large scale optimization problems.

Analysis of JPEN Method
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**Def:** A random vector $X$ is said to have sub-gaussian distribution if for each $t \ge 0$ and $y \in \mathbb{R}^p $ with $\|y\|_2=1$, there exist $0< \tau < \infty $ such that $$\mathbb{P}\{|y^T(X-\mathbb{E}(X))|>t\} \le e^{-t^2/2\tau}$$ Although the JPEN estimators exists for any finite $2 \le n<p<\infty$, for theoretical consistency in operator norm we require $s~log~p=o(n)$ and for Frobenus norm we require $(p+s) ~log~p=o(n)$ where $s$ is the upper bound on the number of non-zero off-diagonal entries in true covariance matrix. For more details, see the remark after Theorem 3.1.\

Covariance Matrix Estimation
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We make the following assumptions about the true covariance matrix $\Sigma_0$.\
**A0.** Let $X:=(X_1,X_2,\cdots,X_p)$ be a mean zero vector with covariance matrix $\Sigma_0$ such that each $X_i/ \sqrt{\Sigma_{0ii}}$ has subgaussian distribution with parameter $\tau$ as defined in (3.1).\
**A1.** With $ E=\{(i,j): \Sigma_{0ij} \neq 0, i \neq j \}, $ the $|E| \le s $ for some positive integer $s$.\
**A2.** There exists a finite positive real number $\bar{k} >0$ such that $ 1/\bar{k} \le \sigma_{min}(\Sigma_0) \le \sigma_{max}(\Sigma_0) \le \bar{k}$.\
Assumption A2 guarantees that the true covariance matrix $\Sigma_0$ is well-conditioned (i.e. all the eigenvalues are finite 