Quantitative stability of classical, entropic, and doubly entropic Wasserstein barycenters
Abstract
In this paper, we show how the recent analysis of (sharp) strong convexity properties of the Kantorovich functional by the third author can be used to establish new global quantitative stability estimates for Wasserstein barycenters and their entropic regularizations with respect to perturbations of the population distribution. For classical Wasserstein barycenters, we improve the state-of-the-art H\"older continuity exponent with respect to the 𝒲1 metric from 1/6 to 1/4, while drastically weakening the assumptions on the population distribution. We also provide an example which shows that the exponent 1/4 cannot be improved. Motivated by computational considerations, we also analyze regularizations of Wasserstein barycenters with \emph{outer} and \emph{inner} entropic penalizations. For outer entropic barycenters, we establish novel higher-order stability results and in particular give a quantitative bound on relative Fisher information. Another contribution of our paper is a strong convexity estimate for an entropic version of the Kantorovich functional via soft Legendre transforms, enabling us to extend our stability results to \emph{doubly} entropic barycenters, as introduced by Chizat.
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BibTeX
@misc{carlier2026quantitative,
title = {{Quantitative stability of classical, entropic, and doubly entropic Wasserstein barycenters}},
author = {Guillaume Carlier and Julien Guérin and Quentin Mérigot},
howpublished = {HAL},
year = {2026},
month = jul,
url = {https://hal.science/hal-05704415v2},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Quantitative stability of classical, entropic, and doubly entropic Wasserstein barycenters
AU - Guillaume Carlier
AU - Julien Guérin
AU - Quentin Mérigot
PY - 2026
DA - 2026/07/26
PB - HAL
UR - https://hal.science/hal-05704415v2
AB - In this paper, we show how the recent analysis of (sharp) strong convexity properties of the Kantorovich functional by the third author can be used to establish new global quantitative stability estimates for Wasserstein barycenters and their entropic regularizations with respect to perturbations of the population distribution. For classical Wasserstein barycenters, we improve the state-of-the-art H\"older continuity exponent with respect to the 𝒲_1 metric from 1/6 to 1/4, while drastically weakening the assumptions on the population distribution. We also provide an example which shows that the exponent 1/4 cannot be improved. Motivated by computational considerations, we also analyze regularizations of Wasserstein barycenters with \emph{outer} and \emph{inner} entropic penalizations. For outer entropic barycenters, we establish novel higher-order stability results and in particular give a quantitative bound on relative Fisher information. Another contribution of our paper is a strong convexity estimate for an entropic version of the Kantorovich functional via soft Legendre transforms, enabling us to extend our stability results to \emph{doubly} entropic barycenters, as introduced by Chizat.
ER -