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. We show that convex exactness depends strongly on the symmetry group of the graphs; for a fixed symmetry group $G$, the DS relaxation either will be convex exact for almost all pairs of isomorphic graphs with symmetry group $G$, or will fail for all such pairs. We show that for reflective groups with at least one full orbit, convex exactness holds almost everywhere. The importance of this result stems from the fact that such groups are the most common symmetry groups in shape matching applications. Finally, we provide some simple examples of nonreflective symmetry groups for which convex exactness always fails. These examples also show that the property of having generic convex exactness is not invariant under group isomorphisms. When convex exactness holds, the isomorphisms of the graphs are the extreme points of the convex solution set and the task of finding an isomorphism in polynomial time is tractable. We discuss several common algorithms capable of retrieving an isomorphism in this case. We also show that interior point algorithms will not return an isomorphism in this case but will return the centroid of the set of isomorphisms, which gives a “fuzzy encoding” of the symmetries of the shape. In fact, for any symmetry group $G$, the centroid solution can be recovered efficiently for almost all pairs of isomorphic graphs with symmetry group $G$.