Mathematical Programming

, Volume 163, Issue 1–2, pp 145–167 | Cite as

Tightness of the maximum likelihood semidefinite relaxation for angular synchronization

  • Afonso S. Bandeira
  • Nicolas BoumalEmail author
  • Amit Singer
Full Length Paper Series A


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.


Angular synchronization Semidefinite programming Tightness of convex relaxation Maximum likelihood estimation 

Mathematics Subject Classification

90C22 (Semidefinite programming) 90C26 (Nonconvex programming, global optimization) 62F10 (Point estimation) 



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 Université catholique de Louvain (Belgium), by a Research in Paris fellowship at Inria and ENS, the “Fonds Spéciaux de Recherche” (FSR UCLouvain), the Chaire Havas “Chaire Economie et gestion des nouvelles données” 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.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  • Afonso S. Bandeira
    • 1
  • Nicolas Boumal
    • 2
    Email author
  • Amit Singer
    • 2
  1. 1.New York UniversityNew York CityUSA
  2. 2.Princeton UniversityPrincetonUSA

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