SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Learning Large-Scale  $\mathrm{MTP}_2$  Gaussian Graphical Models via Bridge-Block Decomposition
Xiwen Wang $^{1}$ , Jiaxi Ying $^{1,2*}$ , Daniel P. Palomar $^{1}$
The Hong Kong University of Science and Technology<sup>1</sup>
HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute2
{xwangew,jx.ying}  $@$  connect.ust.hk, palomar@ust.hk
This paper studies the problem of learning the large-scale Gaussian graphical models that are multivariate totally positive of order two  $(\mathrm{MTP}_2)$ . By introducing the concept of bridge, which commonly exists in large-scale sparse graphs, we show that the entire problem can be equivalently optimized through (1) several smaller-scaled sub-problems induced by a bridge-block decomposition on the thresholded sample covariance graph and (2) a set of explicit solutions on entries corresponding to bridges. From practical aspect, this simple and provable discipline can be applied to break down a large problem into small tractable ones, leading to enormous reduction on the computational complexity and substantial improvements for all existing algorithms. The synthetic and real-world experiments demonstrate that our proposed method presents a significant speed-up compared to the state-of-the-art benchmarks.
In recent decades, the surge in data availability has posed challenges for analyzing and understanding large-scale datasets. A key challenge is examining pairwise relationships among numerous variables, which can be represented using Gaussian graphical models (GGMs) [1] that depict variable connections as graphs. The precision matrix, which is the inverse of the covariance matrix, helps determine the non-zero patterns of GGMs [2, 3]. A traditional approach to estimate the precision matrix is the graphical lasso [4, 5], which is formulated as a regularized log-determinant program. The solutions of graphical lasso possess an appealing property known as sparsity, which severs as a common assumption particularly in large graphical models and has been shown to offer numerous benefits by a significant amount of research. For example, the sparsity can reduce the model size and improve the interpretability of the model by highlighting the most important variables and their relationships, allowing better understanding the underlying causal structure [6-8].
In this paper, we solve the large-scale sparse precision matrix estimation problem for GGMs that follow a multivariate totally positive of order two  $(\mathrm{MTP}_2)$  Gaussian distribution [9, 10], or equivalently, possess a precision matrix that is a symmetric  $M$ -matrix [11], where all off-diagonal elements are non-positive. The so-called  $\mathrm{MTP}_2$  property is a special form of positive dependence and plays an essential role in applications where all interaction potentials are considered non-negative or the focus is on emphasizing positive associations between variables [12-15]. The  $\mathrm{MTP}_2$  property has been applied in various fields. Here are some examples. In finance,  $\mathrm{MTP}_2$  structures are often exploited as the asset returns are often considered positively correlated [16-18]; In machine learning,  $\mathrm{MTP}_2$  GGMs are recognized as attractive Markov random field and have been used in applications such as taxonomic reasoning [19] and psychometrics [13]. The  $\mathrm{MTP}_2$  precision matrix, also referred to
as the generalized graph Laplacian [20-22], along with the combinatorial graph Laplacian [23-27], have found broad applications across a variety of fields due to their unique characteristics. These applications include, but are not limited to, graph Fourier transform [28], electrical circuits analysis [29], image and video coding [30], financial time-series analysis [31, 32], and structured graph learning [33, 34].
The objective of this paper is to build a theoretical foundation and devise effective approaches for learning large-scale, sparse  $\mathrm{MTP}_2$  GGMs. The contributions of this paper are threefold.
- This is the first work in the literature that introduces the notion of bridges in the context of learning  $\mathrm{MTP}_2$  GGMs to the best of our knowledge. Building upon this notion, we prove that the presence of explicit solutions for some entries depends on whether an edge functions as a bridge. Meanwhile, we demonstrate that the optimal solution exhibits a decomposed structure through a vertex-partition known as bridge-block decomposition.
- Based on theoretical findings, we propose an efficient framework that decomposes a large problem into several small tractable ones, each of which can be solved using any existing algorithm. With some negligible extra cost for bridge-block decomposition, the proposed method results in a dimension reduction that significantly cuts down the computational complexity.
- The synthetic and real-world experiments demonstrate that our proposed methods prompt a considerable speed-up for all existing methods and enable the solving of large-scale  $\mathrm{MTP}_2$  GGMs that were previously considered infeasible.
