a positive definite cone. The penalty term $P(A)$ is typically constructed using sparsity inducing matrix norm, also known as sparse learning in the domain of statistical learning. There are two typical configurations of $P(A)$:

-   The $L_0$ norm $\|A\|_{L_0}$ of the matrix $X$, which counts the number of non-zero elements in the matrix. It is also known as the cardinality or the non-zero support of the matrix. Given its definition, it is easy to find that $L_0$ norm based penalty is a strong and intuitive appeal for sparsity structure of the estimated precision matrix. However, it is computationally infeasible to solve exact $L_0$-norm minimization directly, due to the fact that exact $L_0$ norm penalty is discontinuous and non-differentiable. In practice, one either uses a continuous approximation to the form of the $L_0$-penalty, or solve it using a greedy method. Due to the non-convexity of the exact $L_0$ norm penalty, only a local optimum of the feasible solution can be guaranteed. Nevertheless, $L_0$ norm penalty usually leads to much sparser structure in the estimation, while local optimum is good enough for most practical cases.

-   The $L_1$ matrix norm $\|A\|_{L_1} = \sum_{i ,j}^n |A_{ij}|$. $L_1$ norm penalty was firstly introduced into 