Local-global nested graph kernels using nested complexity traces

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Highlights

  • We propose a family of local-global nested graph kernels based on nested representations.

  • The nested aligned kernel is based on the dynamic time warping inspired measures.

  • The nested reproducing kernel is based on the basic H1-reproducing kernel.

  • Both the nested kernels simultaneously reflect the local and global graph characteristics.

  • Experiments demonstrate the effectiveness and efficiency of the proposed graph kernels.

Abstract

In this paper, we propose two novel local-global nested graph kernels, namely the nested aligned kernel and the nested reproducing kernel, drawing on depth-based complexity traces. Both of the nested kernels gauge the nested depth complexity trace through a family of K-layer expansion subgraphs rooted at the centroid vertex, i.e., the vertex with minimum shortest path length variance to the remaining vertices. Specifically, for a pair of graphs, we commence by computing the centroid depth-based complexity traces rooted at the centroid vertices. The first nested kernel is defined by measuring the global alignment kernel, which is based on the dynamic time warping framework, between the complexity traces. Since the required global alignment kernel incorporates the whole spectrum of alignment costs between the complexity traces, this nested kernel can provide rich statistic measures. The second nested kernel, on the other hand, is defined by measuring the basic reproducing kernel between the complexity traces. Since the associated reproducing kernel only requires time complexity O(1), this nested kernel has very low computational complexity. We theoretically show that both of the proposed nested kernels can simultaneously reflect the local and global graph characteristics in terms of the nested complexity traces. Experiments on standard graph datasets abstracted from bioinformatics and computer vision databases demonstrate the effectiveness and efficiency of the proposed graph kernels.

Introduction

In pattern recognition, graph kernels are powerful tools for analyzing structured data represented by graphs [25]. This is because graph kernels not only preserve structural information by implicitly mapping graphs to a high dimensional Hilbert space, but also provide a way of directly applying standard kernel methods for vectorial data (e.g., Support Vector Machines, kernel Principle Component Analysis) to graph structures.

The idea underpinning most existing graph kernels is that of decomposing graphs into substructures and comparing pairs of specific isomorphic substructures, e.g., walks [28], paths [1], and restricted subgraph or subtree substructures [13]. Under this scenario, [3] have developed a family of kernels for comparing point clouds. These kernels are based on a local tree-walk kernel between subtrees, which is defined by a factorization on suitably defined graphical models of the subtrees. Wang and Sahbi [29], on the other hand, have defined a graph kernel for action recognition. They first describe actions in the videos using directed acyclic graphs (DAGs). The resulting kernel is defined as an extending random walk kernel by counting the number of isomorphic walks of DAGs. Harchaoui and Bach [19] have proposed a segmentation graph kernel for images by counting the inexact isomorphic subtree patterns between image segmentation graphs. Other state-of-the-art graph kernels based on substructures include the aligned subtree kernel proposed by Bai et al. [9], the subgraph matching kernel proposed by Kriege and Mutzel [22], the fast depth-based subgraph kernel proposed by Bai and Hancock [6], the optimal assignment kernel proposed by Kriege et al. [23], and the random walk kernel proposed by Kashima et al. [21].

Unfortunately, all the aforementioned graph kernels tend to only capture local characteristics of graphs, since they usually use substructures of limited sizes. As a result, these kernels may fail to reflect global graph characteristics. To overcome this shortcoming, a number of graph kernels based on using the adjacency matrix to capture global graph characteristics have been developed by Johansson et al. [20], Xu et al. [31] and Bai and Hancock [4]. For instance, Johansson et al. [20] have developed a family of global graph kernels based on the Lovász number and its associated orthonormal representation through the adjacency matrix. Xu et al. [31] have proposed a local-global mixed reproducing kernel based on the approximate von Neumann entropy through the adjacency matrix. Bai and Hancock [4] have defined an information theoretic kernel based on the classical Jensen–Shannon divergence between the steady state random walk probability distributions obtained through the adjacency matrix. Recently, there has been increasing interests in continuous-time quantum walks for the analysis of global graph structures [16]. The continuous-time quantum walk is the quantum analogue of the classical continuous-time random walk. Unlike the classical random walk that is governed by a doubly stochastic matrix, the quantum walk is governed by an unitary matrix and is not dominated by the low frequencies of the Laplacian spectrum. Thus, the continuous-time quantum walk is able to better discriminate different graph structures.

