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Sharp comparisons between sliced and standard $1$-Wasserstein distances

Guillaume Carlier, Alessio Figalli, Quentin Mérigot, Yi Wang

TL;DR

This work analyzes quantitative relationships between sliced Wasserstein distances and the standard Wasserstein distance. By leveraging mollification, Kantorovich duality, Radon/Fourier slicing, and integral-geometry tools, the authors prove an improved bound $\mathcal{W}_1(\mu,\nu) \le c_d R^{(d-1)/d} \mathcal{SW}_1(\mu,\nu)^{1/d}$ in odd dimensions (with a logarithmic correction in even dimensions) and establish that the exponent $1/d$ is sharp. They further extend the framework to slicings by $k$-planes, obtaining $\mathcal{W}_1(\mu,\nu) \le C_{d,k} R^{(d-k)/(d-k+1)} \mathcal{SW}_1^k(\mu,\nu)^{1/(d-k+1)}$, with proofs split between even and odd $d-k$ cases. A constructive counterexample confirms the sharpness of the $1/d$ exponent in all dimensions. Overall, the results clarify the reliability of sliced distances as surrogates for Wasserstein distances in high dimensions and connect the analysis to classical integral-geometry tools like the Radon transform and Riesz transforms.

Abstract

Sliced Wasserstein distances are widely used in practice as a computationally efficient alternative to Wasserstein distances in high dimensions. In this paper, motivated by theoretical foundations of this alternative, we prove quantitative estimates between the sliced $1$-Wasserstein distance and the $1$-Wasserstein distance. We construct a concrete example to demonstrate the exponents in the estimate is sharp. We also provide a general analysis for the case where slicing involves projections onto $k$-planes and not just lines.

Sharp comparisons between sliced and standard $1$-Wasserstein distances

TL;DR

This work analyzes quantitative relationships between sliced Wasserstein distances and the standard Wasserstein distance. By leveraging mollification, Kantorovich duality, Radon/Fourier slicing, and integral-geometry tools, the authors prove an improved bound in odd dimensions (with a logarithmic correction in even dimensions) and establish that the exponent is sharp. They further extend the framework to slicings by -planes, obtaining , with proofs split between even and odd cases. A constructive counterexample confirms the sharpness of the exponent in all dimensions. Overall, the results clarify the reliability of sliced distances as surrogates for Wasserstein distances in high dimensions and connect the analysis to classical integral-geometry tools like the Radon transform and Riesz transforms.

Abstract

Sliced Wasserstein distances are widely used in practice as a computationally efficient alternative to Wasserstein distances in high dimensions. In this paper, motivated by theoretical foundations of this alternative, we prove quantitative estimates between the sliced -Wasserstein distance and the -Wasserstein distance. We construct a concrete example to demonstrate the exponents in the estimate is sharp. We also provide a general analysis for the case where slicing involves projections onto -planes and not just lines.
Paper Structure (11 sections, 5 theorems, 108 equations)

This paper contains 11 sections, 5 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Sharp comparison between sliced Wasserstein and Wasserstein distances
  3. Notations and preliminaries
  4. Control of $\mathcal{W}_1$ by $\mathcal{SW}_1$.
  5. Proof of Theorem \ref{['th:ComparisonW_SW']} for odd $d$
  6. Proof of Theorem \ref{['th:ComparisonW_SW']} for even d.
  7. The exponent $\frac{1}{d}$ is sharp
  8. Comparison when slicing is performed by projections onto $k$-planes
  9. Some results from integral geometry
  10. Proof of Theorem \ref{['th:ComparisonW_SWk']} for even $d-k$
  11. Proof of Theorem \ref{['th:ComparisonW_SWk']} for odd $d-k$

Key Result

Theorem 2.1

For any dimension $d\geq 2$, there exists a constant $c_d$ such that for all $R>0$ and all $\mu,\nu$ in $\mathcal{P}(\mathbb{R}^d)$ supported on $B_R$, one has

Theorems & Definitions (9)

  • Theorem 2.1
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Theorem 3.1
  • Lemma 3.2
  • proof