标题： 高维中的最小最大稀疏主子空间估计 显示英文标题

标题： Minimax Sparse Principal Subspace Estimation in High Dimensions

摘要： 我们研究高维稀疏主成分分析，其中$p$（变量数量）可能远大于$n$（观测数量），并分析估计总体协方差矩阵的主特征向量张成子空间的问题。 我们在$0 \leq q \leq 1$的两种不同但相关的$\ell_q$子空间稀疏性概念下，证明了最小最大子空间估计误差的最优非渐近下界和上界。 我们的上界适用于一般的协方差矩阵类，并表明$\ell_q$约束估计可以在不需要严格尖峰协方差条件的情况下达到最优最小最大速率。 显示英文摘要

摘要： We study sparse principal components analysis in high dimensions, where $p$ (the number of variables) can be much larger than $n$ (the number of observations), and analyze the problem of estimating the subspace spanned by the principal eigenvectors of the population covariance matrix. We prove optimal, non-asymptotic lower and upper bounds on the minimax subspace estimation error under two different, but related notions of $\ell_q$ subspace sparsity for $0 \leq q \leq 1$. Our upper bounds apply to general classes of covariance matrices, and they show that $\ell_q$ constrained estimates can achieve optimal minimax rates without restrictive spiked covariance conditions.