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

TL;DR

This paper uncovers a new interference mechanism in Fisher-regularized Wasserstein gradient flows, revealing a paradoxical behavior where dissipation temporarily opposes energy descent, with detailed analysis and simulations confirming the phenomenon.

Contribution

It introduces the Fisher Paradox, analyzes the dynamics on the Gaussian manifold, and provides exact solutions and simulations demonstrating the interference effect in information-regularized gradient flows.

Findings

Identification of a positive cross-dissipation sign below a critical scale

Exact Riccati-type variance equation with three dynamical regimes

Confirmation of the effect beyond Gaussian assumptions through simulations

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.