Soft Graph Matching: Submodular Relaxation and Lovasz Extension

Graph matching aims to align a pair of graphs by minimizing their edge disagreements. As the problem is NP–hard in the worst-case, various methods have been proposed for approximately solving the problem. One popular approach is to relax the combinatorial problem to a continuous formulation, whose solution represents a soft correspondence between the vertex-sets of the graphs. Previous work has primarily motivated such soft matching formulations as an intermediate step towards obtaining hard correspondences. In this paper, we depart from this viewpoint and provide an alternate motivation for soft matching as a means of identifying classes of topologically-invariant subgraphs, which cannot be revealed by hard correspondences. Drawing on this observation, we consider the family of doubly-stochastic relaxations for graph matching and propose a new convex relaxation for the problem. We establish that the objective function of our formulation can be interpreted as the tightest convex relaxation of the combinatorial quadratic graph matching objective function in a certain sense, and describe an efficient first-order algorithm for computing its solution. Through experiments conducted on real-world data, we demonstrate the empirical effectiveness of the algorithm relative to the prevailing relaxations for graph matching.