  Sparse principal component analysis (PCA) involves nonconvex optimization for
which the global solution is hard to obtain. To address this issue, one popular
approach is convex relaxation. However, such an approach may produce suboptimal
estimators due to the relaxation effect. To optimally estimate sparse principal
subspaces, we propose a two-stage computational framework named "tighten after
relax": Within the 'relax' stage, we approximately solve a convex relaxation of
sparse PCA with early stopping to obtain a desired initial estimator; For the
'tighten' stage, we propose a novel algorithm called sparse orthogonal
iteration pursuit (SOAP), which iteratively refines the initial estimator by
directly solving the underlying nonconvex problem. A key concept of this
two-stage framework is the basin of attraction. It represents a local region
within which the `tighten' stage has desired computational and statistical
guarantees. We prove that, the initial estimator obtained from the 'relax'
stage falls into such a region, and hence SOAP geometrically converges to a
principal subspace estimator which is minimax-optimal within a certain model
class. Unlike most existing sparse PCA estimators, our approach applies to the
non-spiked covariance models, and adapts to non-Gaussianity as well as
dependent data settings. Moreover, through analyzing the computational
complexity of the two stages, we illustrate an interesting phenomenon that
larger sample size can reduce the total iteration complexity. Our framework
motivates a general paradigm for solving many complex statistical problems
which involve nonconvex optimization with provable guarantees.
