Title: Group Synchronization on Grids,
Abstract: Group synchronization requires to estimate unknown elements
$({\theta}_v)_{v\in V}$ 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 ${\boldsymbol \theta}_v$ are only determined up to a global
action of the group, we consider the following weak recovery question. Can we
determine the group difference ${\theta}_u^{-1}{\theta}_v$ 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 $d\ge 2$. Viceversa, for some continuous groups, we prove
that weak recovery is impossible for $d=2$. Finally, for strong enough noise,
weak recovery is always impossible.