Stochastic optimal transport in Banach Spaces for regularized estimation of multivariate quantiles

TL;DR

This paper presents a fast stochastic algorithm leveraging Fourier transforms to solve entropic optimal transport problems in Banach spaces, specifically for estimating multivariate quantiles with regularization.

Contribution

It introduces a novel stochastic method using Fourier coefficients and FFT for efficient entropic optimal transport computation in Banach spaces, tailored for multivariate quantile estimation.

Findings

The algorithm converges almost surely in an infinite-dimensional Banach space.

Numerical experiments demonstrate the method's efficiency in computing regularized multivariate quantiles.

Entropic regularization improves the smoothness and estimation quality of multivariate quantiles.

Abstract

We introduce a new stochastic algorithm for solving entropic optimal transport (EOT) between two absolutely continuous probability measures $\mu$ and $\nu$. Our work is motivated by the specific setting of Monge-Kantorovich quantiles where the source measure $\mu$ is either the uniform distribution on the unit hypercube or the spherical uniform distribution. Using the knowledge of the source measure, we propose to parametrize a Kantorovich dual potential by its Fourier coefficients. In this way, each iteration of our stochastic algorithm reduces to two Fourier transforms that enables us to make use of the Fast Fourier Transform (FFT) in order to implement a fast numerical method to solve EOT. We study the almost sure convergence of our stochastic algorithm that takes its values in an infinite-dimensional Banach space. Then, using numerical experiments, we illustrate the performances of our approach on the computation of regularized Monge-Kantorovich quantiles. In particular, we investigate the potential benefits of entropic regularization for the smooth estimation of multivariate quantiles using data sampled from the target measure $\nu$.