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Fisher-Riemann geometry for nonparametric probability densities

Hugo Aimar, Aníbal Chicco Ruiz, Ivana Gómez

TL;DR

The paper develops a Fisher-Riemannian framework for nonparametric probability densities by endowing the probability simplex with the Fisher information metric and solving the resulting geodesic equations. It provides explicit unit-speed geodesic solutions, lifts them to density trajectories via $f(x,t)$, and proves convergence of discrete, dyadic approximations to the continuous geodesics, enabling practical computation of density transport under this geometry. Key contributions include the explicit metric $J( heta)=\theta_{n+1}^{-1}\mathbf{1}\mathbf{1}^T+\mathrm{D}(1/\theta_j)$, the decoupled geodesic ODE $2\theta_k\ddot{\theta}_k+\theta_k^2-(\dot{\theta}_k)^2=0$, closed-form density geodesics $f(x,t)$, and a convergence theorem for dyadic pixelations. Together these results bridge parametric and nonparametric Fisher geometries and provide a computational pathway for density transport in applications such as image processing. $

Abstract

In this article we aim to obtain the Fisher Riemann geodesics for nonparametric families of probability densities as a weak limit of the parametric case with increasing number of parameters.

Fisher-Riemann geometry for nonparametric probability densities

TL;DR

The paper develops a Fisher-Riemannian framework for nonparametric probability densities by endowing the probability simplex with the Fisher information metric and solving the resulting geodesic equations. It provides explicit unit-speed geodesic solutions, lifts them to density trajectories via , and proves convergence of discrete, dyadic approximations to the continuous geodesics, enabling practical computation of density transport under this geometry. Key contributions include the explicit metric , the decoupled geodesic ODE , closed-form density geodesics , and a convergence theorem for dyadic pixelations. Together these results bridge parametric and nonparametric Fisher geometries and provide a computational pathway for density transport in applications such as image processing. $

Abstract

In this article we aim to obtain the Fisher Riemann geodesics for nonparametric families of probability densities as a weak limit of the parametric case with increasing number of parameters.
Paper Structure (6 sections, 12 theorems, 76 equations, 11 figures)

This paper contains 6 sections, 12 theorems, 76 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The Fisher information
  3. The Riemannian geometry in the simplex induced by $J(\theta)=\theta_{n+1}^{-1}\bar{\bar{{1}}}+D(\theta_j^{-1})$
  4. Solutions of the ODE that preserve the unit speed geodesic condition
  5. Fisher geodesic transport of $m$-dimensional densities and the convergence of their dyadic pixelations
  6. Examples and graphics

Key Result

Lemma 2.1

The expected value of the score $\mathcal{S}(\cdot,\theta)$ with respect to $\varphi(\cdot,\theta)$ vanishes.

Figures (11)

  • Figure 1: Several trajectories from $(\frac{1}{3},\frac{1}{3})$ for different initial velocities. Here we consider $t\in[0,\frac{\pi}{2}].$
  • Figure 2: $\tau=\frac{11\pi}{6}$
  • Figure 3: $\tau \in(\frac{11\pi}{6},2\pi)$
  • Figure 4: $\tau \in(\frac{3\pi}{2},\frac{11\pi}{6})$
  • Figure 5: 3-dimensional trajectories
  • ...and 6 more figures

Theorems & Definitions (24)

  • Lemma 2.1
  • proof : Sketch of the proof
  • Lemma 2.2
  • proof
  • Definition 2.1
  • Proposition 2.3
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • ...and 14 more