Wasserstein convergence in Bayesian and frequentist deconvolution models

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 L1-Wasserstein distance between two distributions of the signal to the L1-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 L1-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 L1-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≥1, when only a tail condition is required on the latent mixing density and we derive sharp lower bounds for these problems.