1.1 Notation and Organization
Vectors and matrices are written as lower and upper case bold letters, respectively. An undirected graph is denoted as  $G = (\mathcal{V}, \mathcal{E})$ , where  $\mathcal{V}$  is the set of nodes with size  $|\mathcal{V}| = p$  and  $\mathcal{E} \subset \mathcal{V} \times \mathcal{V}$  is the set of edges. Note that we shall not distinguish between  $(i,j) \in G$  and  $(i,j) \in \mathcal{E}$ . Some graph terminology frequently used throughout the paper are introduced as follows.
- Support graph Given a symmetric matrix  $\mathbf{A} \in \mathbb{S}^p$ , the support graph of  $\mathbf{A}$ , denoted as  $\operatorname{supp}(\mathbf{A})$ , is defined as an undirected graph with the vertex set  $\mathcal{V} = \{1, \dots, p\}$  and the edge set  $\mathcal{E} \in \mathcal{V} \times \mathcal{V}$  such that  $(i,j) \in \mathcal{E}$  if and only if  $A_{ij} \neq 0$  for every two different vertices  $i,j \in \mathcal{V}$ .
- Neighbors The neighbors of a node  $i$  refer to the subset of vertices that are connected to this node, i.e.,  $\mathcal{N}(i) = \{j | (i,j) \in \mathcal{E}\}$ .
- Path and Cycle A path from node  $i_1 \in \mathcal{V}$  to node  $i_T \in \mathcal{V}$  is a sequence (i.e., an ordered set) of edges denoted as  $d_{i_1,i_T}$ , each one incident to the next, i.e.,  $\{(i_t,i_{t + 1})\}_{t = 1}^{T - 1} \subseteq \mathcal{E}$ . A cycle is a path from a node to itself when it does not duplicate include the same edge.
- Partition A partition  $\mathcal{P} = \{\mathcal{V}_1, \ldots, \mathcal{V}_K\}$  is a collection of non-empty, disjoint vertex sets  $\mathcal{V}_1, \ldots, \mathcal{V}_K$  called cluster such that  $\bigcup_{i=1}^{K} \mathcal{V}_k = \mathcal{V}$ .
- Component A component of  $G$  refers to a subset of vertices that are connected to each other by edges, but are not connected to any other vertices in the graph.
The paper is organized as follows. In Section 2, we introduce the problem formulation and related works. In Section 3, we present our main result along with its potential impact. Section 4 presents the numerical experiments to show that the proposed methods can facilitate the performance of existing algorithms and solve large-scale  $\mathrm{MTP}_2$  GGMs problems<sup>2</sup>. In Section 5 we draw the conclusions.
Let  $\pmb{y}$  be a random vector that follows a Gaussian distribution with zero mean and a precision matrix  $\Theta$ , a.k.a. the inverse of covariance matrix  $\pmb{\Sigma}$ . A Gaussian graphical model represents the conditional dependency between random variables via a graph  $G$ , in which nodes correspond to  $\pmb{y}$  and edges between these variables represent conditional dependencies, which can be equivalently characterized by the non-zero patterns of the precision matrix  $\Theta$  [2].
2Codes are available in https://github.com/Xiven1997/mtp2-bbd.
This paper considers estimating the precision matrix  $\Theta$  given  $n$  independent and identically distributed observations  $\{\pmb{y}_1,\dots ,\pmb{y}_n\}$  that follow an MTP2 Gaussian distribution. Equivalently,  $\Theta$  is assumed to be a symmetric  $M$ -matrix, i.e.,  $\Theta_{ij}\leq 0$  for any  $i\neq j$ . The problem is formulated as
$$
\underset {\boldsymbol {\Theta} \in \mathcal {M} ^ {p}} {\text {m i n i m i z e}} - \log \det  (\boldsymbol {\Theta}) + \left\langle \boldsymbol {\Theta}, \boldsymbol {S} \right\rangle + \sum_ {i \neq j} \Lambda_ {i j} | \Theta_ {i j} |, \tag {1}
$$
where  $S$  is the sample covariance matrix,  $\Lambda_{ij} \geq 0$  are the regularization coefficients, the objective is to minimize the negative log-likelihood of the data subject to a weighted  $\ell_1$ -norm penalty on the precision matrix, and  $\mathcal{M}^p$  refers to a set of  $M$ -matrices with dimension  $p$ , i.e.,
$$
\mathcal {M} ^ {p} = \left\{\Theta \in \mathbb {S} ^ {p} \mid \Theta \succ \mathbf {0} \text {a n d} \Theta_ {i j} \leq 0, \forall i \neq j \right\}. \tag {2}
$$
We refer to  $\Theta^{\star}$  as the optimal solution of (1) and to the support graph of  $\Theta^{\star}$ , i.e.,  $\operatorname{supp}(\Theta^{\star})$ , as the optimal graph. Particularly, we define the thresholded sample covariance matrix  $\pmb{T}$  as
$$
T _ {i j} = \left\{ \begin{array}{l l} S _ {i j} - \Lambda_ {i j} & \text {i f} i \neq j \text {a n d} S _ {i j} > \Lambda_ {i j}, \\ 0 & \text {o t h e r w i s e}, \end{array} \right. \tag {3}
$$
and the support graph of  $\pmb{T}$  as the thresholded sample covariance graph (short for thresholded graph). Clearly, entries corresponding to  $S_{ij} \leq \lambda_{ij}$  will be thresholded to zero. Since the thresholded graph holds immense importance throughout the paper, unless explicitly mentioned otherwise, we denote  $G = \operatorname{supp}(T)$ .
