A Dual Method for Outer-Regularized Wasserstein Barycenters with Exact Transport Costs

Outer-regularized Wasserstein barycenters preserve the exact quadratic Wasserstein transport terms while regularizing only the barycenter. We introduce BaFFO, a Back-and-Forth dual ascent method based on exact $c$-transforms and explicit reconstruction of the barycenter from dual potentials. The method is designed for the low-regularization regime and avoids entropic smoothing of the transport problems. Experiments show accurate reconstruction, preservation of sharp connected geometry, and exact sparsity with compact-support regularizers.