Convexity of mutual information along the Ornstein-Uhlenbeck flow

TL;DR

This paper investigates the convexity properties of mutual information along the Ornstein-Uhlenbeck process, showing conditions under which it becomes convex over time and providing counterexamples for early times.

Contribution

It establishes new conditions for the convexity of mutual information along the Ornstein-Uhlenbeck flow, including strong log-concavity and moment conditions, and presents counterexamples.

Findings

Mutual information becomes convex at large times for strongly log-concave initial distributions.

If initial distribution is sufficiently strongly log-concave, mutual information is always convex.

Counterexamples show mutual information can be nonconvex at small times.

Abstract

We study the convexity of mutual information as a function of time along the flow of the Ornstein-Uhlenbeck process. We prove that if the initial distribution is strongly log-concave, then mutual information is eventually convex, i.e., convex for all large time. In particular, if the initial distribution is sufficiently strongly log-concave compared to the target Gaussian measure, then mutual information is always a convex function of time. We also prove that if the initial distribution is either bounded or has finite fourth moment and Fisher information, then mutual information is eventually convex. Finally, we provide counterexamples to show that mutual information can be nonconvex at small time.