1,279 research outputs found
Near-optimal bounds for phase synchronization
The problem of phase synchronization is to estimate the phases (angles) of a
complex unit-modulus vector from their noisy pairwise relative measurements
, where is a complex-valued Gaussian random matrix.
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 relaxation
(SDP) approach or the generalized power method (GPM) to solve it. Numerical
experiments suggest both of these methods succeed with high probability for
up to , yet, existing analyses only
confirm this observation for up to . In this
paper, we bridge the gap, by proving SDP is tight for , 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 track (theoretically) 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 intuitions that use techniques from
statistical mechanics.Comment: 34 pages, 1 figur
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