This paper considers a sparse spiked covariance matrix 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 (Ann Stat 41(4):1780–1815, 2013) where the special case of rank one is considered.

中文翻译：

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稀疏尖峰协方差矩阵的最优估计和秩检测

本文考虑了高维设置下的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极大极小估计以及极大极小秩检测。建立了用于估计谱范数下的尖峰协方差矩阵的最佳收敛速度，这需要与用于估计其他结构化协方差矩阵（例如带状或稀疏协方差矩阵）的技术截然不同的技术。我们还在谱范数下建立了极大极小率，用于估计主子空间，这是主成分分析中的主要关注对象。此外，还获得了秩检测边界的最佳速率。这一结果也解决了 Berthet 和 Rigollet 最近发表的一篇论文（Ann Stat 41(4):1780–1815，