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The geometry of synchronization problems and learning group actions

The geometry of synchronization problems and learning group actions
The geometry of synchronization problems and learning group actions
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 Γ with a flat principal G-bundle over Γ, thus establishing a classification result for synchronization problems using the representation variety of the fundamental group of Γ 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.
0179-5376
1-62
Gao, Tingran
a5642390-2a67-4487-bbce-209b3bdda806
Brodzki, Jacek
b1fe25fd-5451-4fd0-b24b-c59b75710543
Mukherjee, Sayan
13136bd1-6bb6-49f2-a0cf-90e701359f0e
Gao, Tingran
a5642390-2a67-4487-bbce-209b3bdda806
Brodzki, Jacek
b1fe25fd-5451-4fd0-b24b-c59b75710543
Mukherjee, Sayan
13136bd1-6bb6-49f2-a0cf-90e701359f0e

Gao, Tingran, Brodzki, Jacek and Mukherjee, Sayan (2019) The geometry of synchronization problems and learning group actions. Discrete and Computational Geometry, 1-62. (doi:10.1007/s00454-019-00100-2).

Record type: Article

Abstract

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 Γ with a flat principal G-bundle over Γ, thus establishing a classification result for synchronization problems using the representation variety of the fundamental group of Γ 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.

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Gao, Brodzki, Mukherjee - The Geometry of Synchronization Problems and Learning Group Actions - Accepted Manuscript
Restricted to Repository staff only until 14 April 2020.
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More information

Accepted/In Press date: 29 April 2019
e-pub ahead of print date: 28 May 2019
Published date: 28 May 2019

Identifiers

Local EPrints ID: 430926
URI: https://eprints.soton.ac.uk/id/eprint/430926
ISSN: 0179-5376
PURE UUID: 02590661-aab5-454b-8bb2-ca6f4be81c24

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Date deposited: 17 May 2019 16:30
Last modified: 02 Aug 2019 16:30

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