The Noise-sensitivity phase transition in spectral group synchronization over compact groups

Abstract

In Group Synchronization, one attempts to find a collection of unknown group elements from noisy measurements of their pairwise differences. Several important problems in vision and data analysis reduce to group synchronization over various compact groups. Spectral Group Synchronization is a commonly used, robust algorithm for solving group synchronization problems, which relies on diagonalization of a block matrix whose blocks are matrix representations of the measured pairwise differences. Assuming uniformly distributed measurement errors, we present a rigorous analysis of the accuracy and noise sensitivity of spectral group synchronization algorithms over any compact group, up to the rounding error. We identify a Baik-Ben Arous-Péché type phase transition in the noise level, beyond which spectral group synchronization necessarily fails. Below the phase transition, spectral group synchronization succeeds in recovering the unknown group elements, but its performance deteriorates with the noise level. We provide asymptotically exact formulas for the accuracy of spectral group synchronization below the phase transition, up to the rounding error. We also provide a consistent risk estimate, allowing practitioners to estimate the method's accuracy from available measurements. *

Multi-Reference Alignment in High Dimensions: Sample Complexity and Phase Transition

“…We mention in passing that if many measurements are available, one can leverage the redundancy in the data to recover the true relative shifts in challenging environments; see, for example, [15,33,36,40,42].…”

A Unified Approach to Synchronization Problems over Subgroups of the Orthogonal Group

“…In the context of synchronization problems, the spectral estimator was first introduced in [41] for phase synchronization (i.e., G is the group SO(2) of 2D rotations). It has later been generalized to synchronization problems over other subgroups of the orthogonal group, see [1,6,27,33], and even over general compact groups [35]. The main computational cost of this approach comes from two parts.…”

Optimal Orthogonal Group Synchronization and Rotation Group Synchronization

“…After going through the above procedure, the matrix Q in the form of ( 27) is fully determined by its upper triangular elements {r ab } 1≤a<b≤d , and therefore we can write Q = Q(r). The equation (28) guarantees that the rows of Q(r) are orthogonal to each other and (29) implies that each row of Q(r) is a unit vector. The following lemma characterizes a sufficient condition on {r ab } 1≤a<b≤d that implies the existence of Q(r).…”

“…In particular, it is shown by [24] that SDP is tight for O(d) synchronization when σ 2 √ n in the setting of p = 1. The papers [1,5,28,31] have studied spectral methods and its asymptotic error behavior. In terms of statistical estimation error, [22] and [23] have derived error bounds for the generalized power method and the spectral method for an ℓ ∞ -type loss in the setting of p = 1.…”