Abstract:

This paper aims at achieving a simultaneously sparse and low-rank estimator from the semidefinite population covariance matrices. We first benefit from a convex optimization which develops ‘1-norm penalty to encourage the sparsity and nuclear norm to favor the low-rank property. For the proposed estimator, we then prove that with high probability, the Frobenius norm of the estimation rate can be of order O(s log p)/n  under a mild case, where s and p denote the number of nonzero entries and the dimension of the population covariance, respectively and n notes the sample capacity. Finally, an efficient alternating direction method of multipliers with global convergence is proposed to tackle this problem, and merits of the approach are also illustrated by practicing numerical simulations.

Key words: Covariance matrix ,  Sparse and low-rank estimator,  Estimation rate, Alternating direction method of multipliers