Cai Tony, Ma Zongming, Wu Yihong
Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA 19104.
Probab Theory Relat Fields. 2015 Apr 1;161(3-4):781-815. doi: 10.1007/s00440-014-0562-z.
This paper considers a sparse spiked covariancematrix model in the high-dimensional setting and studies the minimax estimation of the covariance matrix and the principal subspace as well as the minimax rank detection. The optimal rate of convergence for estimating the spiked covariance matrix under the spectral norm is established, which requires significantly different techniques from those for estimating other structured covariance matrices such as bandable or sparse covariance matrices. We also establish the minimax rate under the spectral norm for estimating the principal subspace, the primary object of interest in principal component analysis. In addition, the optimal rate for the rank detection boundary is obtained. This result also resolves the gap in a recent paper by Berthet and Rigollet [2] where the special case of rank one is considered.
本文考虑了高维情形下的稀疏尖峰协方差矩阵模型，并研究了协方差矩阵和主子空间的极小极大估计以及极小极大秩检测。建立了在谱范数下估计尖峰协方差矩阵的最优收敛速率，这需要与估计其他结构化协方差矩阵（如实带型或稀疏协方差矩阵）所使用的技术有显著不同的技术。我们还建立了在谱范数下估计主子空间（主成分分析中感兴趣的主要对象）的极小极大速率。此外，得到了秩检测边界的最优速率。该结果也解决了Berthet和Rigollet [2]最近一篇论文中在考虑秩为一的特殊情形时存在的差距。
