Table 1 The tailored graphical lasso algorithm

1

Select the optimal penalty \(\lambda\) for the ordinary graphical lasso problem by StARS with the desired value of \(\beta\) (we propose \(\beta =0.05\)). Let the sigmoid midpoint \(w_0\) be equal to the lower \(\beta\)-quantile of the non-zero weights. Choose a maximum value \(k_{\text{max}}\) to consider, such as 80. Choose a value of the edge penalizing parameter \(\gamma\) in eBIC (\(\text{BIC}_{\gamma }\)) selection criterion

2

For a grid of \(k \in [0,k_{\text{max}}]\):

 \(\bullet\) Let \({\varvec{P}}_k = {\varvec{1}} - g_k( {\varvec{W}})\)

 \(\bullet\) Find \(\lambda _k = \frac{\lambda p^2}{\Vert {\varvec{P}}_k \Vert _1}\)

 \(\bullet\) Find the estimated precision matrix \(\widehat{{\varvec{\Theta }}}_k\) and the corresponding set \(E_k\) of inferred edges, with the weighted graphical lasso using the penalty matrix \(\lambda _k{\varvec{P}}_k\)

 \(\bullet\) Find \(\text{BIC}_{\gamma }^{(k)} (E_k)= -2 l( {\varvec{{\hat{\Theta }}}}_k(E_k) ) + \vert E_k \vert \log {n} + 4 \vert E_k \vert \gamma \log {p}\)

3

Let \(k_{\text{opt}} = \mathop {{\mathrm{arg\,min}}}\limits _{k} \left\{ \text{BIC}_{\gamma }^{(k)} (E_k)\right\}\). The penalty matrix to be used is then \({\varvec{P}}_{k_{\text{opt}}}\), and the common penalty parameter is \(\lambda _{k_{\text{opt}}}\)