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

Guillaume Carlier; Alessio Figalli; Quentin Mérigot; Yi Wang

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arXiv · 2510.16465

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

Guillaume Carlier·Alessio Figalli·Quentin Mérigot·Yi Wang

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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.

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BibTeX RIS

Guillaume Carlier, Alessio Figalli, Quentin Mérigot, Yi Wang. 2025-10-18. Sharp comparisons between sliced and standard $1$-Wasserstein distances. https://arxiv.org/abs/2510.16465

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