PreprintCC BY 4.0arXiv
λ-geodesic convexity of the squared sliced-Wasserstein distance
Abstract
The squared Wasserstein distance with respect to a fixed target measure is not a convex function along Wasserstein geodesics over ℝd when d≥ 2. Still, in dimension d=1, this function is $2$-geodesically convex. We prove that the square of the sliced-Wasserstein distance (which is a distance constructed averaging one-dimensional Wasserstein distances) is λ-geodesically convex (possibly with λ≤ 0) on sets of measures satisfying upper bounds on their density, when the target measure satisfies some given geometric conditions. We also present some consequences of this result in the study of the sliced-Wasserstein flow.
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BibTeX
@misc{cozzi2026λgeodesic,
title = {{$λ$-geodesic convexity of the squared sliced-Wasserstein distance}},
author = {Giacomo Cozzi and Filippo Santambrogio},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20972},
eprint = {2610.20972},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20972},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - λ-geodesic convexity of the squared sliced-Wasserstein distance
AU - Giacomo Cozzi
AU - Filippo Santambrogio
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20972
UR - https://arxiv.org/abs/2610.20972
AB - The squared Wasserstein distance with respect to a fixed target measure is not a convex function along Wasserstein geodesics over ℝ^d when d≥ 2. Still, in dimension d=1, this function is $2$-geodesically convex. We prove that the square of the sliced-Wasserstein distance (which is a distance constructed averaging one-dimensional Wasserstein distances) is λ-geodesically convex (possibly with λ≤ 0) on sets of measures satisfying upper bounds on their density, when the target measure satisfies some given geometric conditions. We also present some consequences of this result in the study of the sliced-Wasserstein flow.
ER -