Tightness of the maximum likelihood semidefinite relaxation for angular synchronization

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 s...

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Veröffentlicht in:Mathematical programming Jg. 163; H. 1-2; S. 145 - 167
Hauptverfasser: Bandeira, Afonso S., Boumal, Nicolas, Singer, Amit
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2017
Springer Nature B.V
Schlagworte:
Algorithms
Analysis
Approximation
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Computer science
Convex analysis
Estimating
Estimating techniques
Full Length Paper
Heuristic
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical models
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Maximum likelihood method
Noise
Numerical Analysis
Optimization
Phases
Semidefinite programming
Studies
Synchronism
Synchronization
Theoretical
Tightness
Angular synchronization
Maximum likelihood estimation
Semidefinite programming
90C22 (Semidefinite programming)
Tightness of convex relaxation
90C26 (Nonconvex programming, global optimization)
62F10 (Point estimation)
ISSN:0025-5610, 1436-4646
Online-Zugang:Volltext
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Zusammenfassung: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.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-016-1059-6