The involvement of  $\mathrm{MTP}_2$  constraints, which require that  $\Theta_{ij} \leq 0$  for all  $i \neq j$ , confers several advantages [11]. For example, all  $M$ -matrices are inverse-positive, i.e.,  $\Theta^{-1} \geq 0$ , and the non-smoothness from  $\ell_1$  norms can be eliminated via  $\sum_{i \neq j} \Lambda_{ij} |\Theta_{ij}| = -\langle \Lambda, \Theta \rangle$ , where  $\Lambda = (\Lambda_{ij})$  and  $\operatorname{diag}(\Lambda) = 0$ . Meanwhile, the  $\mathrm{MTP}_2$  constraints maintain the sparsity in the estimated precision matrix as an implicit regularizer [19]. Throughout the paper, we require the following assumption.
Assumption 2.1. For any  $i \neq j$ , we have  $S_{ij} < \sqrt{S_{ii}S_{jj}}$ .
Assumption 2.1 holds with probability 1 if the number of observations  $n \geq 2$  [13, 19]. Under that assumption, the minimizer of Problem (1) exists and is unique according to [19, Theorem 1].
2.2 Related Works
In literature, devising algorithms for learning  $\mathrm{MTP}_2$  GGMs has garnered considerable attention. For instance, block coordinate descent (BCD) methods [19, 26, 35] update a single row/column of the precision matrix in a cyclic manner by solving non-negative least squares problems. Projection-based methods, such as projected gradient descent [36] and projected quasi-Newton algorithms [37], iteratively take steps along the descent direction and then project the solutions back to the feasible region. Despite these efforts, existing research has not fully explored the potential of exploiting  $\mathrm{MTP}_2$  properties to reduce computational complexity. With complexities of  $\mathcal{O}(p^4)$  for BCD methods and  $\mathcal{O}(p^3)$  for projection-based techniques, addressing large-scale problems continues to be challenging, particularly when manipulating full-dimensional matrices without problem reduction.
Instead of devising efficient algorithms, recently, leveraging the properties of the thresholded sample covariance graph has emerged as a popular approach for learning GGMs. Specifically, the existence of closed-form solution has been established for graphical lasso when the thresholded sample covariance graph is acyclic (i.e., contains no cycles) [38, 39]. However, the applicability of their main results is limited by certain conditions that are difficult to verify. Interestingly, those conditions are unnecessary for the existence of closed-from solutions in  $\mathrm{MTP}_2$  GGMs [20]. Despite the theoretical establishment on closed-form solutions, an exact acyclic structure is rarely observed in practice. Therefore, more research dives into the relationship between supp  $(T)$  and supp  $(\Theta^{\star})$ . Research found that supp  $(\Theta^{\star})$  is a subset of supp  $(T)$  [13], the number of connected components in supp  $(T)$  is identical to that in supp  $(\Theta^{\star})$  [20], and there exist necessary conditions for the presence of edges in supp  $(\Theta^{\star})$  via analyzing the path products on thresholded matrix [13].
