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Kernel Approximation of Fisher-Rao Gradient Flows

Jia-Jie Zhu, Alexander Mielke

TL;DR

A rigorous investigation of Fisher-Rao and Wasserstein type gradient flows concerning their gradient structures, flow equations, and their kernel approximations is presented, developing a principled theoretical framework using tools from PDE gradient flows and optimal transport theory.

Abstract

The purpose of this paper is to answer a few open questions in the interface of kernel methods and PDE gradient flows. Motivated by recent advances in machine learning, particularly in generative modeling and sampling, we present a rigorous investigation of Fisher-Rao and Wasserstein type gradient flows concerning their gradient structures, flow equations, and their kernel approximations. Specifically, we focus on the Fisher-Rao (also known as Hellinger) geometry and its various kernel-based approximations, developing a principled theoretical framework using tools from PDE gradient flows and optimal transport theory. We also provide a complete characterization of gradient flows in the maximum-mean discrepancy (MMD) space, with connections to existing learning and inference algorithms. Our analysis reveals precise theoretical insights linking Fisher-Rao flows, Stein flows, kernel discrepancies, and nonparametric regression. We then rigorously prove evolutionary $Γ$-convergence for kernel-approximated Fisher-Rao flows, providing theoretical guarantees beyond pointwise convergence. Finally, we analyze energy dissipation using the Helmholtz-Rayleigh principle, establishing important connections between classical theory in mechanics and modern machine learning practice. Our results provide a unified theoretical foundation for understanding and analyzing approximations of gradient flows in machine learning applications through a rigorous gradient flow and variational method perspective.

Kernel Approximation of Fisher-Rao Gradient Flows

TL;DR

A rigorous investigation of Fisher-Rao and Wasserstein type gradient flows concerning their gradient structures, flow equations, and their kernel approximations is presented, developing a principled theoretical framework using tools from PDE gradient flows and optimal transport theory.

Abstract

The purpose of this paper is to answer a few open questions in the interface of kernel methods and PDE gradient flows. Motivated by recent advances in machine learning, particularly in generative modeling and sampling, we present a rigorous investigation of Fisher-Rao and Wasserstein type gradient flows concerning their gradient structures, flow equations, and their kernel approximations. Specifically, we focus on the Fisher-Rao (also known as Hellinger) geometry and its various kernel-based approximations, developing a principled theoretical framework using tools from PDE gradient flows and optimal transport theory. We also provide a complete characterization of gradient flows in the maximum-mean discrepancy (MMD) space, with connections to existing learning and inference algorithms. Our analysis reveals precise theoretical insights linking Fisher-Rao flows, Stein flows, kernel discrepancies, and nonparametric regression. We then rigorously prove evolutionary -convergence for kernel-approximated Fisher-Rao flows, providing theoretical guarantees beyond pointwise convergence. Finally, we analyze energy dissipation using the Helmholtz-Rayleigh principle, establishing important connections between classical theory in mechanics and modern machine learning practice. Our results provide a unified theoretical foundation for understanding and analyzing approximations of gradient flows in machine learning applications through a rigorous gradient flow and variational method perspective.

Paper Structure

This paper contains 22 sections, 25 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gradient-flow systems and geodesics
  4. Reproducing kernel Hilbert space
  5. Wasserstein space and gradient flows
  6. Gradient structure for the (regularized) Stein gradient flow
  7. De-Stein: de-kernelized Wasserstein geometry
  8. Flattened Wasserstein, Cahn-Hilliard, and Sobolev discrepancy
  9. Fisher-Rao space and gradient flows
  10. Kernel approximation of Fisher-Rao gradient flows
  11. MMD as de-kernelized Fisher-Rao distance
  12. Riemannian metric perspective
  13. Fisher-Rao-Regularized MMD gradient flow
  14. Flattened Fisher-Rao, Allen-Cahn, and $\varphi$-divergences
  15. Analysis
  16. ...and 7 more sections

Key Result

Theorem 2.2

Suppose the kernel is square-integrable $\|k\|_{L^2_{\rho}}^2:=\\\int k(x, x) d \rho(x)<\infty$ w.r.t. a probability measure $\rho$. Then the inclusion from the the associated RKHS $\mathcal{H}$ to $L^2_{\rho}$, $\operatorname{Id} : \mathcal{H} \rightarrow L^2_{\rho}$, is continuous. Moreover, its a $\mathcal{T}_{k,\rho}$ is Hilbert-Schmidt (i.e., singular values are square-summable). The integral

Theorems & Definitions (43)

  • Remark 1.1: "Hellinger" versus "Fisher-Rao"
  • Definition 2.1: Gradient system
  • Example 2.1: Classical PDE: Allen-Cahn and Cahn-Hilliard
  • Example 2.2: Wasserstein geodesics in Hamiltonian formulation
  • Example 2.3: Fisher-Rao (or Hellinger) geodesics in Hamiltonian formulation
  • Theorem 2.2: Integral operator
  • Lemma 2.3: Kernel ridge regression estimator
  • Lemma 2.4: Alternative optimization problem of KRR estimation
  • Proposition 3.1: Kernel-approximate Wasserstein gradient flow with KRR velocity field
  • Proposition 3.2: Static dual formulation of the De-Stein distance
  • ...and 33 more