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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Published in:	Mathematical programming Vol. 163; no. 1-2; pp. 145 - 167
Main Authors:	Bandeira, Afonso S., Boumal, Nicolas, Singer, Amit
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2017 Springer Nature B.V
Subjects:	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
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Abstract	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.
AbstractList	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. 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.
Author	Boumal, Nicolas Bandeira, Afonso S. Singer, Amit
Author_xml	– sequence: 1 givenname: Afonso S. surname: Bandeira fullname: Bandeira, Afonso S. organization: New York University – sequence: 2 givenname: Nicolas surname: Boumal fullname: Boumal, Nicolas email: nboumal@princeton.edu organization: Princeton University – sequence: 3 givenname: Amit surname: Singer fullname: Singer, Amit organization: Princeton University
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Keywords	Angular synchronization Maximum likelihood estimation Semidefinite programming 90C22 (Semidefinite programming) Tightness of convex relaxation 90C26 (Nonconvex programming, global optimization) 62F10 (Point estimation)
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SubjectTerms	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
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Title	Tightness of the maximum likelihood semidefinite relaxation for angular synchronization
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