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

Abstract

We provide closed-form expressions for standard 1-Wasserstein distance ($W_1$) and Sliced 1-Wasserstein distance ($S{W}_1$) for translated probability measures on ${\mathbb{R}}^d$. Furthermore, we construct an explicit lower bound for the constant $c_d$ established by Carlier, Figalli, Mérigot, and Wang between $W_1$ and $S{W}_1$ [1] for translated probability measures on ${\mathbb{R}}^d$. 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.

6 pages

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