Graph Embedding Using an Edge-Based Wave Kernel

This paper describes a new approach for embedding graphs On pseudo-Riemannian manifolds based on the wave kernel. The wave kernel is the solution of the wave equation on the edges of a graph. Under the embedding, each edge becomes a geodesic on the manifold. The eigensystern of the wave-kernel is determined by the eigenvalues and the eigenfunctions of the normalized adjacency matrix and can be used to solve the edge-based wave equation. By factorising the Gram-matrix for the wave-kernel, we determine the embedding co-ordinates for nodes under the wave-kernel. We investigate the utility of this new embedding as a means of gauging the similarity of graphs. We experiment on sets of graphs representing the proximity of image features in different views of different objects. By applying multidimensional scaling to the similarity matrix we demonstrate that the proposed graph representation is capable of clustering different views of the same object together.