A Dimension-free Computational Upper-bound for Smooth Optimal Transport Estimation

TL;DR

This paper introduces a new estimator for smooth optimal transport that achieves dimension-free statistical and computational rates, overcoming the curse of dimensionality with an innovative sum-of-squares approach.

Contribution

It presents a novel infinite-dimensional sum-of-squares representation enabling a dimension-free estimator for smooth optimal transport.

Findings

Achieves $ ilde{O}( ext{epsilon}^{-2})$ sample complexity independent of dimension.

Computational cost is $ ilde{O}( ext{epsilon}^{-4})$, also dimension-free.

Potentially exponential constants depending on dimension are acknowledged.

Abstract

It is well-known that plug-in statistical estimation of optimal transport suffers from the curse of dimensionality. Despite recent efforts to improve the rate of estimation with the smoothness of the problem, the computational complexity of these recently proposed methods still degrades exponentially with the dimension. In this paper, thanks to an infinite-dimensional sum-of-squares representation, we derive a statistical estimator of smooth optimal transport which achieves a precision $\varepsilon$ from $\tilde{O}(\varepsilon^{-2})$ independent and identically distributed samples from the distributions, for a computational cost of $\tilde{O}(\varepsilon^{-4})$ when the smoothness increases, hence yielding dimension-free statistical and computational rates, with potentially exponentially dimension-dependent constants.