AI 中文总结
该研究针对高维线性回归的经验贝叶斯先验估计问题，提出了计算高效的EBMoM方法，证明其在亚线性样本复杂度下可一致估计先验，且该样本复杂度为最优，优于现有基于似然的方法。
AI 中文摘要
我们研究高维线性回归模型$\boldsymbol{y}=\boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\theta}$中先验的经验贝叶斯估计，其中回归系数独立地取自未知的次高斯先验。与序列模型不同，该设计矩阵耦合了潜在系数，因此要恢复先验需要从噪声和自身的副本中对其进行解卷积。我们提出了经验贝叶斯矩方法（Empirical Bayes Method of Moments，EBMoM），这是一种适用于一般设计的计算高效的方法，它通过下三角估计方程组递归估计先验矩，运行时间为$O(np^2)$。在温和的设计条件下（尤其是广泛的相关随机设计都满足该条件），我们证明当$n/p^{1-o(1)}$时，EBMoM能够一致估计不断增长数量的矩，进而估计先验本身。针对广泛的设计，我们给出了匹配的信息论下界，表明这种亚线性样本复杂度对于非参数先验估计是最优的。这改进了现有基于似然的方法的结果，后者的一致性要求样本量满足$n=\boldsymbol{\theta}(p)$。
英文摘要
We study empirical Bayes estimation of the prior in high-dimensional linear regression $\mathbf{y}=\mathbf{X}\mathbfβ+\mathbf{\varepsilon}$, where the regression coefficients are drawn independently from an unknown sub-Gaussian prior. In contrast to the sequence model, the design matrix couples the latent coefficients, so that recovering the prior requires deconvolving it from both the noise and copies of itself. We introduce the \emph{Empirical Bayes Method of Moments} (EBMoM), a computationally efficient procedure for general designs that recursively estimates the prior moments through a lower-triangular system of estimating equations and runs in time $O(np^2)$. Under mild design conditions, satisfied in particular by a broad class of correlated random designs, we show that EBMoM consistently estimates a growing number of moments and hence the prior itself, provided that $n\geq p^{1-o(1)}$. A matching information-theoretic lower bound, valid for a broad class of designs, shows that this sub-linear sample complexity is optimal for nonparametric prior estimation. This improves on existing results for likelihood-based methods whose consistency requires a linear sample size $n=Ω(p)$.
