Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithm

Part of Advances in Neural Information Processing Systems 32 (NeurIPS 2019)

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Giulia Luise, Saverio Salzo, Massimiliano Pontil, Carlo Ciliberto


<p>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.</p>