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Correctly estimating the discrepancy between two data distributions hasalways been an important task in Machine Learning. Recently, Cuturi proposedthe Sinkhorn distance which makes use of an approximate Optimal Transport costbetween two distributions as a distance to describe distribution discrepancy.Although it has been successfully adopted in various machine learningapplications (e.g. in Natural Language Processing and Computer Vision) sincethen, the Sinkhorn distance also suffers from two unnegligible limitations. Thefirst one is that the Sinkhorn distance only gives an approximation of the realWasserstein distance, the second one is the `divide by zero' problem whichoften occurs during matrix scaling when setting the entropy regularizationcoefficient to a small value. In this paper, we introduce a new Brenierapproach for calculating a more accurate Wasserstein distance between twodiscrete distributions, this approach successfully avoids the two limitationsshown above for Sinkhorn distance and gives an alternative way for estimatingdistribution discrepancy.

Brenier approach for optimal transportation between a quasi-discrete measure and a discrete measure