  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.
