  Principal component analysis (PCA) is possibly one of the most widely used
statistical tools to recover a low-rank structure of the data. In the
high-dimensional settings, the leading eigenvector of the sample covariance can
be nearly orthogonal to the true eigenvector. A sparse structure is then
commonly assumed along with a low rank structure. Recently, minimax estimation
rates of sparse PCA were established under various interesting settings. On the
other side, Bayesian methods are becoming more and more popular in
high-dimensional estimation, but there is little work to connect frequentist
properties and Bayesian methodologies for high-dimensional data analysis. In
this paper, we propose a prior for the sparse PCA problem and analyze its
theoretical properties. The prior adapts to both sparsity and rank. The
posterior distribution is shown to contract to the truth at optimal minimax
rates. In addition, a computationally efficient strategy for the rank-one case
is discussed.
