SlowGuess/NeurIPS_2023_OCR huggingface dataset:
- Significant reduction in computational cost. Foremost, the cost of bridge-block decomposition is cheap. Suppose that we use a BCD solver of complexity  $\mathcal{O}(p^4)$ , then the total cost is reduced to  $\sum_{k=1}^{K} \mathcal{O}(|\mathcal{V}_k|^4) \ll \mathcal{O}(p^4)$ , where  $\sum_{k} |\mathcal{V}_k| = p$ . This can prompt an enormous difference.
- Considerable reduction in memory cost. The memory cost is typically troublesome for large-scale problems as each full-dimensional matrix contains  $p^2$  elements. Theorem 3.3 can avoid generating a number of full-dimensional intermediate variables during computation.
- Potential speed-up via parallel computing. The sub-problems can be optimized independently, which allows parallel computing for significant speed-up.
3.3 Connection to Existing Research
From Theorem 3.3, we obtain a very interesting result that the  $(i,j)$ -th entries of  $\Theta$  admits an explicit solution, i.e.,  $\Theta_{ij} = -T_{ij} / (S_{ii}S_{jj} - T_{ij}^2)$  if edge  $(i,j)$  is a bridge in the thresholded graph  $\operatorname{supp}(T)$ . To the best of our knowledge, this result, shown in Corollary 3.5, is the first sufficient condition for an edge belonging to optimal graph  $\operatorname{supp}(\Theta^\star)$ .
Corollary 3.5. If edge  $(i,j)$  is a bridge in the thresholded graph supp  $(T)$ , then  $(i,j)$  is also a bridge in the optimal graph supp  $(\Theta^{\star})$ .
Corollary 3.5 can be obtained as follows. By the definition of a bridge, from nodes  $i$  to  $j$  the path  $d_{ij} = \{(i,j)\}$  is the only path in the thresholded graph, and removing it would result in an increase in the number of components. Given that  $\mathrm{supp}(\Theta^{\star})\subseteq \mathrm{supp}(T)$  and  $\Theta_{ij}\neq 0$ , it follows that  $d_{ij} = \{(i,j)\}$  remains the unique path from nodes  $i$  to  $j$  in the optimal graph. This indicates that the edge  $(i,j)$  continues to function as a bridge.
In the literature, only some necessary conditions for edges in the thresholded graph supp  $(\pmb{T})$  to be retained in the optimal graph supp  $(\Theta^{\star})$  have been established. For instance, [13] and [20] indicate that supp  $(\Theta^{\star}) \subseteq \operatorname{supp}(\pmb{T})$ , which implies that  $(i,j) \notin \operatorname{supp}(\Theta^{\star})$  if  $(i,j) \notin \operatorname{supp}(\pmb{T})$ .
Another explicit solution is also applicable when  $k$ -th cluster only contains one node, i.e.,  $\mathcal{V}_k = \{i\}$ , then  $\widehat{\Theta}_k = 1 / S_{ii}$ . As a consequence, Theorem 3.3 generalizes to the following corollary for acyclic graph which admits a bridge-block decomposition into  $p$  clusters.
Corollary 3.6. Suppose that the thresholded graph  $\operatorname{supp}(\mathbf{T})$  is acyclic, then the optimal graph  $\operatorname{supp}(\Theta^{\star})$  is also acyclic and  $\Theta^{\star}$  admits a closed-form solution as
$$
\Theta_ {i j} ^ {\star} = \left\{ \begin{array}{l l} \frac {1}{S _ {i i}} \left(1 + \sum_ {m \in \mathcal {N} (i)} \frac {T _ {i m} ^ {2}}{S _ {i i} S _ {m m} - T _ {i m} ^ {2}}\right) & \text {i f} i = j, \\ - \frac {T _ {i j}}{S _ {i i} S _ {j j} - T _ {i j} ^ {2}} & \text {i f} (i, j) \in \mathcal {B}, \\ 0 & \text {o t h e r w i s e .} \end{array} \right. \tag {11}
$$
The proof directly follows Theorem 3.3. Corollary 3.6 coincides with [20, Theorem 3] and [38]. Compared to our results in Theorem 3.3, explicit solutions in literature heavily depend on the graph
structure. Hence, our theory appears as the first to reveal that, fundamentally, it is the edge property, i.e., whether the edge is a bridge, that decides the existence of explicit solutions.
