Sparse and Low-Rank Covariance Matrices Estimation
This provides a method for statistical estimation in high-dimensional data analysis, but it is incremental as it builds on existing norm-based regularization techniques.
The paper tackles the problem of estimating covariance matrices that are both sparse and low-rank by proposing a convex optimization method with l1 and nuclear norm penalties, achieving an estimation rate of O(√(s(log r)/n)) in Frobenius norm under mild conditions.
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 $l_1$-norm penalty to encourage the sparsity and nuclear norm to favor the low-rank property. For the proposed estimator, we then prove that with large probability, the Frobenious norm of the estimation rate can be of order $O(\sqrt{s(\log{r})/n})$ under a mild case, where $s$ and $r$ denote the number of sparse entries and the rank of the population covariance respectively, $n$ notes the sample capacity. Finally an efficient alternating direction method of multipliers with global convergence is proposed to tackle this problem, and meantime merits of the approach are also illustrated by practicing numerical simulations.