We propose a novel formulation for phase synchronization -- the statistical problem of jointly estimating alignment angles from noisy pairwise comparisons -- as a nonconvex optimization problem that enforces consistency among the pairwise comparisons in multiple frequency channels. Inspired by harmonic retrieval in signal processing, we develop a simple yet efficient two-stage algorithm that leverages the multi-frequency information. We demonstrate in theory and practice that the proposed algorithm significantly outperforms state-of-the-art phase synchronization algorithms, at a mild computational costs incurred by using the extra frequency channels. We also extend our algorithmic framework to general synchronization problems over compact Lie groups.
The problem of estimating the phases (angles) of a complex unit-modulus vector @math from their noisy pairwise relative measurements @math , where @math 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 @math up to @math , yet existing analyses only confirm this observation for @math up to @math . In this paper, we bridge the gap by proving that the SDP relaxation is tight for @math , and GPM converges to the global optimum under the same regime. Moreover, we establish a linear convergence rate for GPM, and derive a tight... The little Grothendieck problem consists of maximizing @math źijCijxixj for a positive semidefinite matrix C, over binary variables @math xiź ?1 . In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given @math CźRdn?dn a positive semidefinite matrix, the objective is to maximize @math źijtrCijTOiOjT restricting @math Oi to take values in the group of orthogonal matrices @math Od, where @math Cij denotes the (ij)-th @math d?d block of C. We propose an approximation algorithm, which we refer to as Orthogonal-Cut, to solve the little Grothendieck problem over the group of orthogonal matrices @math Od and show a constant approximation ratio. Our method is based on semidefinite programming. For a given @math dź1, we show a constant approximation ratio of @math źR(d)2, where @math źR(d) is the expected average singular value of a @math d?d matrix with random Gaussian @math N0,1d i.i.d. entries. For @math d=1 we recover the known @math źR(1)2=2 ź approximation guarantee for the classical little Grothendieck problem. Our algorithm and analysis naturally extends to the complex valued case also providing a constant approximation ratio for the analogous little Grothendieck problem over the Unitary Group @math Ud. Orthogonal-Cut also serves as an approximation algorithm for several applications, including the Procrustes problem where it improves over the best previously known approximation ratio of @math 122. The little Grothendieck problem falls under the larger class of problems approximated by a recent algorithm proposed in the context of the non-commutative Grothendieck inequality. Nonetheless, our approach is simpler and provides better approximation with matching integrality gaps. Finally, we also provide an improved approximation algorithm for the more general little Grothendieck problem over the orthogonal (or unitary) group with rank constraints, recovering, when @math d=1, the sharp, known ratios. Consider @math points in @math and @math local coordinate systems that are related through unknown rigid transforms. For each point, we are given (possibly noisy) measurements of its local coordinates in some of the coordinate systems. Alternatively, for each coordinate system, we observe the coordinates of a subset of the points. The problem of estimating the global coordinates of the @math points (up to a rigid transform) from such measurements comes up in distributed approaches to molecular conformation and sensor network localization, and also in computer vision and graphics. The least-squares formulation of this problem, although nonconvex, has a well-known closed-form solution when M=2 (based on the singular value decomposition (SVD)). However, no closed-form solution is known for @math . In this paper, we demonstrate how the least-squares formulation can be relaxed into a convex program, namely, a semidefinite program (SDP). By setting up connections between the uniqueness of this SDP and results fr...  In this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the Hermitian complex form. A special case of the problem that we study corresponds to the max-3-cut model used in a recent paper of Goemans and Williamson J. Comput. System Sci., 68 (2004), pp. 442-470]. We first develop a closed-form formula to compute the probability of a complex-valued normally distributed bivariate random vector to be in a given angular region. This formula allows us to compute the expected value of a randomized (with a specific rounding rule) solution based on the optimal solution of the complex semidefinite programming relaxation problem. In particular, we present an @math -approximation algorithm, and then study the limit of that model, in which the problem remains NP-hard. We show that if the objective