Nonequispaced Fast Fourier Transform Boost for the Sinkhorn Algorithm

TL;DR

This paper introduces an accelerated method using nonequispaced fast Fourier transforms to compute the Wasserstein distance via the Sinkhorn algorithm more efficiently, enabling applications to larger datasets.

Contribution

It presents a novel NFFT-based acceleration for the Sinkhorn algorithm, reducing computational complexity and memory usage for Wasserstein distance approximation.

Findings

Achieves $ ext{O}(n ext{log} n)$ complexity for probability measures with n points.

Avoids expensive matrix allocations, improving efficiency.

Numerical experiments confirm computational advantages and accuracy.

Abstract

This contribution features an accelerated computation of the Sinkhorn's algorithm, which approximates the Wasserstein transportation distance, by employing nonequispaced fast Fourier transforms (NFFT). The algorithm proposed allows approximations of the Wasserstein distance by involving not more than $\mathcal O(n\log n)$ operations for probability measures supported by~$n$ points. Furthermore, the proposed method avoids expensive allocations of the characterizing matrices. With this numerical acceleration, the transportation distance is accessible to probability measures out of reach so far. Numerical experiments using synthetic and real data affirm the computational advantage and superiority.