## The Annals of Statistics

### Minimax sparse principal subspace estimation in high dimensions

#### Abstract

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 introduce two complementary notions of $\ell_{q}$ subspace sparsity: row sparsity and column sparsity. We prove nonasymptotic lower and upper bounds on the minimax subspace estimation error for $0\leq q\leq1$. The bounds are optimal for row sparse subspaces and nearly optimal for column sparse subspaces, they apply to general classes of covariance matrices, and they show that $\ell_{q}$ constrained estimates can achieve optimal minimax rates without restrictive spiked covariance conditions. Interestingly, the form of the rates matches known results for sparse regression when the effective noise variance is defined appropriately. Our proof employs a novel variational $\sin\Theta$ theorem that may be useful in other regularized spectral estimation problems.

#### Article information

Source
Ann. Statist., Volume 41, Number 6 (2013), 2905-2947.

Dates
First available in Project Euclid: 1 January 2014

https://projecteuclid.org/euclid.aos/1388545673

Digital Object Identifier
doi:10.1214/13-AOS1151

Mathematical Reviews number (MathSciNet)
MR3161452

Zentralblatt MATH identifier
1288.62103

#### Citation

Vu, Vincent Q.; Lei, Jing. Minimax sparse principal subspace estimation in high dimensions. Ann. Statist. 41 (2013), no. 6, 2905--2947. doi:10.1214/13-AOS1151. https://projecteuclid.org/euclid.aos/1388545673

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