Generalized Wasserstein barycenters on Riemannian manifolds are absolutely continuous when all input measures are absolutely continuous, for strictly convex cost profiles h with singularity at zero, via a geometric approximation approach.
Riemannian Geometry
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Every closed connected smooth n-manifold M is dominated by the n-skeleton of a finite simplicial complex whose simplex count is bounded by n and the embolic volume of M.
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Absolute continuity of generalized Wasserstein barycenters of finitely many measures
Generalized Wasserstein barycenters on Riemannian manifolds are absolutely continuous when all input measures are absolutely continuous, for strictly convex cost profiles h with singularity at zero, via a geometric approximation approach.
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Topological Complexity and Finite Domination
Every closed connected smooth n-manifold M is dominated by the n-skeleton of a finite simplicial complex whose simplex count is bounded by n and the embolic volume of M.