TY - JOUR

T1 - Near-optimal bounds for phase synchronization∗

AU - Zhong, Yiqiao

AU - Boumal, Nicolas

N1 - Funding Information:
∗Received by the editors March 21, 2017; accepted for publication (in revised form) December 8, 2017; published electronically April 3, 2018. http://www.siam.org/journals/siopt/28-2/M112202.html Funding: The second author is supported by NSF grant DMS-1719558. †Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544 (yiqiaoz@princeton.edu). ‡Department of Mathematics and the Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544 (nboumal@math.princeton.edu). 1Since W is Gaussian, an MLE minimizes the squared Frobenius norm: ‖C − xx∗‖F2 = ‖C‖F2 + ‖xx∗‖F2 − 2x∗Cx. Owing to |xk| = 1 ∀k, this is equivalent to maximizing x∗Cx.

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

KW - Angular synchronization

KW - Eigenvector perturbation bound

KW - Maximum likelihood estimator

KW - Nonconvex optimization

KW - Projected power method

KW - Quadratically constrained quadratic program

KW - Semidefinite programming relaxation

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U2 - 10.1137/17M1122025

DO - 10.1137/17M1122025

M3 - Article

AN - SCOPUS:85049664864

VL - 28

SP - 989

EP - 1016

JO - SIAM Journal on Optimization

JF - SIAM Journal on Optimization

SN - 1052-6234

IS - 2

ER -