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arXiv · 2407.01879

Disintegrated optimal transport for metric fiber bundles

Jun Kitagawa·Asuka Takatsu
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Abstract

We define a new two-parameter family of metrics on subsets of Borel probability measures on general metric fiber bundles, called the $ \textit{disintegrated Monge--Kantorovich metrics}$. This family contains the classical Monge-Kantorovich metrics, linearized optimal transport distance, and fibered Wasserstein distances, and certain cases admit isometric embeddings of the sliced and max-sliced Wasserstein spaces. We prove these metrics are complete, separable (except an endpoint case), and geodesic, with a dual representation. Our results cannot be obtained by applying the theory of $L^q$ maps valued in spaces of probability measures, in fact the $L^q$ map case can be recovered from our results by taking the underlying bundle as a trivial product bundle, and the geodesicness and duality results are new even in the fibered Wasserstein case.

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Jun Kitagawa, Asuka Takatsu. 2026-02-14. Disintegrated optimal transport for metric fiber bundles. https://arxiv.org/abs/2407.01879

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