regularization directly on the entries of the sample covariance matrix.

Much of the recent literature focuses on two broad clases of regularized covariance matrix estimation. i) The one class relies on natural ordering among variables, where one often assumes that the variables far apart are weekly correlated and ii) the other class where there is no assumption on the natural ordering among variables. The first class includes the estimators based on banding and tapering \[@Bickel:3, @Cai1:6\]. These estimators are appropriate for a number of applications for ordered data (time series, spectroscopy, climate data). However for many applications including gene expression data, prior knowledge of any canonical ordering is not available and searching for all permutation of possible ordering would not be feasible. In these situations, an $\ell_1$ penalized estimator becomes more appropriate which yields a permutation-invariant estimate.

To obtain a suitable estimate which is both well-conditioned and sparse, we introduce two regularization terms: **i)** $\ell_1$ penalty for each of the off-diagonal elements of matrix and, **ii)** penalty propotional to the variance of the eigenvalues. The $\ell_1$ minimization problems are well studied in the covariance and inverse covariance matrix estimation literature \[@Freidman:11, @Banerjee:38, @Ravi1:24, @Bein:13, @Maurya:19 etc.\]. @Roth1:26 proposes an $\ell_1$ 