Wasserstein convergence in Bayesian and frequentist deconvolution models

TL;DR

This paper establishes a new inversion inequality for multivariate deconvolution, enabling nearly optimal Bayesian and frequentist estimation of the signal distribution under Wasserstein metrics, especially with Laplace noise.

Contribution

It introduces a superior inversion inequality for multivariate deconvolution and demonstrates its application in Bayesian adaptive estimation and frequentist minimax rates.

Findings

New inversion inequality outperforms existing bounds.

Bayesian posterior concentrates at near-minimax rate in $L^1$-Wasserstein distance.

Kernel deconvolution estimator attains minimax rate under tail conditions.

Abstract

We study the multivariate deconvolution problem of recovering the distribution of a signal from independent and identically distributed observations additively contaminated with random errors (noise) from a known distribution. For errors with independent coordinates having ordinary smooth densities, we derive an inversion inequality relating the $L^1$-Wasserstein distance between two distributions of the signal to the $L^1$-distance between the corresponding mixture densities of the observations. This smoothing inequality outperforms existing inversion inequalities. As an application of the inversion inequality to the Bayesian framework, we consider $1$-Wasserstein deconvolution with Laplace noise in dimension one using a Dirichlet process mixture of normal densities as a prior measure on the mixing distribution (or distribution of the signal). We construct an adaptive approximation of the sampling density by convolving the Laplace density with a well-chosen mixture of normal densities and show that the posterior measure concentrates around the sampling density at a nearly minimax rate, up to a log-factor, in the $L^1$-distance. The same posterior law is also shown to automatically adapt to the unknown Sobolev regularity of the mixing density, thus leading to a new Bayesian adaptive estimation procedure for mixing distributions with regular densities under the $L^1$-Wasserstein metric. We illustrate utility of the inversion inequality also in a frequentist setting by showing that an appropriate isotone approximation of the classical kernel deconvolution estimator attains the minimax rate of convergence for $1$-Wasserstein deconvolution in any dimension $d\geq 1$, when only a tail condition is required on the latent mixing density and we derive sharp lower bounds for these problems