TY - JOUR

T1 - Tightness of the maximum likelihood semidefinite relaxation for angular synchronization

AU - Bandeira, Afonso S.

AU - Boumal, Nicolas

AU - Singer, Amit

N1 - Funding Information:
A. S. Bandeira was supported by AFOSR Grant No. FA9550-12-1-0317. Most of this work was done while he was with the Program for Applied and Computational Mathematics at Princeton University, and some while he was with the Department of Mathematics at the Massachusetts Institute of Technology. N. Boumal was supported by a Belgian F.R.S.-FNRS fellowship while working at the Universit?? catholique de Louvain (Belgium), by a Research in Paris fellowship at Inria and ENS, the "Fonds Sp??ciaux de Recherche" (FSR UCLouvain), the Chaire Havas "Chaire Economie et gestion des nouvelles donn??es" and the ERC Starting Grant SIPA. A. Singer was partially supported by Award Number R01GM090200 from the NIGMS, by Award Numbers FA9550-12-1-0317 and FA9550-13-1-0076 from AFOSR, by Award Number LTR DTD 06-05-2012 from the Simons Foundation, and by the Moore Foundation.

PY - 2017/5/1

Y1 - 2017/5/1

N2 - 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.

AB - 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.

KW - Angular synchronization

KW - Maximum likelihood estimation

KW - Semidefinite programming

KW - Tightness of convex relaxation

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U2 - 10.1007/s10107-016-1059-6

DO - 10.1007/s10107-016-1059-6

M3 - Article

AN - SCOPUS:84981277904

VL - 163

SP - 145

EP - 167

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1-2

ER -