Faster Kernels for Graphs with Continuous Attributes via Hashing

Abstract

Abstract-While state-of-the-art kernels for graphs with discrete labels scale well to graphs with thousands of nodes, the few existing kernels for graphs with continuous attributes, unfortunately, do not scale well. To overcome this limitation, we present hash graph kernels, a general framework to derive kernels for graphs with continuous attributes from discrete ones. The idea is to iteratively turn continuous attributes into discrete labels using randomized hash functions. We illustrate hash graph kernels for the Weisfeiler-Lehman subtree kernel and for the shortest-path kernel. The resulting novel graph kernels are shown to be, both, able to handle graphs with continuous attributes and scalable to large graphs and data sets. This is supported by our theoretical analysis and demonstrated by an extensive experimental evaluation.

A survey on graph kernels

“…Shortest-Path [24] IM + † + † Generalized Shortest-Path [25] IM + + † Graphlet [26] EX --Cycles and Trees [27] EX + ⋆ -Tree Pattern Kernel [28,29] IM + + ⋆ Ordered Directed Acyclic Graphs [30,31] EX + -GraphHopper [32] IM + † + Graph Invariant [33] IM + + Subgraph Matching [34] IM + + Weisfeiler-Lehman Subtree [35] EX + -Weisfeiler-Lehman Edge [35] EX + -Weisfeiler-Lehman Shortest-Path [35] EX + k-dim. Local Weisfeiler-Lehman Subtree [36] EX + -Neighborhood Hash Kernel [37] EX + -Propagation Kernel [38] EX + + Neighborhood Subgraph Pairwise Distance Kernel [39] EX + -Random Walk [22,23,40,3,41,42] IM + + Optimal Assignment Kernel [43] IM + + Weisfeiler-Lehman Optimal Assignment [44] IM + -Pyramid Match [45] IM + -Matchings of Geometric Embeddings [46] IM + + ⋆ Descriptor Matching Kernel [47] IM + + † Graphlet Spectrum [48] EX + -Multiscale Laplacian Graph Kernel [49] IM + + ⋆ † Global Graph Kernel [50] EX --Deep Graph Kernels [19] IM + -Smoothed Graph Kernels [51] IM + ⋆ -Hash Graph Kernel [52] EX + + Depth-based Representation Kernel [53] IM --Aligned Subtree Kernel [54] IM + -…”

Tree++: Truncated Tree Based Graph Kernels

“…The discriptor matching (DM) kernel [39] maps every graph into a set of vectors (descriptors) which integrate both the attribute and neighborhood information of vertices, and then uses a set-of-vector matching kernel [10] to measure graph similarity. HGK [23] is a general framework to extend graph kernels from discrete attributes to continuous attributes. The main idea is to iteratively map continuous attributes to discrete labels by randomized hash functions.…”

A unifying view of explicit and implicit feature maps of graph kernels

“…Kernels limited to graphs with categorical labels can be applied to attributed graphs by discretization of the continuous attributes, see, e.g., Neumann et al (2016). Morris et al (2016) proposed the hash graph kernel framework to obtain efficient kernels for graphs with continuous labels from those proposed for discrete ones. The idea is to iteratively turn continuous attributes into discrete labels using randomized hash functions.…”

“…When k V and k E both are Dirac kernels, each component of the feature vector corresponds to a triple consisting of two vertex labels and a path length. This method of computation has been applied in several experimental comparisons, e.g., Kriege and Mutzel (2012) and Morris et al (2016). This feature map is directly obtained from our results in Sect.…”

“…Synthetic graphs The data sets SyntheticNew and Synthie were synthetically generated to obtain classification benchmarks for graph kernels with attributes. We refer the reader to the publications Feragen et al (2013) 6 and Morris et al (2016), respectively, for the details of the generation process. Additionally, we generated new synthetic graphs in order to systematically vary graph properties of interest like the label diversity, which we expect to have an effect on the running time according to our theoretical analysis.…”