Optimal estimation and rank detection for sparse spiked covariance matrices

Tony Cai, Zongming Ma, Yihong Wu

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)781-815
Number of pages35
JournalProbability Theory and Related Fields
Issue number3-4
StatePublished - Apr 1 2015
Externally publishedYes


  • Covariance matrix
  • Group sparsity
  • Low-rank matrix
  • Minimax rate of convergence
  • Principal subspace
  • Rank detection
  • Sparse principal component analysis

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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