Online Sinkhorn: Optimal Transport distances from sample streams

Part of Advances in Neural Information Processing Systems 33 pre-proceedings (NeurIPS 2020)

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Authors

Arthur Mensch, Gabriel Peyré

Abstract

<p>Optimal Transport (OT) distances are now routinely used as loss functions in ML tasks. Yet, computing OT distances between arbitrary (i.e. not necessarily discrete) probability distributions remains an open problem. This paper introduces a new online estimator of entropy-regularized OT distances between two such arbitrary distributions. It uses streams of samples from both distributions to iteratively enrich a non-parametric representation of the transportation plan. Compared to the classic Sinkhorn algorithm, our method leverages new samples at each iteration, which enables a consistent estimation of the true regularized OT distance. We provide a theoretical analysis of the convergence of the online Sinkhorn algorithm, showing a nearly-1/n asymptotic sample complexity for the iterate sequence. We validate our method on synthetic 1-d to 10-d data and on real 3-d shape data.</p>