# Group Synchronization on Grids

## TL;DR

This paper investigates the possibility of recovering group differences in grid graphs under noisy observations, establishing conditions under which weak recovery is feasible or impossible depending on the dimension and noise level.

## Contribution

It provides the first rigorous analysis of group synchronization on grid graphs, identifying phase transitions for weak recovery based on dimension, group type, and noise.

## Key findings

- Weak recovery possible for d≥3 and certain finite groups in d≥2.
- Weak recovery impossible for some continuous groups in d=2.
- Weak recovery impossible at high noise levels regardless of dimension.

## 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.