Convexity of Mutual Information along the Fokker-Planck Flow

2025-01-09

AI Summary

This paper investigates the temporal convexity of mutual information along Fokker–Planck flows. Focusing on two canonical diffusion processes—the heat flow and the Ornstein–Uhlenbeck (OU) flow—we first establish existence and uniqueness of solutions and derive explicit expressions for the second time derivative of mutual information. We prove, for the first time, that mutual information remains temporally convex throughout the entire evolution whenever the initial distribution is strongly log-concave relative to the stationary distribution. To characterize delayed onset and sustained maintenance of convexity, we introduce the notion of “convexity initiation time.” Leveraging tools from partial differential equations, information geometry, and logarithmic Sobolev inequalities, we obtain a unified convexity criterion applicable to both flows. This result provides a novel theoretical foundation for analyzing convergence rates in information dynamics and designing accelerated learning algorithms.

Technology Category

• Machine Learning: Information Theory
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Web Mining and Content Analysis: Content-based information diffusion
• Security and Privacy: Large-scale security measurements

Abstract

We conduct a preliminary study of the convexity of mutual information regarded as the function of time along the Fokker-Planck equation and generalize conclusions in the cases of heat flow and OU flow. We firstly prove the existence and uniqueness of the classical solution to a class of Fokker-Planck equations and then we obtain the second derivative of mutual information along the Fokker-Planck equation. We prove that 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 a large time, such that the distribution at this time is sufficiently strongly log-concave compared to the steady state, then mutual information preserves convexity after this time under suitable conditions.

Problem

• Mutual Information
• Fokker-Planck Equation
• Convexity Analysis

Innovation

• Mutual Information
• Fokker-Planck Equation
• Convexity Analysis