There have been a number of graph kernels developed using the continuous-time quantum walk. For instance, Bai et al. [8] have developed a quantum kernel by measuring the similarity between two continuous-time quantum walks evolving on a pair of graphs. Specifically, they associate each graph with a mixed quantum state that represents the evolution of the quantum walk. The resulting kernel is computed by measuring the quantum Jensen–Shannon divergence between the associated density matrices. Rossi et al. [26] have developed a quantum kernel by exploiting the relation between the continuous-time quantum walk interferences and the symmetries of a pair of graphs, in terms of the quantum Jensen–Shannon divergence. Both of these quantum kernels employ the Laplacian matrix as the required Hamiltonian operator, and thus can naturally reflect global graph characteristics.

The aim of this work is to overcome the gap between local kernels (i.e., kernels based on local substructures of limited sizes) and the global kernels (i.e., global kernels based on either the adjacency matrix or the continuous-time quantum walk). To this end, we propose two novel local-global nested graph kernels, namely the nested aligned kernel and the nested reproducing kernel, drawing on depth-based complexity traces [6]. Both of the nested kernels gauge the nested depth complexity trace through a family of K-layer expansion subgraphs rooted at the centroid vertex, that has minimum shortest path length variance to the remaining vertices. Specifically, for a pair of graphs, we commence by computing the centroid depth-based complexity traces rooted at the centroid vertices. The first nested kernel is defined by measuring the global alignment kernel, which is developed through the dynamic time warping framework, between the complexity traces [14]. Since the required global alignment kernel incorporates the whole spectrum of alignment costs between the complexity traces, this nested kernel can provide rich statistic measures. The second nested kernel, on the other hand, is defined by measuring the reproducing kernel between the complexity traces [31], [32]. Since the associated reproducing kernel only requires time complexity O(1), this nested kernel has efficient computational complexity. We theoretically show that both of the proposed nested kernels can simultaneously reflect the local and global graph characteristics in terms of the nested complexity traces. Experiments on standard graph datasets abstracted from bioinformatics and computer vision databases demonstrate the effectiveness and efficiency of the proposed graph kernels.

The remainder of this paper is organized as follows. Section 2 reviews the preliminary concepts that will be used in this work. Specifically, we introduce the global alignment kernel through the dynamic time warping framework, the reproducing kernel, the approximate von Neumann entropy, the Shannon entropy associated with steady state random walks, and the centroid depth-based complexity trace. Section 3 defines the proposed local-global nested graph kernels. Section 4 provides the experimental evaluation. Section 5 concludes this work.

Section snippets

Preliminary concepts

In this section, we review some preliminary concepts that will be used in this work. We commence by reviewing the dynamic time warping framework. Specifically, we introduce the global alignment kernel based on this framework [14]. Moreover, we review a reproducing kernel that is an extension of the H1-reproducing kernel to the graph kernel realm. Finally, we review the concept of the depth-based complexity trace that naturally forms a nested sequence of a graph in terms of the entropy measure.

The local-global nested graph kernel

In this section, we introduce two novel local-global nested graph kernels, namely the nested aligned kernel and the nested reproducing kernel, that can reflect both the local and global graph characteristics through the centroid depth-based representations. Specifically, the first nested graph kernel is based on the dynamic time warping measure between the centroid depth-based complexity traces. On the other hand, the second nested graph kernel is based on the basic reproducing kernel between

Experimental evaluations

In this section, we evaluate the performance of the proposed local-global nested graph kernels. We commence the by exhibiting the nest property of the centroid depth-based complexity traces. Finally, we perform the proposed kernels on graph classification tasks.

Conclusion

In this paper, we have proposed two novel local-global nested graph kernels, namely the nested aligned kernel and the nested reproducing kernel. Both of the nested kernels are based on the centroid depth-based complexity traces, that gauge the nested depth complexity trace through a family of K-layer expansion subgraphs rooted at the centroid vertex. Unlike most existing state-of-the-art graph kernels that only probe local or global graph characteristics, the proposed nested kernels

Conflict of interest

None.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (grant nos. 61503422, 61602535, 61773415, 61370123 and 61772057), the Open Projects Program of National Laboratory of Pattern Recognition, the program for innovation research in Central University of Finance and Economics, and the Beijing Natural Science Foundation project (no. 4162037).

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