the standard least square estimation problem by Tibshirani [@TibLASSO], under the name "Lasso regression”. Minimizing the $L_1$ norm based penalty encourages sparse non-zero entries in the estimated precision matrix $A$, which achieves a selection of informative variables for regression and reduces complexity of regression model efficiently. Further extension of the basic $L_1$-norm penalty function allows one assigning different non-negative weight values $\lambda_{ij}$ to different entries $A_{ij}$, as $\sum_{i ,j}^n\lambda_{ij}|A_{ij}|$. This weighted combination can constrain the sparse penalties only on the off-diagonal entries, so as to avoid unnecessary sparsity on the diagonal elements. Furthermore, this extension allows us to introduce prior knowledge about the conditional independence structure of the graph into the joint combination problem.

![Demonstration of $L_1$ norm, exact $L_0$ norm and Seamless $L_0$ norm based penalty function.[]{data-label="fig:L0L1penfuncf"}](L1L0SELO.pdf){width="80.00000%"}

For further understanding of the relation between the exact $L_0$ and $L_1$ norm penalty, we illustrate them with respect to one scalar variable in Figure \[fig:L0L1penfuncf\]. As we can see, within $[-1,1]$, $L_1$-norm penalty plays as a convex envelope of the exact $L_0$-norm penalty. Due to the convexity property of $L_1$ norm penalty, the global optimum of the convex programming problem can be achieved with even linear computational complexity [@TibLASSO; @FrHaTi]. However, 