Tightness of the maximum likelihood semidefinite relaxation for angular synchronization

TL;DR

This paper proves that the convex semidefinite relaxation for the angular synchronization problem is tight with high probability, even under high noise levels, despite the MLE not directly recovering the planted solution.

Contribution

It establishes the tightness of the semidefinite relaxation for angular synchronization in a stochastic noise model, extending beyond cases where the MLE recovers the true signal.

Findings

Semidefinite relaxation is tight with high probability.

Tightness holds even at high noise levels.

MLE does not necessarily recover the planted solution.

Abstract

Maximum likelihood estimation problems are, in general, intractable optimization problems. As a result, it is common to approximate the maximum likelihood estimator (MLE) using convex relaxations. In some cases, the relaxation is tight: it recovers the true MLE. Most tightness proofs only apply to situations where the MLE exactly recovers a planted solution (known to the analyst). It is then sufficient to establish that the optimality conditions hold at the planted signal. In this paper, we study an estimation problem (angular synchronization) for which the MLE is not a simple function of the planted solution, yet for which the convex relaxation is tight. To establish tightness in this context, the proof is less direct because the point at which to verify optimality conditions is not known explicitly. Angular synchronization consists in estimating a collection of $n$ phases, given noisy measurements of the pairwise relative phases. The MLE for angular synchronization is the solution of a (hard) non-bipartite Grothendieck problem over the complex numbers. We consider a stochastic model for the data: a planted signal (that is, a ground truth set of phases) is corrupted with non-adversarial random noise. Even though the MLE does not coincide with the planted signal, we show that the classical semidefinite relaxation for it is tight, with high probability. This holds even for high levels of noise.