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Wasserstein Steepest Descent Flows of Discrepancies with Riesz Kernels

Johannes Hertrich, Manuel Gräf, Robert Beinert, Gabriele Steidl

TL;DR

The paper develops Wasserstein steepest descent flows as a local, tangent-space-based alternative to traditional gradient-flow schemes in the space of probability measures, enabling Euler-forward discretizations and connecting steepest-descent directions to Wasserstein gradients for $\lambda$-convex functionals. It then analyzes flows associated with discrepancies defined by Riesz kernels, deriving analytic forms for interaction-energy flows starting from Dirac measures and revealing a potential transition to non-atomic measures (particle explosions) in higher dimensions. The authors establish directional derivatives, optimality conditions, and connections to equilibrium measures with external fields in potential theory, and they complement theory with numerical simulations of discrepancy flows. These results link Wasserstein dynamics to potential-theoretic concepts and highlight distinct behaviors across kernel orders, including non-convexity and condensation phenomena. Overall, the work broadens the toolbox for particle-based approximations in imaging and related tasks by clarifying when steepest-descent dynamics can be effectively computed and interpreted within the Wasserstein framework.

Abstract

The aim of this paper is twofold. Based on the geometric Wasserstein tangent space, we first introduce Wasserstein steepest descent flows. These are locally absolutely continuous curves in the Wasserstein space whose tangent vectors point into a steepest descent direction of a given functional. This allows the use of Euler forward schemes instead of Jordan--Kinderlehrer--Otto schemes. For $λ$-convex functionals, we show that Wasserstein steepest descent flows are an equivalent characterization of Wasserstein gradient flows. The second aim is to study Wasserstein flows of the maximum mean discrepancy with respect to certain Riesz kernels. The crucial part is hereby the treatment of the interaction energy. Although it is not $λ$-convex along generalized geodesics, we give analytic expressions for Wasserstein steepest descent flows of the interaction energy starting at Dirac measures. In contrast to smooth kernels, the particle may explode, i.e., a Dirac measure becomes a non-Dirac one. The computation of steepest descent flows amounts to finding equilibrium measures with external fields, which nicely links Wasserstein flows of interaction energies with potential theory. Finally, we provide numerical simulations of Wasserstein steepest descent flows of discrepancies.

Wasserstein Steepest Descent Flows of Discrepancies with Riesz Kernels

TL;DR

The paper develops Wasserstein steepest descent flows as a local, tangent-space-based alternative to traditional gradient-flow schemes in the space of probability measures, enabling Euler-forward discretizations and connecting steepest-descent directions to Wasserstein gradients for -convex functionals. It then analyzes flows associated with discrepancies defined by Riesz kernels, deriving analytic forms for interaction-energy flows starting from Dirac measures and revealing a potential transition to non-atomic measures (particle explosions) in higher dimensions. The authors establish directional derivatives, optimality conditions, and connections to equilibrium measures with external fields in potential theory, and they complement theory with numerical simulations of discrepancy flows. These results link Wasserstein dynamics to potential-theoretic concepts and highlight distinct behaviors across kernel orders, including non-convexity and condensation phenomena. Overall, the work broadens the toolbox for particle-based approximations in imaging and related tasks by clarifying when steepest-descent dynamics can be effectively computed and interpreted within the Wasserstein framework.

Abstract

The aim of this paper is twofold. Based on the geometric Wasserstein tangent space, we first introduce Wasserstein steepest descent flows. These are locally absolutely continuous curves in the Wasserstein space whose tangent vectors point into a steepest descent direction of a given functional. This allows the use of Euler forward schemes instead of Jordan--Kinderlehrer--Otto schemes. For -convex functionals, we show that Wasserstein steepest descent flows are an equivalent characterization of Wasserstein gradient flows. The second aim is to study Wasserstein flows of the maximum mean discrepancy with respect to certain Riesz kernels. The crucial part is hereby the treatment of the interaction energy. Although it is not -convex along generalized geodesics, we give analytic expressions for Wasserstein steepest descent flows of the interaction energy starting at Dirac measures. In contrast to smooth kernels, the particle may explode, i.e., a Dirac measure becomes a non-Dirac one. The computation of steepest descent flows amounts to finding equilibrium measures with external fields, which nicely links Wasserstein flows of interaction energies with potential theory. Finally, we provide numerical simulations of Wasserstein steepest descent flows of discrepancies.
Paper Structure (24 sections, 34 theorems, 260 equations, 4 figures)

This paper contains 24 sections, 34 theorems, 260 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Preliminaries
  4. Wasserstein Space
  5. Wasserstein Geodesics
  6. $\lambda$-Convexity along Wasserstein Geodesics
  7. Wasserstein Gradient Flows
  8. Geodesic Directions and Geodesic Tangents
  9. Wasserstein Steepest Descent Flows
  10. Discrepancies
  11. Interaction Energy Flows
  12. Discrepancy Flows
  13. Numerical simulation.
  14. Proof of Lemma \ref{['lem:chain_rule']}
  15. Proofs from Section \ref{['sec:frech-diff']}
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\mu \in \mathcal{P}_2^{r}(\mathbb{R}^d)$ and $\nu \in \mathcal{P}_2(\mathbb{R}^d)$. Then there is a unique plan $\bm \pi \in \Gamma^{\rm{opt}}(\mu, \nu)$ which is induced by a unique measurable optimal transport map $T\colon \mathbb{R}^d \to \mathbb{R}^d$, i.e., and Further, $T = \nabla \psi$, where $\psi\colon \mathbb{R}^d \to (-\infty,+\infty]$ is convex, lower semi-con-tin-u-ous (lsc) an

Figures (4)

  • Figure 1: Halftoning of an image. Gray values are considered as values of a probability density function of a measure which is approximated by an empirical measure such that the discrepancy between both measures becomes small. The halftoned image shows the position of the point measures.
  • Figure 2: 2D particle gradient flow of $\mathcal{D}_{K}^2(\cdot, \delta_{e_1})$ for the Riesz kernel with $r=1$ starting around $\delta_{-e_1}$. The black circles depict the border of $\mathop{\mathrm{supp}}\nolimits \gamma_{\bm v}(t)$ related to the steepest descent direction $\bm v$ at $t=0$ given in \ref{['eq:approx_one_particle_disc_flow_geodesic']}.
  • Figure 3: 3D particle gradient flow of $\mathcal{D}_{K}^2(\cdot, \delta_{e_1})$ for the Riesz kernel with $r=1$ starting around $\delta_{-e_1}$. The left columns show the projection to the $x_1x_2$-plane, the right columns to the $x_3x_2$-plane. The black circles depict the border of $\mathop{\mathrm{supp}}\nolimits \gamma_{\bm v}(t)$ related to the steepest descent direction $\bm v$ at $t=0$ given in \ref{['eq:approx_one_particle_disc_flow_geodesic']}.
  • Figure 4: 2D particle gradient flow of $\mathcal{D}_{K}^2(\cdot, \delta_{e_1})$ for the Riesz kernel with $r=\frac{3}{2}$ starting around $\delta_{-e_1}$.

Theorems & Definitions (60)

  • Theorem 1
  • Proposition 2: BookAmGiSa05
  • Theorem 3: BookAmGiSa05
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • Proposition 7
  • Definition 8
  • ...and 50 more