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 |
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