Bregman-Wasserstein divergence: geometry and applications

Understand

The Bregman-Wasserstein divergence is the optimal transport cost when the underlying cost function is given by a Bregman divergence, and arises naturally in fields such as statistics and machine learning.

• We establish fundamental properties of the Bregman-Wasserstein divergence and propose a novel generalized transport geometry that promotes the Bregman geometry to the space of probability distributions.
• We provide a probabilistic interpretation involving exponential families and define generalized displacement interpolations compatible with the Bregman geometry.
• These interpolations are used to derive a generalized Pythagorean inequality, which is of independent interest.