Pseudo-Riemannian geometry encodes information geometry in optimal transport

Area of Science:

• Mathematics
• Statistics
• Machine Learning

Background:

• Optimal transport and information geometry analyze geometric structures on probability distributions.
• Optimal transport focuses on cost-minimizing movement between distributions.
• Information geometry studies coordinate-invariant properties of statistical inference.

Purpose of the Study:

• To establish a novel differential-geometric relationship between optimal transport and information geometry.
• To explore the implications of this connection for understanding statistical manifolds and divergences.
• To provide new interpretations of existing geometric concepts within this unified framework.

Main Methods:

• Utilizing the pseudo-Riemannian framework developed by Kim and McCann for optimal transport.
• Applying the concept of c-divergence to define divergences via optimal transport maps.
• Analyzing the Ma-Trudinger-Wang (MTW) condition and its geometric interpretation.

Main Results:

• A new differential-geometric connection between optimal transport and information geometry is established.
• The pseudo-Riemannian framework for optimal transport is shown to encode the dualistic structure of statistical manifolds.
• A novel information-geometric interpretation of the MTW tensor is derived.
• Specific cases, like Bregman and log-divergences, reveal constant sectional curvature properties.

Conclusions:

• The unified framework offers new insights into both optimal transport and information geometry.
• The study bridges concepts from differential geometry, statistics, and machine learning.
• This work facilitates a deeper understanding of geometric structures in probability spaces.