Near-Optimal Bounds for Phase Synchronization

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 * + σW , 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 σ up tõ O(n 1/2 ), yet, existing analyses only confirm this observation for σ up to O(n 1/4 ). In this paper, we bridge the gap, by proving SDP 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 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 ∞ perturbation bound for leading eigenvectors. Our result also confirms intuitions that use techniques from statistical mechanics. 5 Suppose the prior is supported and uniformly distributed on {−1, 1} n . By independence, the Bayes-optimal estimator for phase synchronization is a function of Re(C), so we can use informationtheoretic results about the Z 2 synchronization problem.6 In [2, 1], it is suggested information-theoretically exact recovery with high probability is impossible for synchronization over Z 2 if σ > n/(2 − ε) log n. We can use this result to show O( n/ log n) is necessary for a nontrivial ∞ error by putting a uniform prior on {−1, 1} n .

On the Landscape of Synchronization Networks: A Perspective from Nonconvex Optimization

“…The goal is to recover all the group elements up to a global phase from {g i g −1 j } {(i,j)∈E} where E denotes the index set of available observations. This is called the group synchronization problem, see [1,2,6,7,8,13,35,51] for more details.…”

“…Synchronization problems, which aim to recover group elements g i from its relative alignment g i g −1 j (or noisy alignment), are considered on different groups such as phase synchronization [6,51] (corresponding to the group U (1)), Z 2 synchronization [1,7], joint alignment from pairwise differences [13] (corresponding to the cyclic group, i.e., Z/nZ), and general group synchronization problems [35,2]. Many algorithms, some based on convex and nonconvex optimization, are proposed to tackle these problems and proven to work well on group elements retrieval.…”

“…Many algorithms, some based on convex and nonconvex optimization, are proposed to tackle these problems and proven to work well on group elements retrieval. In these aforementioned works, we are mostly influenced by Z 2 and phase synchronization [6,51,1,7,13], especially [7].…”

“…Compared with convex relaxation, iterative methods based on gradient descent and power methods achieve much higher efficiency while still enjoying rigorous theoretic guarantees. However, the discussion on the landscape of corresponding cost function is not covered [51,13]. In [7], the authors studied the Burer-Monteiro method applied to Z 2 synchronization and obtained an objective function in the form of E(θ) (modulo a proper transformation).…”

Principal Component Analysis for Big Data

“…In this review article, we focus on clustering and community detection (Lowrimore and Manton, 2016), ranking, mixture model and manifold learning. Other applications, such as matrix completion (Keshavan, Montanari and Oh, 2010), phase synchronization (Zhong and Boumal, 2017), image segmentation (Dambreville, Rathi and Tannen, 2006) and functional data analysis (Ramsay, 2016), are not discussed here due to space limitations. Motivated by the fact that data are often collected and stored in distant places, many distributed algorithms for PCA have been proposed (Qu et al, 2002;Feldman, Schmidt and Sohler, 2013) and shown to provide strong statistical guarantees .…”