Benign Landscapes of Low-Dimensional Relaxations for Orthogonal Synchronization on General Graphs

Abstract

SIAM Journal on Optimization, Volume 34, Issue 2, Page 1427-1454, June 2024.
Abstract. Orthogonal group synchronization is the problem of estimating [math] elements [math] from the [math] orthogonal group given some relative measurements [math]. The least-squares formulation is nonconvex. To avoid its local minima, a Shor-type convex relaxation squares the dimension of the optimization problem from [math] to [math]. Alternatively, Burer–Monteiro-type nonconvex relaxations have generic landscape guarantees at dimension [math]. For smaller relaxations, the problem structure matters. It has been observed in the robotics literature that, for simultaneous localization and mapping problems, it seems sufficient to increase the dimension by a small constant multiple over the original. We partially explain this. This also has implications for Kuramoto oscillators. Specifically, we minimize the least-squares cost function in terms of estimators [math]. For [math], each [math] is relaxed to the Stiefel manifold [math] of [math] matrices with orthonormal rows. The available measurements implicitly define a (connected) graph [math] on [math] vertices. In the noiseless case, we show that, for all connected graphs [math], second-order critical points are globally optimal as soon as [math]. (This implies that Kuramoto oscillators on [math] synchronize for all [math].) This result is the best possible for general graphs; the previous best known result requires [math]. For [math], our result is robust to modest amounts of noise (depending on [math] and [math]). Our proof uses a novel randomized choice of tangent direction to prove (near-)optimality of second-order critical points. Finally, we partially extend our noiseless landscape results to the complex case (unitary group); we show that there are no spurious local minima when [math].