The Fisher Paradox: Dissipation Interference in Information-Regularized Gradient Flows

2026-03-07

AI Summary

This work investigates Wasserstein gradient flows regularized by Fisher information, uncovering a cross-dissipation term within the dissipation structure whose sign flips when the state width falls below a critical scale, thereby impeding the decay of the free energy functional—an effect termed the “Fisher paradox.” By reducing the dynamics to a Gaussian manifold, the authors derive a variance potential incorporating a logarithmic centrifugal potential, which analytically delineates three dynamical regimes separated by two critical scales. The theoretical predictions exhibit excellent agreement with numerical simulations on a 512-point grid (mean relative error < 5.21×10⁻⁴) and demonstrate universality across bimodal and Laplacian initial conditions, establishing a quantitative link between dissipation delay and the initial information distance.

Technology Category

• Machine Learning: Learning with Manifolds
• Game Theory and Economic Paradigms: Imperfect Information
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We show that Fisher-regularized Wasserstein gradient flows exhibit a previously unrecognized interference mechanism in their dissipation identity: a cross-dissipation term whose sign becomes positive when the state width falls below a critical scale. In this regime the geometric Fisher channel transiently opposes descent of the baseline free-energy functional, producing what we term the Fisher Paradox. Restricting the flow to the Gaussian manifold yields an exact Riccati-type variance equation with a closed-form trajectory, exposing three dynamical regimes separated by two critical scales: sigma = 1 (cross-dissipation sign flip) and sigma = sqrt(epsilon) (Fisher takeover). The variance potential V(u) = u^2 - 2u - epsilon ln(u) contains a logarithmic centrifugal barrier that shifts the equilibrium attractor by Delta sigma approx epsilon/4. The interference persists for a duration t_cross ~ D_KL, linking the dissipation delay directly to the initial information distance. Finite-difference simulations on a 512-point grid confirm all analytical predictions to within 5.21 x 10^-4 mean relative error. Numerical experiments with bimodal and Laplace initial conditions confirm the effect persists beyond Gaussian closure, with direct implications for information-geometric dissipative dynamics.

Problem

• Fisher regularization
• Wasserstein gradient flow
• dissipation interference
• Fisher Paradox
• information geometry

Innovation

• Fisher regularization
• Wasserstein gradient flow
• dissipation interference
• Riccati variance dynamics
• information geometry