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

2026/07/29 by Avery Noroozi
Mathematics · #math.MG #math.OC #math.ST #msc:49Q22 #stat.TH

paper · pdf

6 pages

arxiv created 2026/07/29 · arxiv updated 2026/07/30

Abstract

We provide closed-form expressions for standard 1-Wasserstein distance (W₁) and Sliced 1-Wasserstein distance (SW₁) for translated probability measures on ℝ^d. Furthermore, we construct an explicit lower bound for the constant c_d established by Carlier, Figalli, Mérigot, and Wang between W₁ and SW₁ [1] for translated probability measures on ℝ^d. Sliced Wasserstein distances are computationally efficient and used in image generation and other applications, which raises natural questions in computational efficiency versus theoretical correctness. A lower bound on the constant c_d gives information about the cost of that efficiency.

Citations

• Sharp comparisons between sliced and standard 1-Wasserstein distances
• Fast Approximation of the Generalized Sliced-Wasserstein Distance
• Directional distributions and the half-angle principle

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• Sharp bounds for max-sliced Wasserstein distances
• Central limit theorem for the Sliced 1-Wasserstein distance and the max-Sliced 1-Wasserstein distance
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