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Convergence of Entropic Schemes for Optimal Transport and Gradient Flows

Carlier, Guillaume; Duval, Vincent; Peyré, Gabriel; Schmitzer, Bernhard (2017), Convergence of Entropic Schemes for Optimal Transport and Gradient Flows, SIAM Journal on Mathematical Analysis, 49, 2, p. 1385-1418. 10.1137/15M1050264

Type
Article accepté pour publication ou publié
External document link
https://arxiv.org/abs/1512.02783
Date
2017
Journal name
SIAM Journal on Mathematical Analysis
Volume
49
Number
2
Publisher
SIAM - Society for Industrial and Applied Mathematics
Pages
1385-1418
Publication identifier
10.1137/15M1050264
Metadata
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Author(s)
Carlier, Guillaume
Duval, Vincent cc
Peyré, Gabriel
Schmitzer, Bernhard
Abstract (EN)
Replacing positivity constraints by an entropy barrier is popular to approximate solutions of linear programs. In the special case of the optimal transport problem, this technique dates back to the early work of Schr\odinger. This approach has recently been used successfully to solve optimal transport related problems in several applied fields such as imaging sciences, machine learning and social sciences. The main reason for this success is that, in contrast to linear programming solvers, the resulting algorithms are highly parallelizable and take advantage of the geometry of the computational grid (e.g. an image or a triangulated mesh). The first contribution of this article is the proof of the Γ-convergence of the entropic regularized optimal transport problem towards the Monge-Kantorovich problem for the squared Euclidean norm cost function. This implies in particular the convergence of the optimal entropic regularized transport plan towards an optimal transport plan as the entropy vanishes. Optimal transport distances are also useful to define gradient flows as a limit of implicit Euler steps according to the transportation distance. Our second contribution is a proof that implicit steps according to the entropic regularized distance converge towards the original gradient flow when both the step size and the entropic penalty vanish (in some controlled way)."
Subjects / Keywords
Optimal transport; gradient flows; entropic regularization

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