Manifold Embedding of Graphs Using the Heat Kernel

Abstract

In this paper, we investigate the use of heat kernels as a means of embedding the individual nodes of a graph on a manifold in a vector space. The heat kernel of the graph is found by exponentiating the Laplacian eigen-system over time. We show how the spectral representation of the heat kernel can be used to compute both Euclidean and geodesic distances between nodes. We use the resulting pattern of distances to embed the nodes of the graph on a manifold using multidimensional scaling. The distribution of embedded points can be used to characterise the graph, and can be used for the purposes of graph clustering. Here for the sake of simplicity, we use spatial moments. We experiment with the resulting algorithms on the COIL database, and they are demonstrated to offer a useful margin of advantage over existing alternatives.KeywordsHeat KernelGeodesic DistanceLaplacian MatrixRand IndexLaplacian EigenvalueThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.