  In recent years, sparse principal component analysis has emerged as an
extremely popular dimension reduction technique for high-dimensional data. The
theoretical challenge, in the simplest case, is to estimate the leading
eigenvector of a population covariance matrix under the assumption that this
eigenvector is sparse. An impressive range of estimators have been proposed;
some of these are fast to compute, while others are known to achieve the
minimax optimal rate over certain Gaussian or sub-Gaussian classes. In this
paper, we show that, under a widely-believed assumption from computational
complexity theory, there is a fundamental trade-off between statistical and
computational performance in this problem. More precisely, working with new,
larger classes satisfying a restricted covariance concentration condition, we
show that there is an effective sample size regime in which no randomised
polynomial time algorithm can achieve the minimax optimal rate. We also study
the theoretical performance of a (polynomial time) variant of the well-known
semidefinite relaxation estimator, revealing a subtle interplay between
statistical and computational efficiency.
