Inner entropic regularization ensures strictly convex transport subproblems but can bias the resulting barycenter. We instead consider the outer-regularized problem $\min_{\nu\in\mathcal{P}(\Omega)}\left\{\sum_{i=1}^N\lambda_i W_2^2(\nu,\mu_i)+\gamma\,\mathcal{R}(\nu)\right\}$, in which the regularizer acts on the barycenter measure alone while the transport costs remain unregularized. Our focus is on computing Wasserstein barycenters in the low-regularization regime, where controlling regularization bias requires accurate numerical reconstruction. We prove duality for a general class of convex regularizers and obtain a semi-dual formulation. The barycenter is recovered in closed form from the dual potentials, with the regularization functional determining the reconstruction profile. We maximize the nonsmooth dual objective using a Back-and-Forth scheme combining exact $c$-transforms with gradient ascent in a Sobolev space. Reducing the regularization decreases bias, but it also makes the reconstructed barycenter more sensitive to errors in the dual potentials. Numerical experiments with Gaussian distributions, three-dimensional shapes, and sketches illustrate the behavior of the method in the low-regularization regime.

This talk presents part of joint work with Denys Ruban and Augusto Gerolin.

Bio: Ivan Zhytkevych studied applied mathematics at Kyiv Polytechnic Institute and did his MSc in mathematics at the University of Ottawa. Ivan works on optimal transport and related problems. He is especially interested in mean field games and applications to machine learning. His current work with Denys Ruban and Augusto Gerolin studies the Back-and-Forth method for optimization over probability measures. What he likes most is the computational side of math: taking a method from theory to a solver that runs fast. Right now, he is interested in parallel computing on GPUs and is writing CUDA code for the core steps of optimal transport solvers.