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Computational Optimal Transport and Filtering on Riemannian Manifolds
Daniel Grange (Stony Brook University), Mohammad Al-Jarrah (University of Washington), Ricardo Baptista (California Institute of Technology), Amirhossein Taghvaei (University of Washington), Tryphon T. Georgiou (University of California, Irvine), Sean A. Phillips (United States Air Force Research Laboratory, U.S. Air Force Research Laboratory Space Vehicles Directorate), Allen Tannenbaum (Stony Brook University)
Abstract: In this paper we extend recent developments in computational optimal transport to the setting of Riemannian manifolds. In particular, we show how to learn optimal transport maps from samples that relate probability distributions defined on manifolds. Specializing these maps for sampling conditional probability distributions provides an ensemble approach for solving nonlinear filtering problems defined on such geometries. The proposed computational methodology is illustrated with examples of transport and nonlinear filtering on Lie groups, including the circle S1, the special Euclidean group SE(2), and the special orthogonal group SO(3).