|VDU|. Graph Entropy: Measures of complexity Shannon entropy of random walk : The probability of a steady state random walk on visiting vertex vi is . Shannon entropy of steady state random walk is von Neumann entropy: entropy associated with normalised Laplacian eigenvalues. |V | ˆ i ˆi HV ln 2 i 1 2 Approximated by (Han PRL12) Properties The Jensen-Shannon diffusion kernel for graphs: The Jensen-Shannon diffusion kernel is positive definite (pd). This follows the definitions in [Kondor and Lafferty, 2002, ICML], if a dissimilarity measure between a pair of graphs Gp and Gq satisfies symmetry, then a diffusion kernel associated with the similarity measure is pd. Time Complexity: For a pair of graphs Gp and Gq both having n vertices, computing the Jensen-Shannon diffusion kernel requires time complexity O(n^2). Idea Decompose graph into layered subgraphs from centroid. Use JSD to compare subgraphs. Construct kernel over subgraphs. The Depth-Based Representation of A Graph Subgraphs from the Centroid Vertex For graph G(V,E), construct shortest path matrix matrix SG whose element SG(i, j) are the shortest path lengths between vertices vi and vj . Average-shortest-path vector SV for G(V,E) is a vector with element from vertex vi to the remaining vertices. Centroid vertex for G(V,E) as The K-layer centroid expansion subgraph where Depth-Based Representation For a graph G, we obtain a family of centroid expansion subgraphs , the depthbased representation of G is defined as where H(·) is either the Shannon entropy or the von Neumann entropy. Measures complexity via variation of entropy with depth The Depth-Based Representation An example of the depth-based representation for a graph from the centroid vertex Fast Jensen-Shannon Subgraph Kernel For a pair of graphs Gp(Vp, Ep) and Gq(Vq, Eq), similarity measure is is summed over an entropy-based similarity measure for the K-layer subgraphs. Jensen-Shannon diffusion kernel is the sum of the diffusion kernel measures for all the pairs of K-layer subgraphs Jensen-Shannon subgraph kernel is pd. Because, the proposed subgraph kernel is the sum of the positive Jensen-Shannon diffusion kernel. Times Complexity Subgraph kernel graphs for graphs with n vertices and m edges, has time complexity O(n^2L + mn), where L is the size of the largest layer of the expansion subgraph. Depth–based representation is O(n^2L+mn). Jensen-Shannon diffusion kernel is O(n^2). Observations Advantages a) von Neumann entropy is associated with the degree variance of connected vertices. Subgraph kernel is sensitive to interconnections between vertex clusters. b) For Shannon entropy vertices with large degrees dominate the entropy. Subgraph kernel is suited to characterizing a group of highly interconnected vertices, i.e. a dominant cluster. c) The depth-based representation captures inhomogeneities of complexity with depth. Enables it go gauge structure more finely than straightforwardly applying Jensen-Shannon diffusion kernel to original graphs. d) The proposed subgraph kernel only compares the pairs of subgraphs with the same layer size K. Avoids enumerating all the pairs of subgraphs and renders an efficient computation. e) Overcomes the subgraph size restriction which arises in existing graph kernels. Experiments (New, not in the paper) We evaluate the classification performance of our kernel using 10-fold cross validation associated with C-Support Vector Machine. (Intel i5 3210M 2.5GHz) Classification of graphs abstracted from bioinformatics and computer vision databases. This datasets include: GatorBait (3D shapes), DD, COIL5 (images), CATH1, CATH2. Graph kernels for comparisons include: a) our kernel: 1) using the Shannon entropy (JSSS) 2) using the von Neumann entropy (JSSV) b) Weisfeiler-Lehman subtree kernel (WL), c) the shortest path graph kernel (SPGK), d) the graphlet count kernel (GCGK) Experiments Details of the datasets Experiments Classification Timing Conclusion and Further Work Conclusion Presented a fast version of our Jensen-Shannon kernel. Compares well to alternatives on standard ML datasets. Further Work Hypergraphs, alternative entropies and divergences.
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