Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance

TL;DR

This paper investigates the differential properties of the Sinkhorn approximation of Wasserstein distance, demonstrating its smoothness and providing an efficient gradient computation method, which benefits theoretical guarantees and practical optimization.

Contribution

It characterizes the smoothness of the Sinkhorn distance and offers an explicit, efficient gradient algorithm, enhancing both theoretical understanding and practical applications.

Findings

Sinkhorn distance shares the same smoothness as its regularized version.

An efficient algorithm for computing the gradient of the Sinkhorn distance is proposed.

Preliminary experiments show promising results in learning and optimization tasks.

Abstract

Applications of optimal transport have recently gained remarkable attention thanks to the computational advantages of entropic regularization. However, in most situations the Sinkhorn approximation of the Wasserstein distance is replaced by a regularized version that is less accurate but easy to differentiate. In this work we characterize the differential properties of the original Sinkhorn distance, proving that it enjoys the same smoothness as its regularized version and we explicitly provide an efficient algorithm to compute its gradient. We show that this result benefits both theory and applications: on one hand, high order smoothness confers statistical guarantees to learning with Wasserstein approximations. On the other hand, the gradient formula allows us to efficiently solve learning and optimization problems in practice. Promising preliminary experiments complement our analysis.