This paper advances prior research in two ways. Firstly, unlike previous studies that only provided an explicit solution for  $\Theta_{ij}$  in the case of acyclic thresholded graphs, we reveal that this explicit solution for  $\Theta_{ij}$  consistently applies to all  $(i,j)$  pairs acting as bridges, regardless of whether the thresholded graph is acyclic or non-acyclic. Secondly, we highlight that the optimal graph can be represented in an equivalent decomposed form through a vertex partition, termed as the bridge-block decomposition of the thresholded graph by leveraging  $\mathrm{MTP}_2$  properties.
Figure 1: An illustration of the bridge-block decomposition. Edges (5,6) and (9,10) are identified as bridges since removing either of them increases the number of connected components. After removing the bridge edges, the resulting decomposition  $\mathcal{P}^{\mathrm{bbd}} = \{\{1,2,3,4,5\}, \{7,8,9,10\}, \{11,12,13,14,15,16\}\}$ .
3 Proposed Methods
The goal of this paper is not about developing an algorithm but to shed light on the remarkable properties concealed in the thresholded graph. This section is organized as follows. In Section 3.1, we introduce bridge and bridge-block decomposition. Section 3.2 presents our main result. Then in Section 3.3, we elaborate on the contributions and connections to existing research.
3.1 Bridge and Bridge-Block Decomposition
Bridge is one of the important concepts in graph theory. Technically, an edge is called a bridge if and only if its deletion increases the number of graph components. Therefore, an edge is a bridge only when it is not contained in any cycles [40]. Notably, when a graph consists solely of bridges, it is referred to as acyclic. In this paper, we denote  $\mathcal{B}$  as the set of all bridges, i.e,
$$
\mathcal {B} := \{(i, j) | (i, j) \text {i s a b r i d g e i n} G \}. \tag {4}
$$
Remark 3.1. Bridges are frequently observed, particularly in large-scale sparse graphs [41-43]. This is attributed to the fact that in sparse graphs, only the most significant relationships between variables are retained, and the removal of edges can create additional connected components, giving rise to the presence of bridges.
Remark 3.2. In practice, the set  $\mathcal{B}$  can be efficiently obtained via various bridge-finding algorithms [40, 44]. These algorithms employ a depth-first search approach, resulting in a computational complexity of  $\mathcal{O}(|\mathcal{V}| + |\mathcal{E}|)$ . In the sparse graphs that are of interest to us, the number of edges  $|\mathcal{E}|$  typically scales similarly to the number of nodes  $|\mathcal{V}|$ . As a result, the computational cost of identifying bridges in large-scale sparse graphs remains low.
Using Figure 1 as an illustration, we perform the bridge-block decomposition [45] as follows. By removing all the bridges, we compute the clusters  $\mathcal{V}_k$  corresponding to the components of  $G = (\mathcal{V},\mathcal{E}\setminus \mathcal{B})$ . This process results in a vertex-partition, known as the bridge-block decomposition:
$$
\mathcal {P} ^ {\mathrm {b b d}} = \left\{\mathcal {V} _ {1}, \mathcal {V} _ {2}, \dots , \mathcal {V} _ {K} \right\}, \tag {5}
$$
where  $K$  refers to the number of clusters, also the number of components in the graph  $(\mathcal{V},\mathcal{E}\setminus \mathcal{B})$
Without loss of generality, we define the operator  $\psi$  as  $\psi (i) = k$  if node  $i$  belongs to the  $k$ -th cluster. In the literature, the bridge-block decomposition is also known as the 2-edge-connected decomposition [46, 47], a fundamental concept in graph theory with numerous applications such as community search [48], social network mining [49], and transmission networks [50].
In practice, the cost of obtaining the bridge-block decomposition  $\mathcal{P}^{\mathrm{bbd}}$ , which involves calculating the thresholded graph, bridges, and clusters, is negligible. With the aforementioned preliminary knowledge, we formally introduce our main result as follows.
