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

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.

Figures (7)

• Figure 1: Cross-term $\Delta_\varepsilon(\rho(t))$ (top) and variance $\sigma(t)$ (bottom) for $\varepsilon=0.1$, $\sigma_0=0.5$. The cross-term is positive (orange) while $\sigma<1$, slowing $\mathcal{F}_0$ decay; the sign changes at $\sigma=1$ (dotted). $\sigma(t)$ relaxes toward $\sigma^*\approx1+\varepsilon/4$.
• Figure 2: Baseline free energy $\mathcal{F}_0(\rho(t))$ with ($\varepsilon=0.1$, dashed red) and without ($\varepsilon=0$, solid blue) regularization. The paradox gap (pink shading) confirms transient retardation; both curves converge to $\mathcal{F}_0(\mathcal{N}(0,1))$ (dotted).
• Figure 3: Validation of $C(\rho_\sigma)=(1-\sigma^2)/(2\sigma^4)$. Top: Analytical (purple) vs. finite-difference (cyan, $N=512$). Paradox ($\sigma<1$, orange) and acceleration ($\sigma>1$) regions shown. Bottom: Relative error; mean $5.21\times10^{-4}$.
• Figure 4: Non-Gaussian validation of the Fisher Paradox ($\varepsilon=0.1$, $\sigma_{\mathrm{eff}}=0.5$). (a) Cross-dissipation term $C(\rho(t))=\int\rho\,\nabla\mu_0\!\cdot\!\nabla\mu_F\,dx$ for Gaussian (solid), bimodal mixture (dashed), and Laplace (dotted) initial conditions. All three remain positive (paradox window, shaded) while $\sigma_{\mathrm{eff}}<1$, with identical crossing at $t_{\mathrm{cross}}\approx1.32$; the Laplace cusp spikes to 24.6 at $t{=}0$ (${\approx}4\times$ the Gaussian value of 6.0; clipped; annotated). (b) Paradox gap $\mathcal{F}_0^\varepsilon-\mathcal{F}_0^0$. The gap is negative throughout (regularized system descends $\mathcal{F}_0$ faster while $C>0$, then slower once $C<0$ after $t_{\mathrm{cross}}$) and settles to a permanent offset set by the equilibrium shift $\sigma_\infty=1+\varepsilon/4$. All three initial conditions converge to the same attractor.
• Figure 5: Paradox dependence on $\sigma_0\in[0.3,2.0]$ ($\varepsilon=0.1$). (a) Cross-term $C(\sigma(t))=(1-\sigma^2)/(2\sigma^4)$; blue curves ($\sigma_0<1$) are positive initially (paradox active), red curves ($\sigma_0>1$) negative. (b) Paradox gap $\mathcal{F}_0^\varepsilon-\mathcal{F}_0^0$; negative for $\sigma_0<1$ (retarded free-energy descent), positive for $\sigma_0>1$ (Fisher accelerates dissipation).