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The geometry of synchronization problems and learning group actions. (English) Zbl 1456.05105

Summary: 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 \(\Gamma\) with a flat principal \(G\)-bundle over \(\Gamma\), thus establishing a classification result for synchronization problems using the representation variety of the fundamental group of \(\Gamma\) 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.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
62H30 Classification and discrimination; cluster analysis (statistical aspects)
57R22 Topology of vector bundles and fiber bundles
58A14 Hodge theory in global analysis
28D05 Measure-preserving transformations
35B65 Smoothness and regularity of solutions to PDEs
35J60 Nonlinear elliptic equations
49N90 Applications of optimal control and differential games
49Q20 Variational problems in a geometric measure-theoretic setting

Software:

molaR; t-SNE
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Full Text: DOI arXiv Link

References:

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