Sinkhorn Divergences for Unbalanced Optimal Transport

Abstract

Optimal transport induces the Earth Mover's (Wasserstein) distance between probability distributions, a geometric divergence that is relevant to a wide range of problems. Over the last decade, two relaxations of optimal transport have been studied in depth: unbalanced transport, which is robust to the presence of outliers and can be used when distributions don't have the same total mass; entropy-regularized transport, which is robust to sampling noise and lends itself to fast computations using the Sinkhorn algorithm. This paper combines both lines of work to put robust optimal transport on solid ground. Our main contribution is a generalization of the Sinkhorn algorithm to unbalanced transport: our method alternates between the standard Sinkhorn updates and the pointwise application of a contractive function. This implies that entropic transport solvers on grid images, point clouds and sampled distributions can all be modified easily to support unbalanced transport, with a proof of linear convergence that holds in all settings. We then show how to use this method to define pseudo-distances on the full space of positive measures that satisfy key geometric axioms: (unbalanced) Sinkhorn divergences are differentiable, positive, definite, convex, statistically robust and avoid any "entropic bias" towards a shrinkage of the measures' supports.

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• The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation
• Unbalanced Multi-Marginal Optimal Transport
• On Unbalanced Optimal Transport: An Analysis of Sinkhorn Algorithm
• Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey
• Transfer Operators from Optimal Transport Plans for Coherent Set Detection
• Unbalanced Sobolev Descent
• Entropy-Transport distances between unbalanced metric measure spaces
• Unbalanced Optimal Transport through Non-negative Penalized Linear Regression
• Optimal Transport for Change Detection on LiDAR Point Clouds
• Direct Measure Matching for Crowd Counting
• New Metrics Between Rational Spectra and their Connection to Optimal Transport