Considering a square matrix  $\mathbf{A} \in \mathbb{S}^p$ , we define  $\mathbf{A}_{\mathcal{V}_k} \in \mathbb{S}^{p_k}$  as the principal sub-matrix of  $\mathbf{A}$  keeping the rows and columns indexed by  $\mathcal{V}_k$ , in which  $p_k = |\mathcal{V}_k|$  is the number of nodes in  $k$ -th cluster and
Figure 2: An example of obtaining optimal solution via Theorem 3.3. The thresholded graph of 6 nodes can be partitioned into 2 clusters  $\mathcal{V}_1 = \{1,2,3\}$  and  $\mathcal{V}_2 = \{4,5,6\}$ . The optimal solution  $\Theta \in \mathbb{S}^6$  can be exactly computed via a set of explicit solution  $\{\Theta_{3,4},\Theta_{4,3}\}$ , solutions of smaller-sized sub-problems, i.e.,  $\widehat{\Theta}_1 \in \mathbb{S}^3$  and  $\widehat{\Theta}_2 \in \mathbb{S}^3$ , and zeros in the rest positions.
$$
\Theta_ {\{1, 2, 3 \}} = \widehat {\Theta} _ {1} + \operatorname {d i a g} \left(\zeta_ {1}, \zeta_ {2}, \zeta_ {3}\right).
$$
$$
\Theta_ {\{4, 5, 6 \}} = \widehat {\Theta} _ {2} + \operatorname {d i a g} \left(\zeta_ {4}, \zeta_ {5}, \zeta_ {6}\right).
$$
$$
(i, j) \in \mathcal {B}: \Theta_ {i j} = - T _ {i j} / \left(S _ {i i} S _ {j j} - T _ {i j} ^ {2}\right).
$$
$$
(i, j) \notin \mathcal {B} \text {a n d} \psi (i) \neq \psi (j): \Theta_ {i j} = 0.
$$
we have  $p = \sum_{k} p_{k}$ . For each  $i \in \mathcal{V}_{k}$ , we mark  $\pi(i)$  as its corresponding index in  $\mathcal{V}_{k}$ . Hence, we define  $\widehat{\Theta}_{k}$  as the optimal solution of  $k$ -th sub-problem, i.e.,
$$
\widehat {\boldsymbol {\Theta}} _ {k} = \arg \min  _ {\boldsymbol {\Theta} _ {k} \in \mathcal {M} ^ {p _ {k}}} - \log \det  \left(\boldsymbol {\Theta} _ {k}\right) + \left\langle \boldsymbol {\Theta} _ {k}, \boldsymbol {S} _ {\mathcal {V} _ {k}} - \boldsymbol {\Lambda} _ {\mathcal {V} _ {k}} \right\rangle . \tag {6}
$$
The main result is then given as follows.
Theorem 3.3. Under Assumption 2.1, given the bridge-block decomposition of the thresholded graph  $\text{supp}(\mathbf{T})$  as  $\mathcal{P}^{bbd}$ , and the optimal solution of each sub-problem (46) as  $\widehat{\Theta}_k$ , the optimal solution of Problem (1), i.e.,  $\Theta^{\star}$ , can be obtained as
$$
\boldsymbol {\Theta} _ {i, j} ^ {\star} = \left\{ \begin{array}{l l} {[ \widehat {\boldsymbol {\Theta}} _ {k} ] _ {\pi (i), \pi (i)} + \zeta_ {i}} & \text {i f i = j \in \mathcal {V} _ {k} ,} \\ {[ \widehat {\boldsymbol {\Theta}} _ {k} ] _ {\pi (i), \pi (j)}} & \text {i f i \neq j a n d i , j \in \mathcal {V} _ {k} ,} \\ - T _ {i j} / \left(S _ {i i} S _ {j j} - T _ {i j} ^ {2}\right) & \text {i f (i , j) \in \mathcal {B} ,} \\ 0 & \text {o t h e r w i s e .} \end{array} \right. \tag {7}
$$
in which  $\zeta_{i} = \frac{1}{S_{ii}}\sum_{(i,m)\in \mathcal{B}}\frac{T_{im}^{2}}{S_{ii}S_{mm} - T_{im}^{2}}$  and  $\zeta_{i} = 0$  if  $\forall m:(i,m)\notin \mathcal{B}$
Figure 2 depicts an example of how to apply Theorem 3.3 to obtain the optimal solution more efficiently. Theoretically, the key to show the optimality of (7) is via an explicit expression of the inverse of  $\Theta$ , i.e.,  $R = \Theta^{-1}$  using following theorem.
