Strong Recovery In Group Synchronization

TL;DR

This paper investigates the conditions under which the true group differences in large graph synchronization problems can be exactly recovered from noisy observations, providing both positive and negative theoretical results.

Contribution

It establishes conditions for exact recovery in dense graphs and identifies limitations in sparse graph scenarios for group synchronization.

Findings

Exact recovery is possible in dense graphs with certain priors and observation models.

Recovery becomes impossible in sparse graph regimes under specific conditions.

The results delineate the boundary between feasible and infeasible synchronization scenarios.

Abstract

The group synchronization problem is to estimate unknown group elements at the vertices of a graph when given a set of possibly noisy observations of group differences at the edges. We consider the group synchronization problem on finite graphs with size tending to infinity, and we focus on the question of whether the true edge differences can be exactly recovered from the observations (i.e., strong recovery). We prove two main results, one positive and one negative. In the positive direction, we prove that for a sequence of synchronization problems containing the complete digraph along with a relatively well behaved prior distribution and observation kernel, with high probability we can recover the correct edge labeling. Our negative result provides conditions on a sequence of sparse graphs under which it is impossible to recover the correct edge labeling with high probability.