Group synchronization on grids

Abstract : Group synchronization requires to estimate unknown elements (θv)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 θv are only determined up to a global action of the group, we consider the following weak recovery question. Can we determine the group difference θ−1uθv 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.
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https://hal.archives-ouvertes.fr/hal-01940467
Contributor : Laurent Massoulié <>
Submitted on : Friday, November 30, 2018 - 11:56:05 AM
Last modification on : Friday, June 28, 2019 - 3:01:15 PM

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  • HAL Id : hal-01940467, version 1
  • ARXIV : 1706.08561

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Emmanuel Abbe, Laurent Massoulié, Andrea Montanari, Allan Sly, Nikhil Srivastava. Group synchronization on grids. Mathematical Statistics and Learning, EMS Publishing House, 2018. ⟨hal-01940467⟩

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