Toland duality and particle approximations for signed Wasserstein barycenters
Abstract
We study the problem of minimizing a weighted sum of squared Wasserstein distances with signed coefficients. In the case of a single positive coefficient, we derive two convex dual formulations: one expressed in terms of Brenier potentials, and one as a projection problem in convex order. Such dual formulations arise via Toland duality by exploiting different notions of convexity within the problem. They generalize those recently established for the metric extrapolation problem, which corresponds to signed barycenters between only two measures. For general signed barycenters, we identify necessary optimality conditions that reduce the problem to the case with one positive coefficient. On the numerical side, we propose a grid-free, particle-based algorithm built on entropic regularization and Sinkhorn iterations to compute signed barycenters with arbitrary coefficients.
Related
- Capacity-Constrained Wasserstein Barycenters: Existence, Duality, and Entropic Regularization
- The p-Bures-Wasserstein distance via 3×3 block matrices (with application to p-Bures-Wasserstein barycenters)
- Sliced Wasserstein Barycenters: The Analysis Approach within the Barycentric Coding Model in the Wasserstein Space
- Towards Universal Wasserstein Barycenters through Flow Matching
- Quantitative stability of classical, entropic, and doubly entropic Wasserstein barycenters
BibTeX
@misc{gallouet2026toland,
title = {{Toland duality and particle approximations for signed Wasserstein barycenters}},
author = {Thomas O. Gallouët and Andrea Natale and Gabriele Todeschi},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17073},
eprint = {2609.17073},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05751291v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Toland duality and particle approximations for signed Wasserstein barycenters
AU - Thomas O. Gallouët
AU - Andrea Natale
AU - Gabriele Todeschi
PY - 2026
DA - 2026/09/15
PB - HAL
DO - 10.48550/arxiv.2609.17073
UR - https://hal.science/hal-05751291v1
AB - We study the problem of minimizing a weighted sum of squared Wasserstein distances with signed coefficients. In the case of a single positive coefficient, we derive two convex dual formulations: one expressed in terms of Brenier potentials, and one as a projection problem in convex order. Such dual formulations arise via Toland duality by exploiting different notions of convexity within the problem. They generalize those recently established for the metric extrapolation problem, which corresponds to signed barycenters between only two measures. For general signed barycenters, we identify necessary optimality conditions that reduce the problem to the case with one positive coefficient. On the numerical side, we propose a grid-free, particle-based algorithm built on entropic regularization and Sinkhorn iterations to compute signed barycenters with arbitrary coefficients.
ER -