Near-optimal bounds for phase synchronization^∗

The problem of estimating the phases (angles) of a complex unit-modulus vector z from their noisy pairwise relative measurements C = zz^∗ + σW, where W is a complex-valued Gaussian random matrix, is known as phase synchronization. The maximum likelihood estimator (MLE) is a solution to a unit–modulus-constrained quadratic programming problem, which is nonconvex. Existing works have proposed polynomial-time algorithms such as a semidefinite programming (SDP) relaxation or the generalized power method (GPM). Numerical experiments suggest that both of these methods succeed with high probability for σ up to Õ(n¹/²), yet existing analyses only confirm this observation for σ up to O(n¹/⁴). In this paper, we bridge the gap by proving that the SDP relaxation is tight for σ = O(n/log n), and GPM converges to the global optimum under the same regime. Moreover, we establish a linear convergence rate for GPM, and derive a tighter ∞ bound for the MLE. A novel technique we develop in this paper is to (theoretically) track n closely related sequences of iterates, in addition to the sequence of iterates GPM actually produces. As a by-product, we obtain an ∞ perturbation bound for leading eigenvectors. Our result also confirms predictions that use techniques from statistical mechanics.