Sharp comparisons between sliced and standard 1-Wasserstein distances

2025/10/18 by Guillaume Carlier, Alessio Figalli, Carlier, Guillaume +5 · 1 citation
Mathematics · #39B62 #49Q22 #Commutative Algebra and Its Applications #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Optimization and Control (math.OC) #Point processes and geometric inequalities #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2510.16465

openalex publication_date 2025/10/18 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28

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.

Citations

• Sliced optimal transport: is it a suitable replacement?
• Geometry and analytic properties of the sliced Wasserstein space
• Spherical Sliced-Wasserstein
• Distribution Regression with Sliced Wasserstein Kernels
• Intrinsic Sliced Wasserstein Distances for Comparing Collections of Probability Distributions on Manifolds and Graphs
• Statistical and Topological Properties of Sliced Probability Divergences
• Max-Sliced Wasserstein Distance and its use for GANs
• Generalized Sliced Wasserstein Distances
• On the rate of convergence in Wasserstein distance of the empirical\n measure

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• A Lower Bound on the Sliced Wasserstein Comparison Constant for Translated Measures

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