Exact Recovery with Symmetries for the Doubly-Stochastic Relaxation

Abstract

Graph matching or quadratic assignment, is the problem of labeling the vertices of two graphs so that they are as similar as possible. A common method for approximately solving the NP-hard graph matching problem is relaxing it to a convex optimization problem over the set of doubly stochastic (DS) matrices. Recent analysis has shown that for almost all pairs of isomorphic and asymmetric graphs, the DS relaxation succeeds in correctly retrieving the isomorphism between the graphs. Our goal in this paper is to analyze the case of symmetric isomorphic graphs. This goal is motivated by shape matching applications where the graphs of interest usually have reflective symmetry. For symmetric problems the graph matching problem has multiple isomorphisms and so convex relaxations admit all convex combinations of these isomorphisms as viable solutions. If the convex relaxation does not admit any additional superfluous solution we say that it is convex exact. In this case there are tractable algorithms to retrieve an isomorphism from the convex relaxation. We show that convex exactness depends strongly on the symmetry group of the graphs; For a fixed symmetry group $G$, either the DS relaxation will be convex exact for almost all pairs of isomorphic graphs with symmetry group $G$, or the DS relaxation will fail for all such pairs. We show that for reflective groups with at least one full orbit convex exactness holds almost everywhere, and provide some simple examples of non-reflective symmetry groups for which convex exactness always fails. When convex exactness holds, the isomorphisms of the graphs are the extreme points of the convex solution set. We suggest an efficient algorithm for retrieving an isomorphism in this case. We also show that the "convex to concave" projection method will also retrieve an isomorphism in this case.