EXACT RECOVERY WITH SYMMETRIES FOR PROCRUSTES MATCHING

The Procrustes matching (PM) problem is the problem of finding the optimal rigid motion and labeling of two point sets so that they are as close as possible. Both rigid and nonrigid shape matching problems can be formulated as PM problems. Recently [H. Maron et al., ACM Trans. Graph., 35 (2016), pp. 73:1-73:12] presented a novel convex semidefinite programming relaxation (PM-SDP) for PM which achieves state-of-the-art results on common shape matching benchmarks. In this paper we analyze the successfulness of PM-SDP in solving PM problems without noise (exact PM problems): For asymmetric point sets we show that the PM-SDP relaxation returns the correct solution of PM. Symmetric points sets introduce multiple solutions to the PM problem. Therefore, convex relaxations of the PM problem will necessarily admit unwanted solutions, namely, the convex combinations of the correct solutions. To alleviate this issue, standard approaches regularize the nonconvex problem so that it has a unique solution. In contrast, we show that the solutions of PM are precisely the extreme points of the convex solution set of PM-SDP. We then suggest a random algorithm which returns a solution of PM with probability one, and return all solutions of PM with equal probability. By repeatedly applying this algorithm all solutions of PM can be obtained. In addition, we show that the PM-SDP relaxation retains its exactness in the presence of low noise, is invariant to orthogonal change of coordinates and relabeling of points, and is always nonnegative. Finally, exact PM is shown to be computationally equivalent to the graph isomorphism problem.