Information Geometry for Regularized Optimal Transport and Barycenters of Patterns
Abstract
We propose a new divergence on the manifold of probability distributions, building
upon the entropic regularization of optimal transportation problems. As shown in [7],
regularizing the optimal transport problem with an entropic term is known to bring
several computational benefits. However, because of that regularization, the resulting
approximation of the optimal transport cost does not define a proper distance or divergence
between probability distributions. We have recently tried to introduce a family of
divergences connecting the Wasserstein distance and the KL divergence from the information
geometry point of view (see [3]). However, that proposal was not able to retain
key intuitive aspects of the Wasserstein geometry, such as translation invariances, which
play a key role when used in the more general problem of computing optimal transport
barycenters. The divergence we propose in this work is able to retain such properties
and admits an intuitive interpretation.
upon the entropic regularization of optimal transportation problems. As shown in [7],
regularizing the optimal transport problem with an entropic term is known to bring
several computational benefits. However, because of that regularization, the resulting
approximation of the optimal transport cost does not define a proper distance or divergence
between probability distributions. We have recently tried to introduce a family of
divergences connecting the Wasserstein distance and the KL divergence from the information
geometry point of view (see [3]). However, that proposal was not able to retain
key intuitive aspects of the Wasserstein geometry, such as translation invariances, which
play a key role when used in the more general problem of computing optimal transport
barycenters. The divergence we propose in this work is able to retain such properties
and admits an intuitive interpretation.
Research Areas
