Quantitative Stability of the Pushforward Operation by an Optimal Transport Map

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Guillaume Carlier, Alex Delalande, Quentin Mérigot

Published: 2025, Last Modified: 26 Jan 2026Found. Comput. Math. 2025EveryoneRevisionsBibTeXCC BY-SA 4.0
Abstract: We study the quantitative stability of the mapping that to a measure associates its pushforward measure by a fixed (non-smooth) optimal transport map. We exhibit a tight Hölder-behavior for this operation under minimal assumptions. Our proof essentially relies on a new bound that quantifies the size of the singular sets of a convex and Lipschitz continuous function on a bounded domain.