In conclusion, our proposed methods generalize the existing results with more profound understandings, offering significant potential for solving large-scale precision matrix estimation problems.
4 Numerical Simulations
We conduct experiments on synthetic and real-world data to evaluate how the proposed method accelerates the convergence of existing algorithms compared to algorithms that do not exploit it. All experiments were conducted on 2.60GHZ Xeon Gold 6348 machines and Linux OS. All methods are implemented in MATLAB and the state-of-the-art methods we consider include
- BCD: Block Coordinate Descent [19] of complexity  $\mathcal{O}(p^4)$ .
- PGD: Projected Gradient Descent [36] of complexity  $\mathcal{O}(p^3)$ .
- PQN-LBFGS: A projected quasi-Newton method with limited memory BFGS [51]. The complexity is  $\mathcal{O}((m + p)p^2)$ , where  $m$  is the number of iterations stored for approximating the Hessian.
- FPN: Fast projected Newton-like method [37] of complexity  $\mathcal{O}(p^3)$ .
Note that all methods converge to the optimal solution. The target here is not to compare these methods but to evaluate how our proposed framework promotes the efficiency of these methods.
4.1 Synthetic Data Experiments
We use the processes described in [19] to generate the data. We begin with an underlying graph that has an adjacency matrix  $\mathbf{A} \in \mathbb{S}^p$ , and define  $\Theta = \delta \mathbf{I} - \mathbf{A}$ , where  $\delta = 1.05 \cdot \lambda_{\max}(\mathbf{A})$  and  $\lambda_{\max}(\mathbf{A})$  represents the largest eigenvalue of  $\mathbf{A}$ . His ensures that  $\Theta$  is a positive definite matrix with off-diagonal elements being negative, making  $\Theta$  a randomly generated  $M$ -matrix. Next, we normalize  $\Theta$  by substituting it with  $D\Theta D$ , where  $D$  is a suitably chosen diagonal matrix, ensuring that  $\mathrm{diag}(\Theta^{-1}) = 1$ . We then sample  $n = 10p$  data points from a Gaussian distribution  $\mathcal{N}(\mathbf{0},\Theta^{-1})$  and calculate the sample covariance matrix as  $S$ .
Following [25], we construct the regularization matrix  $\Lambda$ . Briefly, given an initial estimate  $\Theta^{(0)}$ , we set  $\Lambda_{ij} = \chi / (\Theta_{ij}^{(0)} + \epsilon)$  when  $i \neq j$  and  $\Lambda_{ij} = 0$  when  $i = j$ . Here,  $\chi > 0$  determines the sparsity level and  $\varepsilon$  is a small positive constant, such as  $10^{-3}$ . Generally, a large penalty is expected on  $\Theta_{ij}$  when  $\Theta_{ij}^{(0)}$  is small. We efficiently compute  $\Theta^{(0)}$  using the following equation:
$$
\Theta_ {i j} ^ {(0)} = - T _ {i j} \backslash \left(S _ {i i} S _ {j j} - T _ {i j} ^ {2}\right), \quad \forall i \neq j, \tag {12}
$$
and then appropriately adjust the value of  $\chi$  so that  $\operatorname{supp}(\mathbf{T})$  has only one connected component, while  $\operatorname{supp}(\Theta^{\star})$  can roughly recover the underlying structure.
We consider two scale-free random graphs as underlying with their typical structures in Figure 4.
1. Barabasi-Albert (BA) graph [52, 53] of order one. BA models generate random scale-free networks via preferential attachment, suitable for modeling various networks like the Internet, world wide web, protein interactions, citations, and social/online networks.
2. Stochastic Block Model (SBM) of networks [54], a.k.a. the community graph [55]. Stochastic block models serve as fundamental tools in network science, creating random networks based on community structures, where nodes within the same group are more likely to form connections.
