{"ID":2853406,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.16465","arxiv_id":"2510.16465","title":"Sharp comparisons between sliced and standard $1$-Wasserstein distances","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.","short_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 di...","url_abs":"https://arxiv.org/abs/2510.16465","url_pdf":"https://arxiv.org/pdf/2510.16465v1","authors":"[\"Guillaume Carlier\",\"Alessio Figalli\",\"Quentin Mérigot\",\"Yi Wang\"]","published":"2025-10-18T12:18:17Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"math.MG\",\"math.OC\"]","methods":"[]","has_code":false}