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Second order models for optimal transport and cubic splines on the Wasserstein space

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SplinesWasserstein_final(1).pdf (1.289Mb)
Date
2019
Dewey
Analyse
Sujet
multimarginal optimal transport; Wasserstein; Cubic splines
Journal issue
Foundations of Computational Mathematics
Number
19
Publication date
2019
Article pages
1113–1143
Publisher
Springer
DOI
http://dx.doi.org/10.1007/s10208-019-09425-z
URI
https://basepub.dauphine.fr/handle/123456789/17941
Collections
  • CEREMADE : Publications
Metadata
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Author
Benamou, Jean-David
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Gallouët, Thomas
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Vialard, François-Xavier
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Type
Article accepté pour publication ou publié
Abstract (EN)
On the space of probability densities, we extend the Wasserstein geodesics to the case of higher-order interpolation such as cubic spline interpolation. After presenting the natural extension of cubic splines to the Wasserstein space, we propose simpler approach, similarly to Brenier's generalized Euler solutions. Our method is based on the relaxation of the variational problem on the path space. We propose an efficient implementation based on multimarginal optimal transport and entropic regularization in 1D and 2D. Our framework also enables extrapolation in the Wasserstein geodesic via a natural convex relaxation.

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