Minimax rates of convergence for Wasserstein deconvolution with supersmooth errors in any dimension

TL;DR

This paper establishes optimal convergence rates for estimating probability measures in Wasserstein distance when data is contaminated with supersmooth noise, extending results to any dimension.

Contribution

It provides the first comprehensive analysis of Wasserstein deconvolution rates with supersmooth errors in arbitrary dimensions, leveraging lower bounds from one-dimensional cases.

Findings

Derived optimal convergence rates for Wasserstein deconvolution with supersmooth errors.

Extended lower bounds from one-dimensional deconvolution to higher dimensions.

Demonstrated the impact of supersmooth error distributions on estimation accuracy.

Abstract

The subject of this paper is the estimation of a probability measure on ${\mathbb R}^d$ from data observed with an additive noise, under the Wasserstein metric of order $p$ (with $p\geq 1$). We assume that the distribution of the errors is known and belongs to a class of supersmooth distributions, and we give optimal rates of convergence for the Wasserstein metric of order $p$. In particular, we show how to use the existing lower bounds for the estimation of the cumulative distribution function in dimension one to find lower bounds for the Wasserstein deconvolution in any dimension.