In the experiments, we consider  $p = 5000$  and  $p = 20000$  and calculate the relative errors as follows:
$$
\operatorname {R E} (\Theta) = | f (\Theta) - f \left(\Theta^ {\star}\right) | / | f \left(\Theta^ {\star}\right) |, \tag {13}
$$
where we compare the relative errors at each iteration against the computational time. Here,  $\Theta^{\star}$  represents the optimal solution, and  $f$  signifies the objective function of (1). For methods employing bridge-block decomposition (bdd), we first calculate  $\widehat{\Theta}_k^{(s)}$  for each cluster at the  $s$ -th iteration, followed by calculating the intermediate solution  $\Theta^{(s)}$  using (7). The computational time at the  $s$ -th iteration is then the sum of the costs to obtain  $\mathcal{P}^{\mathrm{bdd}}$ , calculate all  $\widehat{\Theta}_k^{(s)}$ , and compute  $\Theta^{(s)}$  via (7).
Figure 3: Relative errors of the objective values versus the computational time. Colors are used for distinguishing the state-of-the-art methods. Solid circle symbols stand for methods accelerated by bridge-block decomposition (bdd) and square symbols stand for methods without acceleration.
The results for  $p = 5000$  and  $p = 20000$  are displayed in Figure 3, with all results averaged over five realizations. We highlight that the extra computational cost compared to methods without acceleration is negligible as shown in Table 1. From the figures, we observe that:
- Significant Speed-up: Our proposed framework substantially accelerates state-of-the-art methods by one to four orders of magnitude.
- Solving Otherwise Infeasible Problems: When  $p = 20000$ , existing methods cannot optimize the problem within  $10^{4}$  seconds. In contrast, our framework greatly speeds up convergence, making it possible to solve large-scale problems.
- Higher Speed-up for Sparser Graph: Proposed method achieves greater acceleration on the BA graph than the SBM graph, as the former exhibits a stronger sparsity pattern.
- Higher Speed-up for Methods of Higher-complexity: Figure 3 demonstrates that our method has a more significant impact on improving the efficiency of the BCD method, which has higher computational complexity and thus benefits more from dimension reduction.
To further shed light on the factors that determine the magnitudes of improvement, in the next sub-section, we conduct additional experiments that dive into how bridge-block decomposition boosts the performance of existing methods subject to different structures of the thresholded graph.
Table 1: Average computational time (seconds) of extra cost.
Conducting Bridge-block decomposition	Computing Θ from {Θk}Kk=1
p = 5k	p = 20k	p = 5k	p = 20k
BA	0.2	2.2	0.0	0.0
SBM	0.3	2.3	0.1	0.2
(1) BA graph
(2) SBM graph
Figure 4: Structures of BA graph and SBM graph.
Figure 5: Ratios of improvements for BCD method with respect to computational time.
4.2 Further Experiments on Ratios of Improvement
We investigate a scenario where the underlying graph is a community graph with a block-tridiagonal adjacency matrix, and each cluster contains an equal number of nodes  $(p_1 = |\mathcal{V}_k|, \forall k)$ . Without loss of generality, we assume that the internal edges within clusters form a cycle. To generate  $S$  and  $\Lambda$ , we follow the procedures in Section 4.1. We then modify  $\Lambda$  to  $\alpha (\mathbf{11}^T - \mathbf{A}) + (1 - \alpha) \Lambda$ , ensuring that  $\operatorname{supp}(T) = \operatorname{supp}(\mathbf{A})$ , where  $\alpha \in [0,1]$  and  $\mathbf{A}$  is the adjacency matrix of the considered graph.
We experiment with various values of  $K$  (the number of clusters) and  $p_1$  (the number of nodes in each cluster). For each configuration, we calculate the ratio of improvement, defined as the quotient of the time required to achieve  $\mathrm{RE}(\Theta) < 10^{-6}$  without using bridge-block decomposition to the time when applying the decomposition. We conduct multiple trials of the BCD method and display the results in Figure 5. The results in Figure 5 suggest that the number of sub-graphs  $K$  is a primary factor influencing the ratio values. The improvement is deemed tremendous, especially for large-scale settings. The involvement of bridge-block decomposition enables solving an  $\mathrm{MTP}_2$  GGMs with hundreds of millions of variables as long as we can conduce bridge-block decomposition. Consequently, many previously unattainable real-life applications can now be optimized.
