penalized log-likelihood estimator and shows that estimator is consistent in Frobenius norm at the rate of $O_P\Big(\sqrt{\{(p+s)~log~p\}/{n}}\Big)$, as both $p$ and $n$ approach to infinity. Here $s$ is the number of non-zero off-diagonal elements in the true covariance matrix. In another interesting paper @Bein:13 propose an estimator of covariance matrix as penalized maximum likelihood estimator with a weighted lasso type penalty. In these optimization problems, the $\ell_1$ penalty results in sparse and a permutation-invariant estimator as compared to other $l_q, q \neq 1$ penalties. Another advantage is that the $\ell_1$ norm is a convex function which makes it suitable for large scale optimization problems. A number of fast algorithms exist in the literature for covariance and inverse covariance matrix estimation \[(@Freidman:11, @Roth:25\]. The eigenvalues variance penalty overcomes the over-dispersion in the sample covariance matrix so that the estimator remains well-conditioned.

@Ledoit:17 proposed an estimator of covariance matrix as a linear combination of sample covariance and identity matrix. Their estimator of covariance matrix is well-conditioned but it is not sparse. @Roth:25 proposed estimator of covariance matrix based on quadratic loss function and $\ell_1$ penalty with a log-barrier on the determinant of covariance matrix. The log-determinant barrier is a valid technique to 