估计协方差矩阵的特征向量和特征空间：一致性的最优界和条件

中文摘要 AI 辅助

设\(X = [ \xi_1, \,\, \xi_2,...\,\,,\xi_d]^\top\)是具有（隐藏）协方差矩阵\(M = (m_{ij})_{1 \leq i, j \leq d}\)的大维度\(d\)（\(d \rightarrow \infty\)）的零均值随机向量，其中\(m_{ij} = m_{ji} = \textbf{Cov}(\xi_i, \xi_j)\)。设\(X_1, X_2, \dots, X_n\)是\(X\)的\(n\)个独立同分布样本。考虑样本协方差矩阵\(\textstyle \tilde{M}:= \frac{1}{n} \sum_{i=1}^{n} X_i X_i^\top\)。实际中常用\(\tilde M\)的特征向量和特征空间估计\(M\)的。本文在对\(M\)的温和假设下，针对广泛的参数\(d\)和\(n\)，提供了最优误差分析，得到了匹配量级的上下界。作为推论，得到了估计量一致性的新的充要条件，其中仅要求样本数量\(n\)线性依赖于\(M\)的有效秩，其可比维度\(d\)小得多。

英文摘要

Let $X = [ ξ_1, \,\, ξ_2,...\,\, ,ξ_d]^\top$ be a zero-mean random vector of large dimension $d$ ($d \rightarrow \infty$) with (hidden) covariance matrix $M = (m_{ij})_{1 \leq i, j \leq d},$ where $m_{ij} = m_{ji} = \textbf{Cov}(ξ_i, ξ_j).$ Let $X_1, X_2, \dots, X_n$ be $n$ iid samples of $X$. Consider the sample covariance matrix $$\textstyle \tilde{M} := \frac{1}{n} \sum_{i=1}^{n} X_i X_i^\top.$$ In practice, one frequently uses the eigenvectors and eigenspaces of $\tilde M$ as estimators for those of $M$. A central task is to provide an error analysis for these estimators. In this paper, we provide an optimal error analysis, obtaining upper and lower bounds of matching order of magnitude, for a wide range of parameters $d$ and $n$, under mild assumptions on $M$. As corollaries, we obtain new necessary and sufficient conditions for the consistency of the estimators. In these conditions, we only require the number of samples $n$ to depend linearly on the effective rank of $M$, which can be much smaller than the dimension $d$.

发表机构

• The University of Hong Kong（香港大学）

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