data distribution, given the fixed empirical covariance matrix. When the derived inverse estimation is sparse, it becomes easier to compute marginal distribution of each random variable and conduct statistical inference. To achieve that goal, optimizations methods have been developed based on $L_0$ or $L_1$ norm penalty for the estimation of $C^{-1}$, to enhance its sparse structure constraint on the estimated inverse of covariance matrix and discover underlying conditionally independent parts.

Let $\hat C \in \mathbb{R}^{n \times n}$ be the empirical covariance matrix of $n$ random variables (represented as the nodes in the graph model). The sparsity penalized maximum likelihood estimation $A$ of the precision matrix $C^{-1}$ can be derived by solving the following positive definite cone program: $$\label{eq:logdetopt}
 A = \operatorname*{argmin}_{ X \succ 0}   -{\cal L}(X) + \lambda P(X)$$ where $$\label{def:LLcov}
{\cal L}(A) {\stackrel{\mbox{\upshape\tiny def}}{=}}\log\det(A)-\operatorname{Tr}(A\hat C),$$ is the log likelihood of the distribution defined by $A$, $\log\det$ being the logarithm of the determinant, and $P(A)$ is a sparsity inducing regularization term [@FrHaTi]. $\lambda$ is the regularization coefficient balancing the data-fitting oriented likelihood and sparsity penalty. Since the precision matrix of joint normal distribution should be positive definite, any feasible solution to this optimization problem is thus required to locate within 