The exact timing of these algorithm also depends upon the implementation, platform etc. (we did our computations in $R$ on a AMD 2.8GHz processor). Following the approach [@Bickel1:4], the optimal tuning parameter $(\lambda,\gamma)$ was obtained by minimizing the $5-$fold cross validation error $$(1/5) \sum_{i=1}^5\|\hat{\Sigma}_i^v-\Sigma_i^{-v}\|_1,$$ where $\hat{\Sigma}_i^v$ is JPEN estimate of the covariance matrix

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![image](timing.png){width="40.00000%"}

based on $v=4n/5$ observations, $\Sigma_i^{-v}$ is the sample covariance matrix using $(n/5)$ observations. Figure 4.1 illustrates the total computational time taken to estimate the covariance matrix by $Glasso,~PDSCE$ and $JPEN$ algorithms for different values of $p$ for Toeplitz type of covariance matrix on log-log scale (see section $\S 5$ for Toeplitz type of covariance matrix). Although the proposed method requires optimization over a grid of values of $(\lambda,\gamma) \in \hat{\mathscr{S}}^{K}_1$, our algorithm is very fast and easily scalable to large scale data analysis problems.

Simulation Results
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We compare the performance of the proposed method to other existing methods on simulated data for five types of covariance and inverse covariance matrices.\
**(i) Hub Graph:** Here the rows/columns of $\Sigma_0$ are partitioned into J equally-sized disjoint groups: $ \{ V_1 \cup  V_2~  \cup, ..., \cup~ V_J\} = \{1,2,...,p\},$ each group is associated with a 