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Convexity of Mutual Information along the Fokker-Planck Flow

Jiayang Zou, Luyao Fan, Jiayang Gao, Jia Wang

TL;DR

This work analyzes how mutual information evolves along the Fokker-Planck flow, extending known convexity results from heat and OU flows to a broader FP setting. By formulating FP as a Wasserstein gradient flow and deriving the second-time derivative of mutual information, the authors show convexity under conditions where the initial density is sufficiently relatively log-concave with respect to the steady state, with ΔV ≤ (1/2) ||∇V||^2. They establish existence and uniqueness of classical FP solutions, provide Schrödinger-equation-based representations, and derive explicit derivative formulas involving backward Fisher information. The results generalize convexity properties to FP channels, including unbounded domains, and offer a framework for future exploration of weaker assumptions and broader applicability in stochastic analysis and information theory.

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.

Convexity of Mutual Information along the Fokker-Planck Flow

TL;DR

This work analyzes how mutual information evolves along the Fokker-Planck flow, extending known convexity results from heat and OU flows to a broader FP setting. By formulating FP as a Wasserstein gradient flow and deriving the second-time derivative of mutual information, the authors show convexity under conditions where the initial density is sufficiently relatively log-concave with respect to the steady state, with ΔV ≤ (1/2) ||∇V||^2. They establish existence and uniqueness of classical FP solutions, provide Schrödinger-equation-based representations, and derive explicit derivative formulas involving backward Fisher information. The results generalize convexity properties to FP channels, including unbounded domains, and offer a framework for future exploration of weaker assumptions and broader applicability in stochastic analysis and information theory.

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.
Paper Structure (16 sections, 17 theorems, 103 equations)

This paper contains 16 sections, 17 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Existence and Uniqueness of Classical Solutions to the FP Flow
  3. Log-convavity and Entropy Related Quantities
  4. Log-concavity of Solutions to Parabolic Systems
  5. Fundamental Quantities and Mutual Versions
  6. Convexity of Mutual Information
  7. Derivatives of Mutual Information along the FP Flow
  8. Convexity of Mutual Information along the FP Flow
  9. Discussion and Future Work
  10. Some Applications of Stochastic Analysis in PDE
  11. Proof of Theorem \ref{['Thm:Existence and Uniqueness to the Cauchy Problem']}
  12. Proof of Lemma \ref{['derivatives of entropy']}
  13. Proof of Theorem \ref{['derivatives of mutual information']}
  14. Proof of Lemma \ref{['step1']}
  15. Proof of Lemma \ref{['step2']}
  16. ...and 1 more sections

Key Result

Theorem 1

If Assumption Assumption for potential and Assumption for initial hold, the Fokker-Planck flow FPflow in the entire space has the unique classical solution where $c\left( x \right) =\frac{1}{4}\left\| \nabla V \right\| ^2-\frac{1}{2}\Delta V , \gamma_0=e^{\frac{V}{2}}\mu_0$. Furthermore, $\mu(x,t)\in C^{1,\infty}((0,\infty)\times \mathbb{R}^N).$ Then this solution can be expressed as the convolut

Theorems & Definitions (38)

  • Remark 1
  • Theorem 1
  • proof
  • Definition 1
  • Definition 1
  • Lemma 1
  • Corollary 1
  • Definition 2
  • Definition 3
  • Definition 4
  • ...and 28 more