TY - JOUR

T1 - Angular synchronization by eigenvectors and semidefinite programming

AU - Singer, A.

N1 - Funding Information:
The project described was supported by Award Number DMS-0914892 from the NSF, by Award Number FA9550-09-1-0551 from AFOSR, and by Award Number R01GM090200 from the National Institute of General Medical Sciences. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute of General Medical Sciences or the National Institutes of Health.

PY - 2011/1

Y1 - 2011/1

N2 - 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-θjmod 2π. 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.

AB - 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-θjmod 2π. 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.

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U2 - 10.1016/j.acha.2010.02.001

DO - 10.1016/j.acha.2010.02.001

M3 - Article

C2 - 21179593

AN - SCOPUS:78649320169

SN - 1063-5203

VL - 30

SP - 20

EP - 36

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

IS - 1

ER -