Sparse PCA: Optimal rates and adaptive estimation

T. Tony Cai, Zongming Ma, Yihong Wu

Research output: Contribution to journalArticlepeer-review


Principal component analysis (PCA) is one of the most commonly used statistical procedures with a wide range of applications. This paper considers both minimax and adaptive estimation of the principal subspace in the high dimensional setting. Under mild technical conditions, we first establish the optimal rates of convergence for estimating the principal subspace which are sharp with respect to all the parameters, thus providing a complete characterization of the difficulty of the estimation problem in term of the convergence rate. The lower bound is obtained by calculating the local metric entropy and an application of Fano's lemma. The rate optimal estimator is constructed using aggregation, which, however, might not be computationally feasible. We then introduce an adaptive procedure for estimating the principal subspace which is fully data driven and can be computed efficiently. It is shown that the estimator attains the optimal rates of convergence simultaneously over a large collection of the parameter spaces. A key idea in our construction is a reduction scheme which reduces the sparse PCA problem to a highdimensional multivariate regression problem. This method is potentially also useful for other related problems.

Original languageEnglish (US)
Pages (from-to)3074-3110
Number of pages37
JournalAnnals of Statistics
Issue number6
StatePublished - Dec 2013
Externally publishedYes


  • Adaptive estimation
  • Aggregation
  • Covariance matrix
  • Eigenvector
  • Group sparsity
  • Low-rank matrix
  • Minimax lower bound
  • Optimal rate of convergence
  • Principal component analysis
  • Thresholding

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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