Sparse PCA: Optimal rates and adaptive estimation
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In this paper, the authors considered both minimax and adaptive estimation of the principal subspace in the high dimensional setting and established the optimal rates of convergence for estimating the subspace which are sharp with respect to all the parameters, thus providing a complete characterization of the difficulty of the estimation problem in terms of the convergence rate.Abstract:
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 high-dimensional multivariate regression problem. This method is potentially also useful for other related problems.read more
Citations
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Journal ArticleDOI
Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation
TL;DR: Minimax rates of convergence for estimating several classes of structured covariance and precision matrices, including bandable, Toeplitz, and sparse covariance matrices as well as sparse precisionMatrices, are given under the spectral norm loss.
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Optimal detection of sparse principal components in high dimension
TL;DR: In this article, a finite sample analysis of the detection levels for sparse principal components of a high-dimensional covariance matrix is performed, based on a sparse eigenvalue statistic.
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Minimax bounds for sparse PCA with noisy high-dimensional data
TL;DR: A lower bound on the minimax risk of estimators under the l2 loss is established, in the joint limit as dimension and sample size increase to infinity, under various models of sparsity for the population eigenvectors.
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Random matrix theory in statistics: A review
Debashis Paul,Alexander Aue +1 more
TL;DR: An overview of random matrix theory is given with the objective of highlighting the results and concepts that have a growing impact in the formulation and inference of statistical models and methodologies.
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Fantope Projection and Selection: A near-optimal convex relaxation of sparse PCA
TL;DR: A novel convex relaxation of sparse principal subspace estimation based on the convex hull of rank-d projection matrices (the Fantope) is proposed and implies the near-optimality of DSPCA (d'Aspremont et al. [1]) even when the solution is not rank 1.
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