Differentiable maps between Wasserstein spaces

2020/10/05 by Bernadette Lessel, Lessel, Bernadette, Thomas Schick +1 · 2 citations
Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Ophthalmology and Eye Disorders

paper · pdf · doi:10.48550/arxiv.2010.02131

openalex publication_date 2020/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A notion of differentiability is being proposed for maps between Wasserstein spaces of order 2 of smooth, connected and complete Riemannian manifolds. Due to the nature of the tangent space construction on Wasserstein spaces, we only give a global definition of differentiability, i.e. without a prior notion of pointwise differentiability. With our definition, however, we recover the expected properties of a differential. Special focus is being put on differentiability properties of pushforward maps induced by smooth maps between the underlying manifolds, and on convex mixing of differentiable maps, with an explicit construction of the differential.

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