Sinkhorn Distances: Lightspeed Computation of Optimal Transport

Optimal transport distances are a fundamental family of distances for probability measures and histograms of features. Despite their appealing theoretical properties, excellentperformance in retrieval tasks and intuitive formulation, their computation involvesthe resolution of a linear program whose cost can quickly become prohibitive whenever the size of the support of these measures or the histograms' dimensionexceeds a few hundred. We propose in this work a new family of optimal transport distances that look at transport problems from a maximumentropy perspective.We smooth the classic optimal transport problem with an entropic regularization term, and show that the resulting optimum is also a distance whichcan be computed through Sinkhorn's matrix scaling algorithm at a speed that is several orders of magnitude faster than that of transport solvers. We also show that this regularized distance improves upon classic optimal transport distances on the MNIST classification problem.