Regularized Optimal Transport and the ROT Mover's Distance

Abstract : This paper presents a unified framework for smooth convex regularization of discrete optimal transport problems. In this context, the regularized optimal transport turns out to be equivalent to a matrix nearness problem with respect to Bregman divergences. Our framework thus naturally generalizes a previously proposed regularization based on the Boltzmann-Shannon entropy related to the Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We call the regularized optimal transport distance the rot mover's distance in reference to the classical earth mover's distance. We develop two generic schemes that we respectively call the alternate scaling algorithm and the non-negative alternate scaling algorithm, to compute efficiently the regularized optimal plans depending on whether the domain of the regularizer lies within the non-negative orthant or not. These schemes are based on Dykstra's algorithm with alternate Bregman projections, and further exploit the Newton-Raphson method for separable divergences. We enhance the separable case with a sparse extension to deal with high data dimensions. We also instantiate our proposed framework and discuss the inherent specificities for well-known regularizers and statistical divergences in the machine learning and information geometry communities. Finally, we demonstrate our methods with experiments on synthetic and audio data that illustrate the effect of different regularizers and penalties on the output solutions.
Type de document :
Pré-publication, Document de travail
2017
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https://hal.archives-ouvertes.fr/hal-01540866
Contributeur : Nicolas Papadakis <>
Soumis le : vendredi 16 juin 2017 - 16:37:44
Dernière modification le : samedi 17 juin 2017 - 01:14:21

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  • HAL Id : hal-01540866, version 1
  • ARXIV : 1610.06447

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Citation

Arnaud Dessein, Nicolas Papadakis, Jean-Luc Rouas. Regularized Optimal Transport and the ROT Mover's Distance. 2017. <hal-01540866>

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