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

arXiv: https://arxiv.org/abs/2302.00982 · HTML: https://mlsanity.com/paper/2302.00982/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 47 references that are in the map.
- First posted on arXiv 2 Feb 2023.

## Builds on (most cited first, 17 of 17)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Wasserstein-2 Generative Networks](https://mlsanity.com/paper/1909.13082/) (2019)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Faster Wasserstein Distance Estimation with the Sinkhorn Divergence](https://mlsanity.com/paper/2006.08172/) (2020)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Supervised Training of Conditional Monge Maps](https://mlsanity.com/paper/2206.14262/) (2022)
- [Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections](https://mlsanity.com/paper/2107.01718/) (2021)
- [Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression](https://mlsanity.com/paper/1303.6149/) (2013)
- [Stability of Entropic Optimal Transport and Schr"odinger Bridges](https://mlsanity.com/paper/2106.03670/) (2021)
- [Convergence of a Newton algorithm for semi-discrete optimal transport](https://mlsanity.com/paper/1603.05579/) (2016)
- [Near-optimal estimation of smooth transport maps with kernel sums-of-squares](https://mlsanity.com/paper/2112.01907/) (2021)
- [Center-outward quantiles and the measurement of multivariate risk](https://mlsanity.com/paper/1912.04924/) (2019)
- [Multivariate rank via entropic optimal transport: sample efficiency and generative modeling](https://mlsanity.com/paper/2111.00043/) (2021)

## Abstract (arXiv)

We introduce a new stochastic algorithm for solving entropic optimal transport (EOT) between two absolutely continuous probability measures μ and nu. Our work is motivated by the specific setting of Monge-Kantorovich quantiles where the source measure μ 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.