Large Covariance Matrix Estimation Based on Factor Models via Nonconvex Optimization

In this paper, we study the problem of large covariance matrix estimation based on the factor model assumption, in which case the covariance matrix is represented by a combination of a low-rank matrix and a sparse matrix. We formulate the estimation problem as a nonconvex problem, and an iterative optimization algorithm is proposed to obtain the estimator. The algorithm starts at the widely used optimization-free estimator called principal orthogonal complement thresholding (POET), and then a refined estimator is obtained by iterative optimization. We name the obtained nonconvex estimator POET with refining iteratively (POETRY). Theoretically, we prove that POETRY achieves a superior statistical rate compared to POET, matching the minimax rate of convergence for factor-based covariance estimation. Additionally, our algorithm exhibits significantly lower per-iteration computational complexity compared to existing convex relaxation-based methods. Numerical experiments validate the superiority of our algorithm over the state-of-the-art ones and corroborate our theory.