Graph Matching through Entropic Manifold Alignment

In this paper we cast the problem of graph matching as one of non-rigid manifold alignment. The low dimensional manifolds are from the commute time embedding and are matched though coherent point drift. Although there have been a number of attempts to realise graph matching in this way, in this paper we propose a novel information-theoretic measure of alignment, the so-called symmetrized normalized-entropy-square variation. We succesfully test this dissimilarity measure between manifolds on a a challenging database. The measure is estimated by means of the bypass Leonenko entropy functional. In addition we prove that the proposed measure induces a positive definite kernel between the probability density functions associated with the manifolds and hence between graphs after deformation. In our experiments we find that the optimal embedding is associated to the commute time distance and we also find that our approach, which is purely topological, outperforms several state-of-the-art graph-based algorithms for point matching.