  In this paper, we propose a cone projected power iteration algorithm to
recover the first principal eigenvector from a noisy positive semidefinite
matrix. When the true principal eigenvector is assumed to belong to a convex
cone, the proposed algorithm is fast and has a tractable error. Specifically,
the method achieves polynomial time complexity for certain convex cones
equipped with fast projection such as the monotone cone. It attains a small
error when the noisy matrix has a small cone-restricted operator norm. We
supplement the above results with a minimax lower bound of the error under the
spiked covariance model. Our numerical experiments on simulated and real data,
show that our method achieves shorter run time and smaller error in comparison
to the ordinary power iteration and some sparse principal component analysis
algorithms if the principal eigenvector is in a convex cone.
