PASCAL - Pattern Analysis, Statistical Modelling and Computational Learning

Graph characteristics from the heat kernel trace
Bai Xiao, Richard Wilson and Edwin Hancock
Pattern Recognition Volume 42, Number 11, pp. 2589-2606, 2009. ISSN 0031-3203


Graph structures have been proved important in high level-vision since they can be used to represent structural and relational arrangements of objects in a scene. One of the problems that arises in the analysis of structural abstractions of objects is graph clustering. In this paper, we explore how permutation invariants computed from the trace of the heat kernel can be used to characterize graphs for the purposes of measuring similarity and clustering. The heat kernel is the solution of the heat equation and is a compact representation of the path-length distribution on a graph. The trace of the heat kernel is given by the sum of the Laplacian eigenvalues exponentiated with time. We explore three different approaches to characterizing the heat kernel trace as a function of time. Our first characterization is based on the zeta function, which from the Mellin transform is the moment generating function of the heat kernel trace. Our second characterization is unary and is found by computing the derivative of the zeta function at the origin. The third characterization is derived from the heat content, i.e. the sum of the elements of the heat kernel. We show how the heat content can be expanded as a power series in time, and the coefficients of the series can be computed using the Laplacian spectrum. We explore the use of these characterizations as a means of representing graph structure for the purposes of clustering, and compare them with the use of the Laplacian spectrum. Experiments with the synthetic and real-world databases reveal that each of the three proposed invariants is effective and outperforms the traditional Laplacian spectrum. Moreover, the heat-content invariants appear to consistently give the best results in both synthetic sensitivity studies and on real-world object recognition problems.

EPrint Type:Article
Project Keyword:Project Keyword UNSPECIFIED
Subjects:Computational, Information-Theoretic Learning with Statistics
Machine Vision
Learning/Statistics & Optimisation
Theory & Algorithms
Information Retrieval & Textual Information Access
ID Code:6839
Deposited By:Edwin Hancock
Deposited On:08 March 2010