  Correctly estimating the discrepancy between two data distributions has
always been an important task in Machine Learning. Recently, Cuturi proposed
the Sinkhorn distance which makes use of an approximate Optimal Transport cost
between two distributions as a distance to describe distribution discrepancy.
Although it has been successfully adopted in various machine learning
applications (e.g. in Natural Language Processing and Computer Vision) since
then, the Sinkhorn distance also suffers from two unnegligible limitations. The
first one is that the Sinkhorn distance only gives an approximation of the real
Wasserstein distance, the second one is the `divide by zero' problem which
often occurs during matrix scaling when setting the entropy regularization
coefficient to a small value. In this paper, we introduce a new Brenier
approach for calculating a more accurate Wasserstein distance between two
discrete distributions, this approach successfully avoids the two limitations
shown above for Sinkhorn distance and gives an alternative way for estimating
distribution discrepancy.
