PreprintarXiv licencearXiv
On the stability of proximal operators in Wasserstein spaces under different notions of convexity
Abstract
The proximal operator is a fundamental tool in variational analysis and optimization. In the setting of a Hilbert space, given a proper, lower semicontinuous convex functional, its proximal operator is non-expansive, that is, 1-Lipschitz continuous. In the Wasserstein setting, the contraction properties of this operator have been investigated from different perspectives by Carlen and Craig and by Adve and Mészáros, among others, and are not completely understood. In this paper, we study the stability properties of proximal maps, with a particular focus on non-expansivity, under various notions of convexity of the functional that can be considered in the Wasserstein space.
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BibTeX
@misc{marino2026stability,
title = {{On the stability of proximal operators in Wasserstein spaces under different notions of convexity}},
author = {Simone Di Marino and Sara Farinelli and Emanuele Naldi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.08209},
eprint = {2607.08209},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08209},
}
RIS
TY - UNPB
TI - On the stability of proximal operators in Wasserstein spaces under different notions of convexity
AU - Simone Di Marino
AU - Sara Farinelli
AU - Emanuele Naldi
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2607.08209
UR - https://arxiv.org/abs/2607.08209
AB - The proximal operator is a fundamental tool in variational analysis and optimization. In the setting of a Hilbert space, given a proper, lower semicontinuous convex functional, its proximal operator is non-expansive, that is, 1-Lipschitz continuous. In the Wasserstein setting, the contraction properties of this operator have been investigated from different perspectives by Carlen and Craig and by Adve and Mészáros, among others, and are not completely understood. In this paper, we study the stability properties of proximal maps, with a particular focus on non-expansivity, under various notions of convexity of the functional that can be considered in the Wasserstein space.
ER -