SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\Theta_ {i, j} = \left\{ \begin{array}{l l} {\left[ \widehat {\Theta} _ {k} \right] _ {\pi (i), \pi (i)} + \zeta_ {i}} & \text {i f} i = j \in \mathcal {V} _ {k}, \\ {\left[ \widehat {\Theta} _ {k} \right] _ {\pi (i), \pi (j)}} & \text {i f} i \neq j \text {a n d} i, j \in \mathcal {V} _ {k}, \\ - T _ {i j} / \left(S _ {i i} S _ {j j} - T _ {i j} ^ {2}\right) & \text {i f} (i, j) \in \mathcal {E} _ {e}, \\ 0 & \text {o t h e r w i s e}, \end{array} \right. \tag {50}
$$
in which  $\mathcal{E}_e$  refers to a set of external edges among clusters, i.e.,
$$
\mathcal {E} _ {e} = \left\{\left(i, j\right) \mid (i, j) \in \operatorname {s u p p} (\boldsymbol {T}), \psi (i) \neq \psi (j) \right\}, \tag {51}
$$
$\zeta_{i}$  is computed as
$$
\zeta_ {i} = \frac {1}{S _ {i i}} \sum_ {(i, m) \in \mathcal {E} _ {e}} \frac {T _ {i m} ^ {2}}{S _ {i i} S _ {m m} - T _ {i m} ^ {2}}, \tag {52}
$$
and  $\widehat{\Theta}_k$  is computed as the optimal solution of  $k$ -th sub-problem:
$$
\widehat {\boldsymbol {\Theta}} _ {k} = \arg \min  _ {\boldsymbol {\Theta} _ {k} \in \mathcal {M} ^ {p _ {k}}} - \log \det  \left(\boldsymbol {\Theta} _ {k}\right) + \left\langle \boldsymbol {\Theta} _ {k}, \boldsymbol {S} _ {\mathcal {V} _ {k}} - \boldsymbol {\Lambda} _ {\mathcal {V} _ {k}} \right\rangle . \tag {53}
$$
Remark C.2. The initial point is the optimal solution if the vertex-partition  $\mathcal{P}$  is a bridge-block decomposition of the optimal graph.
Remark C.3. When the vertex partition  $\mathcal{P}$  is given as  $\mathcal{P} = \{\{1\}, \{2\}, \ldots, \{p\}\}$ , i.e.,  $\mathcal{V}_k = \{k\}$  for  $k = 1, \ldots, p$ , the initial point (50) reduces to a closed-form expression:
$$
\Theta_ {i, j} = \left\{ \begin{array}{l l} \frac {1}{S _ {i i}} \left(1 + \sum_ {(i, m) \in \operatorname {s u p p} (\mathbf {T})} \frac {T _ {i m} ^ {2}}{S _ {i i} S _ {m m} - T _ {i m} ^ {2}}\right) & \text {i f} i = j, \\ - T _ {i j} / \left(S _ {i i} S _ {j j} - T _ {i j} ^ {2}\right) & \text {i f} (i, j) \in \operatorname {s u p p} (\mathbf {T}), \\ 0 & \text {o t h e r w i s e .} \end{array} \right. \tag {54}
$$
In practice, utilizing the warm-start strategy can also enhance the performance of certain numerical algorithms.
Figure 8: Average log-likelihood in out-of-sample data for different precision matrix estimation models as a function of the sparsity promoting hyperparameter  $\lambda$ .
Figure 9: Estimated Graphical Lasso of Crop data set at their highest average likelihood. Red edges represent positive conditional correlations, while blue edges represent negative ones.
D Additional Comments on Real-World Experiments
To show that  $\mathrm{MTP}_2$  assumption approximately holds for Crop dataset, we selected 20 random subsets from the Crop dataset. For each subset, we computed the Graphical Lasso and the  $\mathrm{MTP}_2$  graphical model for different values of using the first 10 observations. The remaining 36 observations were used to calculate the out-of-sample log-likelihood, which was then averaged across all datasets. This process allows us to evaluate how well these models generalize to unseen data.
As depicted in Figure 8 of the attached PDF, the  $\mathrm{MTP}_2$  graphical model outperforms the Graphical Lasso, providing a higher test log-likelihood. We present one instance of the estimated graphical lasso model in Figure 9. It reveals that most conditional correlations are positive (red edges,  $90\%$ ), with a few being negative (blue edges,  $10\%$ ). This pattern implies strong positive dependence in the Crop data, aligning with the characteristics of  $\mathrm{MTP}_2$ .
These results are not unexpected, given that the Crop dataset comprises multiple clusters. Within the same cluster, we expect data points to exhibit greater similarity compared to those in different clusters. This situation signifies a form of positive dependence, thereby justifying the plausibility of the  $\mathrm{MTP}_2$  assumption in this problem.