Iterative Bregman Projections for Regularized Transportation Problems
Benamou, Jean-David; Carlier, Guillaume; Cuturi, Marco; Nenna, Luca; Peyré, Gabriel (2015), Iterative Bregman Projections for Regularized Transportation Problems, SIAM Journal on Scientific Computing, 37, 2, p. A1111–A1138. 10.1137/141000439
TypeArticle accepté pour publication ou publié
External document linkhttps://arxiv.org/abs/1412.5154v1
Journal nameSIAM Journal on Scientific Computing
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Abstract (EN)This article details a general numerical framework to approximate so-lutions to linear programs related to optimal transport. The general idea is to introduce an entropic regularization of the initial linear program. This regularized problem corresponds to a Kullback-Leibler Bregman di-vergence projection of a vector (representing some initial joint distribu-tion) on the polytope of constraints. We show that for many problems related to optimal transport, the set of linear constraints can be split in an intersection of a few simple constraints, for which the projections can be computed in closed form. This allows us to make use of iterative Bregman projections (when there are only equality constraints) or more generally Bregman-Dykstra iterations (when inequality constraints are in-volved). We illustrate the usefulness of this approach to several variational problems related to optimal transport: barycenters for the optimal trans-port metric, tomographic reconstruction, multi-marginal optimal trans-port and in particular its application to Brenier's relaxed solutions of in-compressible Euler equations, partial un-balanced optimal transport and optimal transport with capacity constraints.
Subjects / KeywordsKullback-Leibler Bregman divergence projection; Transportation Problems; Iterative Bregman Projections
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