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A tree-based kernel for graphs with continuous attributes

Giovanni Da San Martino, Nicolò Navarin, Alessandro Sperduti

TL;DR

This work proposes a graph kernel for complex and continuous nodes’ attributes, whose features are tree structures extracted from specific graph visits, and presents an approximated variant of the kernel, which reduces its complexity significantly.

Abstract

The availability of graph data with node attributes that can be either discrete or real-valued is constantly increasing. While existing kernel methods are effective techniques for dealing with graphs having discrete node labels, their adaptation to non-discrete or continuous node attributes has been limited, mainly for computational issues. Recently, a few kernels especially tailored for this domain, and that trade predictive performance for computational efficiency, have been proposed. In this paper, we propose a graph kernel for complex and continuous nodes' attributes, whose features are tree structures extracted from specific graph visits. The kernel manages to keep the same complexity of state-of-the-art kernels while implicitly using a larger feature space. We further present an approximated variant of the kernel which reduces its complexity significantly. Experimental results obtained on six real-world datasets show that the kernel is the best performing one on most of them. Moreover, in most cases the approximated version reaches comparable performances to current state-of-the-art kernels in terms of classification accuracy while greatly shortening the running times.

A tree-based kernel for graphs with continuous attributes

TL;DR

This work proposes a graph kernel for complex and continuous nodes’ attributes, whose features are tree structures extracted from specific graph visits, and presents an approximated variant of the kernel, which reduces its complexity significantly.

Abstract

The availability of graph data with node attributes that can be either discrete or real-valued is constantly increasing. While existing kernel methods are effective techniques for dealing with graphs having discrete node labels, their adaptation to non-discrete or continuous node attributes has been limited, mainly for computational issues. Recently, a few kernels especially tailored for this domain, and that trade predictive performance for computational efficiency, have been proposed. In this paper, we propose a graph kernel for complex and continuous nodes' attributes, whose features are tree structures extracted from specific graph visits. The kernel manages to keep the same complexity of state-of-the-art kernels while implicitly using a larger feature space. We further present an approximated variant of the kernel which reduces its complexity significantly. Experimental results obtained on six real-world datasets show that the kernel is the best performing one on most of them. Moreover, in most cases the approximated version reaches comparable performances to current state-of-the-art kernels in terms of classification accuracy while greatly shortening the running times.

Paper Structure

This paper contains 13 sections, 5 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Related work
  4. Ordered Decomposition DAG Kernels for Graphs with Discrete Labels
  5. Graph Kernels for Continuous Node Labels
  6. A First Kernel For Graphs With Continuous Node Labels
  7. Our Proposal
  8. Computation speedup with RBF kernel approximation
  9. Experiments
  10. Experimental setup
  11. Experimental results
  12. Computational times
  13. Conclusions

Figures (3)

  • Figure 1: ODD kernel summary. 1) Decomposition of a graph into its DDs; 2) definition of a total ordering among the children of each node; 3) generation of an explicit FeatureMap extracting all proper subtrees (ST kernel) from the set of Ordered DDs.
  • Figure 2: Sketch of an algorithm to compute the FeatureMap of a graph. The notation is Python-style: {} is an HashMap, and the in operator applied to an HashMap performs the lookup of the element in it.
  • Figure 3: Sketch of an algorithm for computing the ODDCL$_{ST}$ kernel. $\lambda$ is a kernel parameter.