Benign landscapes of low-dimensional relaxations for orthogonal synchronization on general graphs

TL;DR

This paper investigates the optimization landscape of low-dimensional relaxations for orthogonal group synchronization on graphs, establishing conditions under which all second-order critical points are globally optimal, with implications for robotics and oscillators.

Contribution

It proves that for all connected graphs, second-order critical points are globally optimal when the relaxation dimension p is at least r+2, improving previous bounds and extending results to noisy and complex cases.

Findings

Second-order critical points are globally optimal for p ≥ r+2 in noiseless cases.

Relaxation dimension p ≥ r+2 suffices for synchronization on general graphs.

Results extend to the complex (unitary) case, with no spurious local minima when 2p ≥ 3r.

Abstract

Orthogonal group synchronization is the problem of estimating $n$ elements $Z_1, \ldots, Z_n$ from the $r \times r$ orthogonal group given some relative measurements $R_{ij} \approx Z_i^{}Z_j^{-1}$. The least-squares formulation is nonconvex. To avoid its local minima, a Shor-type convex relaxation squares the dimension of the optimization problem from $O(n)$ to $O(n^2)$. Alternatively, Burer--Monteiro-type nonconvex relaxations have generic landscape guarantees at dimension $O(n^{3/2})$. For smaller relaxations, the problem structure matters. It has been observed in the robotics literature that, for SLAM 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 $Y_1, \ldots, Y_n$. For $p \geq r$, each $Y_i$ is relaxed to the Stiefel manifold $\mathrm{St}(r, p)$ of $r \times p$ matrices with orthonormal rows. The available measurements implicitly define a (connected) graph $G$ on $n$ vertices. In the noiseless case, we show that, for all connected graphs $G$, second-order critical points are globally optimal as soon as $p \geq r+2$. (This implies that Kuramoto oscillators on $\mathrm{St}(r, p)$ synchronize for all $p \geq r + 2$.) This result is the best possible for general graphs; the previous best known result requires $2p \geq 3(r + 1)$. For $p > r + 2$, our result is robust to modest amounts of noise (depending on $p$ and $G$). 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 $2p \geq 3r$.