Group synchronization on grids

Abstract

Group synchronization requires to estimate unknown elements $({\mathbf{\theta }}_v{)}_{v\in V}$ of a compact group $\mathfrak{G}$ associated to the vertices of a graph $G=(V,E)$, using noisy observations of the group differences associated to the edges. This model is relevant to a variety of applications ranging from structure from motion in computer vision to graph localization and positioning, to certain families of community detection problems.

We focus on the case in which the graph $G$ is the $d$-dimensional grid. Since the unknowns ${\mathbf{\theta }}_v$ are only determined up to a global action of the group, we consider the following weak recovery question. Can we determine the group difference ${\mathbf{\theta }}_u^{-1}{\mathbf{\theta }}_v$ between far apart vertices $u,v$ better than by random guessing? We prove that weak recovery is possible (provided the noise is small enough) for $d\ge 3$ and, for certain finite groups, for $d\ge 2$. Vice-versa, for some continuous groups, we prove that weak recovery is impossible for $d=2$. Finally, for strong enough noise, weak recovery is always impossible.