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The Geometry of Synchronization Problems and Learning Group Actions
Tingran Gao, Jacek Brodzki, Sayan Mukherjee
We develop a geometric framework, based on the classical theory of fibre bundles, to characterize the cohomological nature of a large class of synchronization-type problems in the context of graph inference and combinatorial optimization. We identify each synchronization problem in topological group G on connected graph ΓΓ\Gamma with a flat principal G-bundle over ΓΓ\Gamma , thus establishing a classification result for synchronization problems using the representation variety of the fundamental group of ΓΓ\Gamma into G. We then develop a twisted Hodge theory on flat vector bundles associated with these flat principal G-bundles, and provide a geometric realization of the graph connection Laplacian as the lowest-degree Hodge Laplacian in the twisted de Rham–Hodge cochain complex. Motivated by these geometric intuitions, we propose to study the problem of learning group actions—partitioning a collection of objects based on the local synchronizability of pairwise correspondence relations—and provide a heuristic synchronization-based algorithm for solving this type of problems. We demonstrate the efficacy of this algorithm on simulated and real datasets.