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

TL;DR

This paper introduces a Brenier-based method for more accurately computing the Wasserstein distance between a quasi-discrete and a discrete distribution, overcoming limitations of the Sinkhorn distance.

Contribution

The paper proposes a novel Brenier approach that improves Wasserstein distance estimation by avoiding approximation errors and numerical issues inherent in Sinkhorn distance.

Findings

The Brenier approach provides more accurate Wasserstein distance estimates.

It successfully avoids the divide-by-zero problem in matrix scaling.

The method outperforms Sinkhorn distance in accuracy.

Abstract

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.