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Metric extrapolation in the Wasserstein space

Thomas O. Gallouët, Andrea Natale, Gabriele Todeschi

TL;DR

This work introduces metric extrapolation in the Wasserstein space as a variational problem that extends minimizing geodesics to arbitrary times by allowing negative coefficients. It establishes two equivalent convex formulations—Toland duality and a weak/barycentric OT framework—and develops an Entropic-Sinkhorn-based numerical scheme for atomic measures. Theoretical results include existence, uniqueness, strong duality, and connections to $H^1$ projection via Gamma-convergence, plus explicit one-dimensional reductions and special cases. Comprehensive numerical experiments demonstrate shape extrapolation and comparisons to gradient-flow dynamics, highlighting the method's ability to extrapolate probability measures while preserving structure and momentum where appropriate.

Abstract

In this article we study a variational problem providing a way to extend for all times minimizing geodesics connecting two given probability measures, in the Wasserstein space. This is simply obtained by allowing for negative coefficients in the classical variational characterization of Wasserstein barycenters. We show that this problem admits two equivalent convex formulations: the first can be seen as a particular instance of Toland duality and the second is a barycentric optimal transport problem. We propose an efficient numerical scheme to solve the latter formulation based on entropic regularization and a variant of Sinkhorn algorithm.

Metric extrapolation in the Wasserstein space

TL;DR

This work introduces metric extrapolation in the Wasserstein space as a variational problem that extends minimizing geodesics to arbitrary times by allowing negative coefficients. It establishes two equivalent convex formulations—Toland duality and a weak/barycentric OT framework—and develops an Entropic-Sinkhorn-based numerical scheme for atomic measures. Theoretical results include existence, uniqueness, strong duality, and connections to projection via Gamma-convergence, plus explicit one-dimensional reductions and special cases. Comprehensive numerical experiments demonstrate shape extrapolation and comparisons to gradient-flow dynamics, highlighting the method's ability to extrapolate probability measures while preserving structure and momentum where appropriate.

Abstract

In this article we study a variational problem providing a way to extend for all times minimizing geodesics connecting two given probability measures, in the Wasserstein space. This is simply obtained by allowing for negative coefficients in the classical variational characterization of Wasserstein barycenters. We show that this problem admits two equivalent convex formulations: the first can be seen as a particular instance of Toland duality and the second is a barycentric optimal transport problem. We propose an efficient numerical scheme to solve the latter formulation based on entropic regularization and a variant of Sinkhorn algorithm.
Paper Structure (22 sections, 13 theorems, 175 equations, 5 figures)

This paper contains 22 sections, 13 theorems, 175 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Geodesics in the Wasserstein space
  3. Toland's duality
  4. Weak optimal transport formulation
  5. Related works
  6. Structure of the paper
  7. Main properties and Toland duality
  8. Existence and uniqueness of solutions
  9. Dual formulation
  10. Barycentric Optimal Transport formulation
  11. Relation with the $H^1$ projection on convex functions
  12. One-dimensional setting
  13. A $\Gamma$-convergence result
  14. Particular solutions
  15. Extrapolation when $\nu_1$ is a Dirac mass
  16. ...and 7 more sections

Key Result

Lemma 2.1

Let $\nu_0,\nu_1 \in \mathcal{P}_2(\mathbb{R}^d)$, and consider the functional $\mathcal{F}_t(\nu_0,\nu_1; \cdot):\mathcal{P}_2(\mathbb{R}^d) \rightarrow\mathbb{R}$ defined in eq:diffW2fun. For any $\mu_0, \mu_1 \in \mathcal{P}_2(\mathbb{R}^d)$, let $\mu:[0,1]\rightarrow \mathcal{P}_2(\mathbb{R}^d),

Figures (5)

  • Figure 1: An example in which a locally length-minimizing geodesic passing by $\nu_0 = \nu_0^++\nu_0^-$ and $\nu_1 = \nu_1^+ +\nu_1^-$ exists for all times but the metric extrapolation does not coincide with it.
  • Figure 2: Example of extrapolation. The four images in the center are pairwise extrapolated in both positive and negative direction, from $t=1$ (blue) to $t=3$ (red).
  • Figure 3: Comparison between the metric extrapolation (top row) and the Lagrangian extrapolation (bottom row), from $\nu_0$ (first column from the left) to $\nu_1$ (second column from the left). Time grows from left to right, from $t=1$ (blue) to $t=4$ (red).
  • Figure 4: A comparison between the gradient flow $\nu_\mathrm{gf}$ (dashed blue line) and metric extrapolation $\nu_t$ (solid red line) on the same time interval, for the same $\nu_0$ and two different initial conditions for $\nu_1$.
  • Figure 5: Example of gradient flow of the opposite Wasserstein distance (top row) from $\nu_0$ (first column from the left) with initial condition $\nu_1$ (second column from the left) and comparison with the metric extrapolation (bottom row). Time grows from left to right, from $t=1$ (blue) to $t=4$ (red).

Theorems & Definitions (32)

  • Lemma 2.1: Theorem 3.4 in Matthes2019bdf2
  • Remark 2.2: Link with extrapolation
  • Lemma 2.3: Stability of the extrapolation
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • ...and 22 more