Wasserstein convergence in Bayesian and frequentist deconvolution models

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 having known distribution. For errors with ordinary smooth distribution, we recast the multidimensional problem as a one-dimensional problem leveraging the equivalence between the ${\mathit{L}}^1$-Wasserstein and the max-sliced ${\mathit{L}}^1$-Wasserstein metrics and derive an inversion inequality relating the ${\mathit{L}}^1$-Wasserstein distance between two distributions of the signal to the ${\mathit{L}}^1$-distance between the corresponding mixture densities of the observations. This smoothing inequality outperforms existing inversion inequalities. We apply it to derive ${\mathit{L}}^1$-Wasserstein rates of convergence for the distribution of the signal. As an application to the Bayesian framework, we consider ${\mathit{L}}^1$-Wasserstein deconvolution with the Laplace noise in dimension one using a Dirichlet process mixture of normal densities as a prior measure for the mixing distribution. We construct an adaptive approximation of the sampling density by convolving the Laplace density with a well-chosen mixture of Gaussian densities and show that the posterior measure contracts at a nearly minimax-optimal rate, up to a log-factor, in the ${\mathit{L}}^1$-distance. The rate automatically adapts to the unknown Sobolev regularity of the mixing density, thus leading to a new Bayesian adaptive estimation procedure over the full scale of regularity levels. We illustrate the utility of the inversion inequality also in a frequentist setting by showing that a minimum distance estimator attains the minimax convergence rates for ${\mathit{L}}^1$-Wasserstein deconvolution in any dimension $\mathit{d}\ge 1$, lower bounds being derived here.