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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. In this type of problems, the pairwise interaction between adjacent vertices in the graph is of a "non-scalar" nature, typically taking values in a group; the "consistency" among these non-scalar pairwise interactions provide information for the dataset from which the graph is constructed. We model these data as a fibre bundle equipped with a connection, and consider a horizontal diffusion process on the fibre bundle driven by a standard diffusion process on the base manifold of the fibre bundle; the spectral information of the horizontal diffusion decouples the base manifold structure from the observed non-scalar pairwise interactions. We demonstrate an application of this framework on evolutionary anthropology.