Theorem 3.4. Given  $S, T \in \mathbb{S}^p$ , the bridge-block decomposition  $\mathcal{P}^{bdd} = \{\mathcal{V}_1, \dots, \mathcal{V}_K\}$  of  $\text{supp}(\mathbf{T})$  and a set of matrices  $\{\widehat{\Theta}_k \in \mathbb{S}_{++}^{|\mathcal{V}_k|}\}_{k=1}^K$ , the inverse of  $\Theta$  in the form of (7) is derived as
$$
R _ {i j} = \left\{ \begin{array}{l l} {\left[ \widehat {\boldsymbol {R}} _ {k} \right] _ {\pi (i), \pi (j)}} & \text {i f i , j \in \mathcal {V} _ {k} ,} \\ T _ {i j} & \text {i f (i , j) \in \mathcal {B} ,} \\ \sqrt {S _ {i i} S _ {j j}} \cdot g _ {i j} (\boldsymbol {R}) & \text {o t h e r w i s e ,} \end{array} \right. \tag {8}
$$
where  $\widehat{\pmb{R}}_k = [\widehat{\Theta}_k]^{-1}$  and for each  $i$ ,  $j$  in different clusters,  $g_{ij}$  is given as
$$
g _ {i j} (\boldsymbol {R}) = \prod_ {t = 0} ^ {2 T} R _ {u _ {t}, u _ {t + 1}} / \sqrt {S _ {u _ {t} , u _ {t}} S _ {u _ {t + 1} , u _ {t + 1}}}, \quad \text {w h e r e} u _ {0} \triangleq i, u _ {2 T + 1} \triangleq j, \tag {9}
$$
in which  $T - 1$  is the number of bridges in  $d_{ij}$ ,  $g_{ij} = 0$  if  $d_{ij} = \emptyset$ , and  $u$  denotes a sequence of 'incident' bridges, i.e.,  $d_{ij} \cap \mathcal{B} = \{(u_1, u_2), \ldots, (u_{2T-1}, u_{2T})\}$  following the orders of  $d_{ij}$  while preserving the elements that are bridges.
We note that all terms in (9) can be computed via (8) and Theorem 3.4 itself generally holds for arbitrary  $S, T$ , and  $\{\widehat{\Theta}_k\}_{k=1}^K$ . Detailed proofs and discussions are deferred to Appendix A.
Theorem 3.4 is theoretically critical as it provides an explicit form of the inverse of  $\Theta$ , which was previously difficult to compute. It is vital for demonstrating the optimality of  $\Theta$  by connecting the KKT system of the original large-scale problem to the KKT systems of smaller sub-problems through the bridge-block decomposition. Consequently, together with  $\mathrm{MTP}_2$  properties, we show that  $\Theta$  in the form of (7) satisfies the KKT condition of Problem (1) in Appendix B.
The role of  $\mathrm{MTP}_2$  constraints: Although Theorem 3.4 can be applied without requiring  $\mathrm{MTP}_2$  properties, it is important to highlight that these properties serve as sufficient conditions to demonstrate
the optimality of  $\Theta = R^{-1}$ . By incorporating the  $\mathrm{MTP}_2$  constraints, the non-smoothness in graphical lasso is eliminated, resulting in simplified optimality conditions as shown below:
$$
\forall i: \quad - R _ {i i} + S _ {i i} = 0 \quad - R _ {i i} + S _ {i i} = 0 \tag {10a}
$$
$$
\forall \Theta_ {i j} \neq 0: - R _ {i j} + S _ {i j} + \lambda_ {i j} \cdot \operatorname {s i g n} (\Theta_ {i j}) = 0 \quad \Longrightarrow \quad - R _ {i j} + S _ {i j} - \lambda_ {i j} = 0 \tag {10b}
$$
$$
\forall \Theta_ {i j} = 0: \quad | - R _ {i j} + S _ {i j} | \leq \lambda_ {i j} \quad - R _ {i j} + S _ {i j} - \lambda_ {i j} \leq 0 \tag {10c}
$$
Simultaneously,  $\pmb{R}$  becomes non-negative  $(\geq 0)$ . As a result, The most challenging part of the KKT conditions  $-R_{ij} + S_{ij} - \lambda_{ij} \leq 0$  holds under  $\mathrm{MTP}_2$  properties. In Appendix C, we will elaborate the details and explore additional conditions under which we can extend the applicability of Theorem 3.3 to the traditional graphical lasso.
Proposed solving framework: In practical applications, it is often more efficient to employ the bridge-block decomposition technique instead of directly manipulating full-sized matrices. This approach involves breaking down the problem into smaller isolated sub-problems. By addressing these sub-problems individually, we can compute the optimal solution by utilizing the solutions obtained from each sub-problem, as outlined in Theorem 3.3. This approach offers several advantages: