Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithm
This provides a method for computing barycenters in optimal transport, which is incremental as it builds on existing Frank-Wolfe strategies for Sinkhorn divergences.
The paper tackles the problem of estimating barycenters of probability distributions using the Sinkhorn divergence by proposing a Frank-Wolfe algorithm that incrementally populates support without pre-allocation, achieving proven convergence rates for discrete and continuous distributions.
We present a novel algorithm to estimate the barycenter of arbitrary probability distributions with respect to the Sinkhorn divergence. Based on a Frank-Wolfe optimization strategy, our approach proceeds by populating the support of the barycenter incrementally, without requiring any pre-allocation. We consider discrete as well as continuous distributions, proving convergence rates of the proposed algorithm in both settings. Key elements of our analysis are a new result showing that the Sinkhorn divergence on compact domains has Lipschitz continuous gradient with respect to the Total Variation and a characterization of the sample complexity of Sinkhorn potentials. Experiments validate the effectiveness of our method in practice.