  In this paper, we study the estimation of the $k$-dimensional sparse
principal subspace of covariance matrix $\Sigma$ in the high-dimensional
setting. We aim to recover the oracle principal subspace solution, i.e., the
principal subspace estimator obtained assuming the true support is known a
priori. To this end, we propose a family of estimators based on the
semidefinite relaxation of sparse PCA with novel regularizations. In
particular, under a weak assumption on the magnitude of the population
projection matrix, one estimator within this family exactly recovers the true
support with high probability, has exact rank-$k$, and attains a $\sqrt{s/n}$
statistical rate of convergence with $s$ being the subspace sparsity level and
$n$ the sample size. Compared to existing support recovery results for sparse
PCA, our approach does not hinge on the spiked covariance model or the limited
correlation condition. As a complement to the first estimator that enjoys the
oracle property, we prove that, another estimator within the family achieves a
sharper statistical rate of convergence than the standard semidefinite
relaxation of sparse PCA, even when the previous assumption on the magnitude of
the projection matrix is violated. We validate the theoretical results by
numerical experiments on synthetic datasets.
