Unbalanced Optimal Transport and Maximum Mean Discrepancies: Interconnections and Rapid Evaluation

TL;DR

This paper introduces computationally efficient methods for comparing measures with different masses using unbalanced optimal transport and maximum mean discrepancies, leveraging fast Fourier transforms to significantly reduce computation time.

Contribution

It presents novel accelerated algorithms for unbalanced optimal transport and maximum mean discrepancies, establishing their interconnections and enabling efficient analysis of large, high-dimensional datasets.

Findings

Reduced computational complexity from O(n^2) to O(n log n)

Established theoretical links between transportation distances and MMD

Enabled scalable analysis of large datasets

Abstract

This contribution presents substantial computational advancements to compare measures even with varying masses. Specifically, we utilize the nonequispaced fast Fourier transform to accelerate the radial kernel convolution in unbalanced optimal transport approximation, built upon the Sinkhorn algorithm. We also present accelerated schemes for maximum mean discrepancies involving kernels. Our approaches reduce the arithmetic operations needed to compute distances from $\mathcal O(n^2)$ to $\mathcal O(n\log n)$, opening the door to handle large and high-dimensional datasets efficiently. Furthermore, we establish robust connections between transportation problems, encompassing Wasserstein distance and unbalanced optimal transport, and maximum mean discrepancies. This empowers practitioners with compelling rationale to opt for adaptable distances.