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

翻译：我们研究多元反卷积问题：从独立同分布观测中恢复信号分布，这些观测受到已知分布随机误差（噪声）的加性污染。当误差具有独立坐标且服从普通光滑密度时，我们推导出一个反演不等式，将信号的两个分布之间的$L^1$-Wasserstein距离与对应观测混合密度之间的$L^1$-距离联系起来。该平滑不等式优于现有反演不等式。作为该反演不等式在贝叶斯框架中的应用，我们考虑一维拉普拉斯噪声下的$1$-Wasserstein反卷积，采用正态密度的狄利克雷过程混合作为混合分布（即信号分布）的先验测度。通过将拉普拉斯密度与精心选择的正态混合密度进行卷积，我们构建了采样密度的自适应逼近，并证明后验测度以$L^1$-距离下接近极小极大最优速率（对数因子范围内）集中于采样密度。该后验分布还能自动适应混合密度的未知Sobolev正则性，从而在$L^1$-Wasserstein度量下为具有正则密度的混合分布提出一种新的贝叶斯自适应估计程序。我们还在频率学派框架下展示了该反演不等式的效用：通过证明经典核反卷积估计量的适当保序逼近在任意维度$d\geq 1$上达到$1$-Wasserstein反卷积的极小极大收敛速率（仅需潜在混合密度满足尾部条件），并为这些问题推导出尖锐下界。