is to maximize a positive semidefinite Hermitian form, then the randomization-rounding procedure guarantees a worst-case performance ratio of @math , which is better than the ratio of @math for its counterpart in the real case due to Nesterov. Furthermore, if the objective matrix is real-valued positive semidefinite with nonpositive off-diagonal elements, then the performance ratio improves to 0.9349. Maximum likelihood estimation problems are, in general, intractable optimization problems. As a result, it is common to approximate the maximum likelihood estimator (MLE) using convex relaxations. In some cases, the relaxation is tight: it recovers the true MLE. Most tightness proofs only apply to situations where the MLE exactly recovers a planted solution (known to the analyst). It is then sufficient to establish that the optimality conditions hold at the planted signal. In this paper, we study an estimation problem (angular synchronization) for which the MLE is not a simple function of the planted solution, yet for which the convex relaxation is tight. To establish tightness in this context, the proof is less direct because the point at which to verify optimality conditions is not known explicitly. Angular synchronization consists in estimating a collection of n phases, given noisy measurements of the pairwise relative phases. The MLE for angular synchronization is the solution of a (hard) non-bipartite Grothendieck problem over the complex numbers. We consider a stochastic model for the data: a planted signal (that is, a ground truth set of phases) is corrupted with non-adversarial random noise. Even though the MLE does not coincide with the planted signal, we show that the classical semidefinite relaxation for it is tight, with high probability. This holds even for high levels of noise. We estimate @math phases (angles) from noisy pairwise relative phase measurements. The task is modeled as a nonconvex least-squares optimization problem. It was recently shown that this problem can be solved in polynomial time via convex relaxation, under some conditions on the noise. In this paper, under similar but more restrictive conditions, we show that a modified version of the power method converges to the global optimum. This is simpler and (empirically) faster than convex approaches. Empirically, they both succeed in the same regime. Further analysis shows that, in the same noise regime as previously studied, second-order necessary optimality conditions for this quadratically constrained quadratic program are also sufficient, despite nonconvexity. An estimation problem of fundamental interest is that of phase (or angular) synchronization, in which the goal is to recover a collection of phases (or angles) using noisy measurements of relative phases (or angle offsets). It is known that in the Gaussian noise setting, the maximum likelihood estimator (MLE) is an optimal solution to a nonconvex quadratic optimization problem and can be found with high probability using semidefinite programming (SDP), provided that the noise power is not too large. In this paper, we study the estimation and convergence performance of a recently proposed low-complexity alternative to the SDP-based approach, namely, the generalized power method (GPM). Our contribution is twofold. First, we show that the sequence of estimation errors associated with the GPM iterates is bounded above by a decreasing sequence. As a corollary, we show that all iterates achieve an estimation error that is on the same order as that of an MLE. Our result holds under the least restrictive assumpti... The angular synchronization problem is to obtain an accurate estimation (up to a constant additive phase) for a set of unknown angles θ1,…,θn from m noisy measurements of their offsets θi−θjmod2π. Of particular interest is angle recovery in the presence of many outlier measurements that are uniformly distributed in [0,2π) and carry no information on the true offsets. We introduce an efficient recovery algorithm for the unknown angles from the top eigenvector of a specially designed Hermitian matrix. The eigenvector method is extremely stable and succeeds even when the number of outliers is exceedingly large. For example, we successfully estimate n=400 angles from a full set of m=(4002) offset measurements of which 90 are outliers in less than a second on a commercial laptop. The performance of the method is analyzed using random matrix theory and information theory. We discuss the relation of the synchronization problem to the combinatorial optimization problem Max-2-Lin mod L and present a semidefinite relaxation for angle recovery, drawing similarities with the Goemans–Williamson algorithm for finding the maximum cut in a weighted graph. We present extensions of the eigenvector method to other synchronization problems that involve different group structures and their applications, such as the time synchronization problem in distributed networks and the surface reconstruction problems in computer vision and optics.