Convexity of Mutual Information along the Fokker-Planck Flow

TL;DR

This paper investigates the convexity properties of mutual information along the Fokker-Planck flow, extending previous results for heat and Ornstein-Uhlenbeck flows, and establishes conditions for convexity preservation based on log-concavity.

Contribution

It generalizes the convexity analysis of mutual information to the Fokker-Planck flow and provides conditions for convexity preservation related to initial distribution properties.

Findings

Mutual information is convex along the Fokker-Planck flow under certain conditions.

Existence and uniqueness of classical solutions to the Fokker-Planck equations are established.

Convexity preservation depends on the initial distribution's strong log-concavity.

Abstract

We study the convexity of mutual information as a function of time along the Fokker-Planck flow. The results are generalizations of that along heat flow and Ornstein-Ulenbeck flow, which were established by A. Wibisono and V. Jog. We prove the existence and uniqueness of the classical solutions to a class of Fokker-Planck equations and then we obtain the second derivative of mutual information along the Fokker-Planck equation. If the initial distribution is sufficiently strongly log-concave compared to the steady state, then mutual information always preserves convexity under suitable conditions. In particular, if there exists some time point at which the distribution is sufficiently strongly log-concave, then mutual information will preserve convexity after that time.