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- W2598300585 abstract "The problem of phase synchronization is to estimate the phases (angles) of a complex unit-modulus vector $z$ from their noisy pairwise relative measurements $C = zz^* + sigma W$, where $W$ 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 $sigma$ up to $tilde{mathcal{O}}(n^{1/2})$, yet, existing analyses only confirm this observation for $sigma$ up to $mathcal{O}(n^{1/4})$. In this paper, we bridge the gap, by proving SDP is tight for $sigma = mathcal{O}(sqrt{n /log n})$, and GPM converges to the global optimum under the same regime. Moreover, we establish a linear convergence rate for GPM, and derive a tighter $ell_infty$ bound for the MLE. A novel technique we develop in this paper is to track (theoretically) $n$ closely related sequences of iterates, in addition to the sequence of iterates GPM actually produces. As a by-product, we obtain an $ell_infty$ perturbation bound for leading eigenvectors. Our result also confirms intuitions that use techniques from statistical mechanics." @default.
- W2598300585 created "2017-04-07" @default.
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- W2598300585 date "2018-01-01" @default.
- W2598300585 modified "2023-10-18" @default.
- W2598300585 title "Near-Optimal Bounds for Phase Synchronization" @default.
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- W2598300585 doi "https://doi.org/10.1137/17m1122025" @default.
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