Optimal Estimation and Rank Detection for Sparse Spiked Covariance Matrices.
Optimal Estimation and Rank Detection for Sparse Spiked Covariance Matrices.
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DOI：
10.1007/s00440-014-0562-z
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发表时间：
2015-04-01
影响因子：
2
通讯作者：
Wu Y
中科院分区：
文献类型：
--
作者：
Cai T;Ma Z;Wu Y
本文研究了高维稀疏尖峰协方差矩阵模型，研究了协方差矩阵和主子空间的Minimax估计以及Minimax秩检测。建立了谱范数下估计尖峰协方差矩阵的最优收敛速度，这需要与估计其他结构协方差矩阵（如可带或稀疏协方差矩阵）的技术显著不同的技术。我们还建立了谱范数下的极大极小率，用于估计主子空间，主子空间是主成分分析中感兴趣的主要对象。此外，最佳速率的秩检测边界得到。这一结果也解决了差距在最近的一篇论文中Berthet和Rigollet的特殊情况下的秩一被认为是。
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 where the special case of rank one is considered.