Information Geometry for Regularized Optimal Transport and Barycenters of Patterns.

Area of Science:

• Information Geometry
• Optimal Transport Theory
• Probability Distributions

Background:

• Entropic regularization of optimal transport offers computational benefits but yields approximations that are not true divergences.
• Previous attempts to bridge Wasserstein and Kullback-Leibler divergences lacked key geometric properties like translation invariance.

Purpose of the Study:

• To propose a new divergence measure on the manifold of probability distributions.
• To develop a divergence that retains intuitive geometric properties, specifically translation invariance.
• To provide a measure suitable for applications like optimal transport barycenters.

Main Methods:

• Building upon entropic regularization of optimal transportation problems.
• Leveraging concepts from information geometry.
• Developing a novel mathematical framework for divergence measures.

Main Results:

• A new divergence measure for probability distributions is proposed.
• The proposed divergence retains translation invariance, a key property of Wasserstein geometry.
• The new divergence admits an intuitive interpretation.

Conclusions:

• The new divergence offers an improvement over existing entropic regularized transport costs.
• This work provides a valuable tool for analyzing probability distributions with desirable geometric properties.
• The proposed divergence is well-suited for advanced applications like computing optimal transport barycenters.