Minimax Rates of Estimation for Sparse PCA in High Dimensions
نویسندگان
چکیده
We study sparse principal components analysis in the high-dimensional setting, where p (the number of variables) can be much larger than n (the number of observations). We prove optimal, non-asymptotic lower and upper bounds on the minimax estimation error for the leading eigenvector when it belongs to an lq ball for q ∈ [0, 1]. Our bounds are sharp in p and n for all q ∈ [0, 1] over a wide class of distributions. The upper bound is obtained by analyzing the performance of lqconstrained PCA. In particular, our results provide convergence rates for l1-constrained PCA.
منابع مشابه
Sparse CCA: Adaptive Estimation and Computational Barriers
Canonical correlation analysis (CCA) is a classical and important multivariate technique for exploring the relationship between two sets of variables. It has applications in many fields including genomics and imaging, to extract meaningful features as well as to use the features for subsequent analysis. This paper considers adaptive and computationally tractable estimation of leading sparse can...
متن کاملRate-optimal Posterior Contraction for Sparse Pca
Principal component analysis (PCA) is possibly one of the most widely used statistical tools to recover a low rank structure of the data. In the high-dimensional settings, the leading eigenvector of the sample covariance can be nearly orthogonal to the true eigenvector. A sparse structure is then commonly assumed along with a low rank structure. Recently, minimax estimation rates of sparse PCA ...
متن کاملRate-optimal Posterior Contraction for Sparse Pca By
Principal component analysis (PCA) is possibly one of the most widely used statistical tools to recover a low-rank structure of the data. In the highdimensional settings, the leading eigenvector of the sample covariance can be nearly orthogonal to the true eigenvector. A sparse structure is then commonly assumed along with a low rank structure. Recently, minimax estimation rates of sparse PCA w...
متن کاملSparse PCA: Optimal Rates and Adaptive Estimation
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...
متن کاملMinimax Sparse Principal Subspace Estimation in High Dimensions
We study sparse principal components analysis in high dimensions , where p (the number of variables) can be much larger than n (the number of observations), and analyze the problem of estimating the subspace spanned by the principal eigenvectors of the population covariance matrix. We prove optimal, non-asymptotic lower and upper bounds on the minimax subspace estimation error under two differe...
متن کامل