Wasserstein convergence in Bayesian and frequentist deconvolution models
研究了从含已知分布误差的观测中恢复信号分布的多变量反卷积问题，推导出L1-Wasserstein距离的反演不等式，并应用于贝叶斯和频率学派框架，得到自适应最优收敛速率。
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 L1-Wasserstein and the max-sliced L1-Wasserstein metrics and 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. We apply it to derive L1-Wasserstein rates of convergence for the distribution of the signal. As an application to the Bayesian framework, we consider L1-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 L1-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 L1-Wasserstein deconvolution in any dimension d≥1, lower bounds being derived here.