4.3 Real-word Data Experiment
We consider learning the  $\mathrm{MTP}_2$  GGM for the Crop Image dataset available from the UCR Time Series Archive [56]. This dataset comprises  $p = 24000$  pixels, where each pixel corresponds to a time series record capturing spectral information in  $n = 46$  instances. These instances represent geometrically and radiometrically corrected satellite images. By examining this dataset, we can observe the temporal changes in the observed areas through the recorded measurements. To facilitate our analysis, the data has been pre-processed by [57] using an indexing technique. This preprocessing step enables us to work with compressed informative indexes instead of handling large image files.
Our goal is to perform graph-based clustering to the time series data that contains 46 observations, denoted as  $\{\pmb{y}_1,\dots ,\pmb{y}_{46}\}$  using  $\mathrm{MTP_2}$  GGMs, where  $\pmb {y}_i\in \mathbb{R}^{24000}$ . The data includes 24 classes corresponding to different indexed land covers, such as corn, barley, or lake. It is important to note that these labels are not involved in the clustering process but help evaluate the quality of the final graph-based clusters by computing the graph modularity [58]. Via analyzing the data, in Appendix D, we discuss that why learning  $\mathrm{MTP_2}$  GGMs is statistically meaningful in the context of clustering and show that the  $\mathrm{MTP_2}$  assumptions approximately hold for this dataset.
The initial estimate  $\Theta^{(0)}$  is computed using the method proposed in [59]. The regularization matrix  $\Lambda$  is determined using the approach in Section 4.1 with  $\epsilon = 0.01$  and  $\chi = 0.2$ . We then compute the sample correlation matrix as  $S \in \mathbb{S}^{24000}$ . This results in a precision matrix estimation problem involving  $5.76 \times 10^8$  variables to optimize, which is quite challenging for state-of-the-art methods without bridge-block decomposition.
Figure 6b shows the results of empirical convergence, and Figure 6a visualizes a local structure of  $\mathrm{supp}(\Theta)^{\star}$  (modularity  $= 0.6849$ ). In line with previous findings, our suggested approach greatly speeds up the convergence of all current algorithms by a minimum of three orders of magnitude. This demonstrates immense potential for managing large-scale sparse graphical models.
(a) Visualizations.
(b) Experimental results on Crop data  $(p = 24000)$ .
Figure 6: Visualizations and experimental results of sparse  $\mathrm{MTP}_2$  GGMs for Crop data set. (a) A local structure of optimal graph with 2008 nodes. Nodes with matching labels are assigned the same color and connected by a matching edge color, while different groups of nodes are connected by gray edges. (b) Results of convergence to learn sparse  $\mathrm{MTP}_2$  GGMs for Crop data set.
While our paper's primary focus is not to uncover insights into the estimated  $\mathrm{MTP}_2$  GGMs for a deeper understanding of the underlying data, interesting phenomena can still be observed in the structures presented in Figure 6a. Notably, we find that the majority of edges exist within the same crop type, while connections between nodes associated with different crops are relatively sparse. This observation is valuable for clustering processes and aligns with our expectations, as stronger positive dependencies are often observed within the same class, while dependencies between different classes tend to be weaker.
Moreover, our graphical representation reveals more nuanced patterns. For instance, in Figure 6a of our manuscript, we observe a dense network of edges between two crop types, 'temporary meadow' and 'pasture', indicating a significantly stronger conditional dependency compared to other categories. This observation further supports our understanding. Hence, the inferred conditional dependency structure encapsulates the inherent interrelationships among different crops. As a result, our proposed framework based on bridge-block decomposition effectively facilitates learning in large-scale  $\mathrm{MTP}_2$  graphical models.
5 Conclusions and Discussions
Real-world sparse Gaussian graphical models often comprise subsets of variables that are densely connected to one another, while variables in different clusters maintain weak connections. However, standard estimation algorithms do not account for this property. In this paper, by introducing the concept of bridge, we leverage these characteristics to reveal an interesting finding: the optimal solution for Problem (1) equivalently admits a decomposed form via the bridge-block decomposition of the thresholded graph. We provide theoretical insights into the separability of optimal solutions and the existence of explicit expressions based on edge properties, specifically whether an edge is a bridge. This method surpasses conventional approaches that depend on unique graph structures, such as determining if a graph is acyclic. From the practical aspect, our proposed method offers a handy architect for accelerating any existing algorithms by handling a large-scale learning problem via a number of small and tractable sub-problems. Although our method is mainly developed for sparse large-scale graphs and therefore seems inapplicable for dense graph, as elaborated in the appendix C, it can also provide a quick way to obtain a solution, which can serve as either a starting point (warm-start) for numerical algorithms or an alternative for acquiring approximate solutions. Overall, our simple and provable approach demonstrates exceptional performance and paves a novel way for more extensive designs of large-scale  $\mathrm{MTP}_2$  Gaussian graphical models.
