that of $\hat{\Sigma}_{S}$.

Estimation of Inverse Covariance Matrix
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We extend the JPEN approach to estimate a well-conditioned and sparse inverse covariance matrix. Similar to the covariance matrix estimation, we first propose an estimator for inverse covariance matrix based on regularized inverse correlation matrix and discuss its rate of convergence in Frobenious and operator norm.\
**Notation:** We shall use $Z$ and $\Omega$ for inverse correlation and inverse covariance matrix respectively.\
**Assumptions:** We make the following assumptions about the true inverse covariance matrix $\Omega_0$. Let $\Sigma_0=\Omega_0^{-1}$.\
**B0.** Same as the assumption $A0$.\
**B1.** With $ H=\{(i,j): \Omega_{0ij} \neq 0, i \neq j \}$, the $|H| \le s $, for some positive integer $s$.\
**B2.** There exist $ 0< \bar{k} < \infty $ large enough such that $ (1/{\bar{k}}) \le \sigma_{min}(\Omega_0) \le \sigma_{max}(\Omega_0) \le \bar{k}$.\
Let $\hat{R}_K$ be a JPEN estimator for the true correlation matrix. By Lemma 3.2, $\hat{R}_K$ is positive definite. Define the JPEN estimator of inverse correlation matrix as the solution to the following optimization problem, $$\hat{Z}_{K}=\operatorname*{arg\,min}_{Z=Z^T,tr(Z)=tr(\hat{R}_K^{-1})|(\lambda,\gamma) \in \hat{\mathscr{S}}^{{K}}_2}\Big [ \|Z-\hat{R}_K^{-1} \|^2~+~\lambda\|Z^-\|_1~+~\gamma\sum_{i=1}^{p} \{\sigma_i(Z)- \bar{\sigma}(Z)\}^2 \Big ]$$ where

$$\begin{aligned}
\hat{\mathscr{S}}^{{K}}_2& = \Big \{(\lambda,\gamma): \lambda,\gamma >0, \lambda \asymp \gamma \asymp \sqrt{\frac{log~ p}{n}}, \forall \epsilon >0, \\
& ~~~~~~~~~~~~ \sigma_{min}\{(\hat{R}_K^{-1}+\gamma t_1 I)-\frac{\lambda}{2}*sign(\hat{R}_K^{-1}+\gamma 