Nonconvex phase synchronization

TL;DR

This paper demonstrates that a modified power method can efficiently solve the nonconvex phase synchronization problem under certain noise conditions, matching the success of convex relaxation methods but with greater simplicity and speed.

Contribution

It introduces a modified power method that converges to the global optimum in phase synchronization, under conditions similar to those for convex relaxation, with empirical and theoretical validation.

Findings

Modified power method converges to the global optimum under certain noise conditions.

Empirically, the method is faster and as successful as convex relaxation.

Second-order optimality conditions are sufficient for global optimality in this nonconvex setting.

Abstract

We estimate $n$ phases (angles) from noisy pairwise relative phase measurements. The task is modeled as a nonconvex least-squares optimization problem. It was recently shown that this problem can be solved in polynomial time via convex relaxation, under some conditions on the noise. In this paper, under similar but more restrictive conditions, we show that a modified version of the power method converges to the global optimum. This is simpler and (empirically) faster than convex approaches. Empirically, they both succeed in the same regime. Further analysis shows that, in the same noise regime as previously studied, second-order necessary optimality conditions for this quadratically constrained quadratic program are also sufficient, despite nonconvexity.