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Approximation of Splines in Wasserstein Spaces

Jorge Justiniano, Martin Rumpf, Matthias Erbar

TL;DR

This work develops a time-discrete variational framework for splines in the Wasserstein-2 space to interpolate probability measures. By discretizing the spline energy with local Wasserstein distances to Wasserstein barycenters and regularizing with the path energy, the authors prove existence and analyze Gaussian-consistent behavior, including Mosco convergence in the diagonal-covariance Gaussian setting. They also provide a fully discrete, Sinkhorn-based algorithm for computing splines and demonstrate robustness through numerical experiments and texture synthesis applications. The approach unifies Eulerian spline theory in optimal transport with practical computational tools, enabling smooth interpolation of distributions and feature-driven textures in images and videos.

Abstract

This paper investigates a time discrete variational model for splines in Wasserstein spaces to interpolate probability measures. Cubic splines in Euclidean space are known to minimize the integrated squared acceleration subject to a set of interpolation constraints. As generalization on the space of probability measures the integral over the squared acceleration is considered as a spline energy and regularized by addition of the usual action functional. Both energies are then discretized in time using local Wasserstein-2 distances and the generalized Wasserstein barycenter. The existence of time discrete regularized splines for given interpolation conditions is established. On the subspace of Gaussian distributions, the spline interpolation problem is solved explicitly and consistency in the discrete to continuous limit is shown. The computation of time discrete splines is implemented numerically, based on entropy regularization and the Sinkhorn algorithm. A variant of the iPALM method is applied for the minimization of the fully discrete functional. A variety of numerical examples demonstrate the robustness of the approach and show striking characteristics of the method. As a particular application the spline interpolation for synthesized textures is presented.

Approximation of Splines in Wasserstein Spaces

TL;DR

This work develops a time-discrete variational framework for splines in the Wasserstein-2 space to interpolate probability measures. By discretizing the spline energy with local Wasserstein distances to Wasserstein barycenters and regularizing with the path energy, the authors prove existence and analyze Gaussian-consistent behavior, including Mosco convergence in the diagonal-covariance Gaussian setting. They also provide a fully discrete, Sinkhorn-based algorithm for computing splines and demonstrate robustness through numerical experiments and texture synthesis applications. The approach unifies Eulerian spline theory in optimal transport with practical computational tools, enabling smooth interpolation of distributions and feature-driven textures in images and videos.

Abstract

This paper investigates a time discrete variational model for splines in Wasserstein spaces to interpolate probability measures. Cubic splines in Euclidean space are known to minimize the integrated squared acceleration subject to a set of interpolation constraints. As generalization on the space of probability measures the integral over the squared acceleration is considered as a spline energy and regularized by addition of the usual action functional. Both energies are then discretized in time using local Wasserstein-2 distances and the generalized Wasserstein barycenter. The existence of time discrete regularized splines for given interpolation conditions is established. On the subspace of Gaussian distributions, the spline interpolation problem is solved explicitly and consistency in the discrete to continuous limit is shown. The computation of time discrete splines is implemented numerically, based on entropy regularization and the Sinkhorn algorithm. A variant of the iPALM method is applied for the minimization of the fully discrete functional. A variety of numerical examples demonstrate the robustness of the approach and show striking characteristics of the method. As a particular application the spline interpolation for synthesized textures is presented.
Paper Structure (20 sections, 12 theorems, 106 equations, 10 figures, 1 algorithm)

This paper contains 20 sections, 12 theorems, 106 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Organization.
  3. Background
  4. Review of Optimal Transport
  5. Wasserstein spaces as a Riemannian Manifold
  6. Splines in Wasserstein Spaces
  7. Definition of splines
  8. Variational discretization of splines
  9. Gaussian E-splines
  10. The case of general Gaussian distributions
  11. The case of Gaussian distributions with diagonal covariance matrices
  12. Convergence of discrete Gaussian E-splines
  13. Fully discrete Wasserstein splines and numerical results
  14. Algorithmic foundations
  15. Numerical results
  16. ...and 5 more sections

Key Result

Lemma 3.9

Let $\Omega$ be as in Subsection ssec:reviewOT and let $(\mu_n)_n\subseteq\mathcal{P}(\Omega)$ be tight, and $(\nu_n)_n\subseteq\mathcal{P}(\Omega)$. If $\sup_{n}\mathcal{W}^2(\mu_n,\nu_n)\leq C<\infty$, then $(\nu_n)_n$ is also tight.

Figures (10)

  • Figure 1: A comparison of the different spline models sampled at nine equidistant times in 1D for Gaussian probability distributions as interpolation constraints depicted in grey. Sampled random variables are drawn as black dots, and their optimal sample trajectories are depicted by the connecting black curves: top left: continuous P-spline (red); top middle: continuous E-spline/T-spline (orange) sampled at nine equidistant times; top right: standard deviations for both the P-spline (red) and E-spline/T-spline (orange). Orange dots represent the discrete values obtained with our method; bottom left: continuous T-spline (blue) sampled at nine equidistant times; bottom middle: continuous E-spline (orange) sampled at nine equidistant times; bottom right: standard deviations for both the T-spline (blue) and E-spline (orange). Orange dots represent the discrete values obtained with our method.
  • Figure 2: First two rows: A comparison of discrete piecewise geodesic interpolation (first row) and discrete spline interpolation (second row) for $\delta=0$ is shown for key frame distributions framed in red. The optimization was done on $\mathcal{P}^{G,d}_2$. Third row: Same as second row, except the optimization was performed in the full space $\mathcal{P}_2(\mathbb{R}^d)$. Bottom row: Difference between spline and piece-wise geodesic interpolations. Top right: Plot of the center of masses as a polygonal curve in $\mathbb{R}^2$. Bottom right: Plot of the standard deviations as a polygonal curve in $\mathbb{R}^2$.
  • Figure 3: Discrete piecewise geodesic interpolation (top) and discrete spline interpolation for $\delta=0$ (middle) of three key frames with constant density on an annulus for the first and constant density on a disk for the second and third (framed in red). Bottom: Contribution of each time-step $k=1,\ldots,K-1$ to the spline energy, i.e. $\mathcal{W}^2(\mu_k,\textup{Bar}(\mu_{k-1},\mu_{k+1}))$ for the spline interpolation (orange) and piecewise geodesic interpolation (green).
  • Figure 4: Piecewise geodesic (top) and spline interpolation (middle) are shown for key frames (framed in red) consisting of a thin annulus-shaped distribution and two equal thin square-shaped distributions, using the color map ${0}\space$$\space4\mathrm{e}{-4}$. Bottom: Difference between the spline and piecewise geodesic interpolations, using the color map ${-6}\mathrm{e}{-5}\space$$\space6\mathrm{e}{-5}$.
  • Figure 5: The key frames represent two Gaussians that are far apart from each other (first and fourth key frames) and close to each other (second and third key frames). Piecewise geodesic (top) and spline (bottom) interpolations are shown.
  • ...and 5 more figures

Theorems & Definitions (39)

  • Definition 2.1: Wasserstein distance
  • Definition 3.1
  • Example 3.2: Euclidean space
  • Definition 3.3
  • Remark 3.4
  • Definition 3.5
  • Remark 3.6
  • Definition 3.7: Discrete spline energy
  • Definition 3.8: Regularized discrete spline interpolations
  • Lemma 3.9
  • ...and 29 more