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Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport

Moritz Piening, Robert Beinert

TL;DR

Numerical experiments validate DSW as a scalable substitute for the WoW distance and show that DSW minimization is equivalent to WoW minimization for discretized meta-measures, while avoiding unstable higher-order moments and computational savings.

Abstract

Wasserstein distances define a metric between probability measures on arbitrary metric spaces, including meta-measures (measures over measures). The resulting Wasserstein over Wasserstein (WoW) distance is a powerful, but computationally costly tool for comparing datasets or distributions over images and shapes. Existing sliced WoW accelerations rely on parametric meta-measures or the existence of high-order moments, leading to numerical instability. As an alternative, we propose to leverage the isometry between the 1d Wasserstein space and the quantile functions in the function space $L_2([0,1])$. For this purpose, we introduce a general sliced Wasserstein framework for arbitrary Banach spaces. Due to the 1d Wasserstein isometry, this framework defines a sliced distance between 1d meta-measures via infinite-dimensional $L_2$-projections, parametrized by Gaussian processes. Combining this 1d construction with classical integration over the Euclidean unit sphere yields the double-sliced Wasserstein (DSW) metric for general meta-measures. We show that DSW minimization is equivalent to WoW minimization for discretized meta-measures, while avoiding unstable higher-order moments and computational savings. Numerical experiments on datasets, shapes, and images validate DSW as a scalable substitute for the WoW distance.

Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport

TL;DR

Numerical experiments validate DSW as a scalable substitute for the WoW distance and show that DSW minimization is equivalent to WoW minimization for discretized meta-measures, while avoiding unstable higher-order moments and computational savings.

Abstract

Wasserstein distances define a metric between probability measures on arbitrary metric spaces, including meta-measures (measures over measures). The resulting Wasserstein over Wasserstein (WoW) distance is a powerful, but computationally costly tool for comparing datasets or distributions over images and shapes. Existing sliced WoW accelerations rely on parametric meta-measures or the existence of high-order moments, leading to numerical instability. As an alternative, we propose to leverage the isometry between the 1d Wasserstein space and the quantile functions in the function space . For this purpose, we introduce a general sliced Wasserstein framework for arbitrary Banach spaces. Due to the 1d Wasserstein isometry, this framework defines a sliced distance between 1d meta-measures via infinite-dimensional -projections, parametrized by Gaussian processes. Combining this 1d construction with classical integration over the Euclidean unit sphere yields the double-sliced Wasserstein (DSW) metric for general meta-measures. We show that DSW minimization is equivalent to WoW minimization for discretized meta-measures, while avoiding unstable higher-order moments and computational savings. Numerical experiments on datasets, shapes, and images validate DSW as a scalable substitute for the WoW distance.

Paper Structure

This paper contains 32 sections, 17 theorems, 66 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Wasserstein Distances
  3. Sliced Wasserstein Distances
  4. Slicing Infinite Dimensional Banach Spaces
  5. Slicing the 1D Wasserstein Space
  6. Double-Slicing the Wasserstein Space
  7. Numerical Experiments
  8. Shape Classification via Local Distance Distributions
  9. Optimal Transport Dataset Distance
  10. Comparing Distributions of Point Clouds
  11. Comparing Image Distributions via their Patch Distributions
  12. Conclusion
  13. Non-Spherical Sliced Wasserstein Distance on Banach Spaces
  14. Double-Sliced Wasserstein Distance
  15. Metric Properties
  16. ...and 17 more sections

Key Result

Theorem 3.1

For $\xi \in \mathcal{P}_2(U^*)$, the $\xi$-based SW distance defines a well-defined pseudo-metric. If$\operatorname{supp} \xi \cap \operatorname{span} v \not \in \bigl\{ \emptyset, \{0\} \bigr\}$ for all $v \in U^* \setminus\{0\}$, then equation eq:sw-Banach defines a metric on $\mathcal{P}_2(U)$.

Figures (12)

  • Figure 1: Left: Impact of projection number $S$, grid size $R$, and kernel parameter $\sigma$ on MNIST-2000 SQW classification. Right: Scatter plots and correlation between TLB (1d WoW, horizontal axis) and SQW (vertical axis) for different parameters on MNIST-2000.
  • Figure 2: Scatter plots and correlations ($\uparrow$) between the s-OTDD and the OTDD (top) and our DSW ('Ours') and the OTDD (bottom) for MNIST (\ref{['subfig:MNIST']}), FashionMNIST (\ref{['subfig:FashionMNIST']}), and CIFAR-10 (\ref{['subfig:CIFAR']}).
  • Figure 3: OT-NNA, WoW, and our DSW ('Ours')between target and reference point cloud batches for varying numbers of shapes $M$ in target batch (\ref{['subfig:target_shapes']}), for Gaussian noise $\sigma_{\text{Noise}}$ for target points (\ref{['subfig:gaussian_target']}) and varying target point cloud resolution $m$ (\ref{['subfig:point_num_target']}). Fixed reference values are marked in red.
  • Figure 4: Comparing synthetic texture image batches via Euclidean Wasserstein and our sliced patch-based distance based on varying 'lacunarity' (\ref{['subfig:lac']}) and 'persistence' (\ref{['subfig:persistence']}). Both distances are minimized for 'true' parameters (red), but our DSW ('Ours') distance leads to clearer discrimination.
  • Figure 5: Plots of the support functions of the two empirical measure pairs from Section \ref{['subsection:slicing_functions']}.
  • ...and 7 more figures

Theorems & Definitions (29)

  • Theorem 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Corollary 3.3.1
  • Theorem 4.1
  • Lemma A.1
  • proof
  • Proposition A.2
  • proof
  • Theorem A.3
  • ...and 19 more