Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities

Understand

The dynamics of probability density functions have been extensively studied in computational science and engineering to understand physical phenomena and facilitate algorithmic design.

• Of particular interest are dynamics formulated as gradient flows of energy functionals under the Wasserstein metric.
• The development of functional inequalities, such as the log-Sobolev inequality, plays a pivotal role in analyzing the convergence of these dynamics.
• This paper aims to extend the success of functional inequality techniques to dynamics that are gradient flows under the Fisher-Rao metric, with various $f$-divergences serving as energy functionals.