## Abstract

Group synchronization requires to estimate unknown elements (θ_{v})v∈V of a compact group 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 θ_{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 (formula presented) 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 ≥ 3 and, for certain finite groups, for d ≥ 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.

Original language | English (US) |
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Pages (from-to) | 227-256 |

Number of pages | 30 |

Journal | Mathematical Statistics and Learning |

Volume | 1 |

Issue number | 3-4 |

DOIs | |

State | Published - 2018 |

## All Science Journal Classification (ASJC) codes

- Computational Theory and Mathematics
- Signal Processing
- Statistics and Probability
- Theoretical Computer Science

## Keywords

- Graphs
- community detection
- group synchronization
- weak recovery