Wasserstein convergence in Bayesian and frequentist deconvolution models
贝叶斯与频率学派反卷积模型中的 Wasserstein 收敛AI 解读 ↓
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贝叶斯与频率学派反卷积模型中的 Wasserstein 收敛
- 研究问题
- 在观测被已知分布的加性误差污染时，如何恢复多维信号分布，并刻画 Wasserstein 距离下的估计收敛速度？
- 方法与思路
- 作者借助 Wasserstein 距离与最大切片距离的等价关系构造反演不等式，再用于贝叶斯后验收缩和频率学派最小距离估计。
- 材料陈述的贡献
- 摘要称，新不等式支持收敛速率分析；一维 Laplace 噪声下得到自适应贝叶斯结果，多维最小距离估计达到所述极小极大速率。
- 阅读关注点
- 核对普通光滑误差的定义、距离等价的条件与常数，并区分一般多维结论和一维贝叶斯应用中的损失函数及对数因子。
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- 仅据版本未注明的 OpenAlex 索引摘要；具体正则条件与速率公式未说明，不能把一维 Laplace 贝叶斯结果推广到所有误差分布。
原文摘要
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.