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

Benamou, Jean-David; Gallouët, Thomas; Vialard, François-Xavier (2019), Second order models for optimal transport and cubic splines on the Wasserstein space, Foundations of Computational Mathematics, 19, p. 1113–1143. 10.1007/s10208-019-09425-z

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Type
Article accepté pour publication ou publié
Date
2019
Journal name
Foundations of Computational Mathematics
Number
19
Publisher
Springer
Pages
1113–1143
Publication identifier
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Author(s)
Benamou, Jean-David
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Gallouët, Thomas
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Vialard, François-Xavier
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
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.
Subjects / Keywords
multimarginal optimal transport; Wasserstein; Cubic splines