Continuum Limit of Posteriors in Graph Bayesian Inverse Problems
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We consider the problem of recovering a function input of a differential equation formulated on an unknown domain M. We assume to have access to a discrete domain M_n={x_1, ..., x_n}⊂ M, and to noisy measurements of the output solution at p< n of those points. We introduce a graph-based Bayesian inverse problem, and show that the graph-posterior measures over functions in M_n converge, in the large n limit, to a posterior over functions in M that solves a Bayesian inverse problem with known domain. The proofs rely on the variational formulation of the Bayesian update, and on a new topology for the study of convergence of measures over functions on point clouds to a measure over functions on the continuum. Our framework, techniques, and results may serve to lay the foundations of robust uncertainty quantification of graph-based tasks in machine learning. The ideas are presented in the concrete setting of recovering the initial condition of the heat equation on an unknown manifold.
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