Fields Colloquium

Entropic Brenier maps are regularized analogues of Brenier maps (optimal transport maps) which converge to Brenier maps as the regularization parameter shrinks (Pooladian and Niles-Weed, 2021). In this work, we prove quantitative stability bounds between entropic Brenier maps under variations of the target measure. In particular, when all measures have bounded support, we establish the optimal Lipschitz constant for the mapping from probability measures to entropic Brenier maps. This provides an exponential improvement to a result of Carlier, Chizat, and Laborde (2024). As an application, we prove near-optimal bounds for the stability of semi-discrete unregularized Brenier maps for a family of discrete target measures.

Aram-Alexandre Pooladian is a PhD student at New York University where he is supervised by Jonathan Niles-Weed. His research interests are at the intersection of theoretical and mathematical developments in optimal transport, and the development of large-scale algorithms for probabilistic inference. His research has been funded by NSERC, the NSF, Google, and Meta.