  Sparse PCA is a widely used technique for high-dimensional data analysis. In
this paper, we propose a new method called low-rank principal eigenmatrix
analysis. Different from sparse PCA, the dominant eigenvectors are allowed to
be dense but are assumed to have a low-rank structure when matricized
appropriately. Such a structure arises naturally in several practical cases:
Indeed the top eigenvector of a circulant matrix, when matricized appropriately
is a rank-1 matrix. We propose a matricized rank-truncated power method that
could be efficiently implemented and establish its computational and
statistical properties. Extensive experiments on several synthetic data sets
demonstrate the competitive empirical performance of our method.