[1] H. Rue and L. Held, "Gaussian Markov random fields: theory and applications," Monographs on Statistics and Applied Probability, vol. 104, 2005.
[2] S. L. Lauritzen, Graphical models. Clarendon Press, 1996, vol. 17.
[3] J. Whittaker, Graphical models in applied multivariate statistics. Wiley Publishing, 2009.
[4] O. Banerjee, L. El Ghaoui, and A. d'Aspremont, "Model selection through sparse maximum likelihood estimation for multivariate Gaussian or binary data," The Journal of Machine Learning Research, vol. 9, pp. 485-516, 2008.
[5] J. Friedman, T. Hastie, and R. Tibshirani, "Sparse inverse covariance estimation with the graphical lasso," Biostatistics, vol. 9, no. 3, pp. 432-441, 2008.
[6] M. Yuan and Y. Lin, "Model selection and estimation in the Gaussian graphical model," Biometrika, vol. 94, no. 1, pp. 19-35, 2007.
[7] H. Liu, K. Roeder, and L. Wasserman, "Stability approach to regularization selection for high dimensional graphical models," Advances in Neural Information Processing Systems, vol. 23, 2010.
[8] P. Danaher, P. Wang, and D. M. Witten, “The joint graphical lasso for inverse covariance estimation across multiple classes,” Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol. 76, no. 2, pp. 373–397, 2014.
[9] E. Bølviken, "Probability inequalities for the multivariate normal with non-negative partial correlations," Scandinavian Journal of Statistics, pp. 49-58, 1982.
[10] D. M. Malioutov, J. K. Johnson, and A. S. Willsky, "Walk-sums and belief propagation in Gaussian graphical models," The Journal of Machine Learning Research, vol. 7, pp. 2031-2064, 2006.
[11] R. J. Plemmons, “M-matrix characterizations. I-nonsingular M-matrices,” Linear Algebra and its Applications, vol. 18, no. 2, pp. 175–188, 1977.
[12] S. Fallat, S. Lauritzen, K. Sadeghi, C. Uhler, N. Wermuth, and P. Zwiernik, “Total positivity in Markov structures,” The Annals of Statistics, pp. 1152–1184, 2017.
[13] S. Lauritzen, C. Uhler, and P. Zwiernik, "Maximum likelihood estimation in Gaussian models under total positivity," The Annals of Statistics, vol. 47, no. 4, pp. 1835-1863, 2019.
[14] F. Röttger, S. Engelke, and P. Zwiernik, "Total positivity in multivariate extremes," The Annals of Statistics, vol. 51, no. 3, pp. 962-1004, 2023.
[15] D. Rossell and P. Zwiernik, “Dependence in elliptical partial correlation graphs,” *Electronic Journal of Statistics*, vol. 15, no. 2, pp. 4236–4263, 2021.
[16] A. Müller and M. Scarsini, “Archimedean copulae and positive dependence,” Journal of Multivariate Analysis, vol. 93, no. 2, pp. 434–445, 2005.
[17] R. Agrawal, U. Roy, and C. Uhler, "Covariance matrix estimation under total positivity for portfolio selection," Journal of Financial Econometrics, vol. 20, no. 2, pp. 367-389, 2022.
[18] R. Zhou, J. Ying, and D. P. Palomar, "Covariance matrix estimation under low-rank factor model with nonnegative correlations," IEEE Transactions on Signal Processing, vol. 70, pp. 4020-4030, 2022.
[19] M. Slawski and M. Hein, "Estimation of positive definite M-matrices and structure learning for attractive Gaussian markov random fields," Linear Algebra and its Applications, vol. 473, pp. 145-179, 2015.
In what follows, we present technical proofs omitted in the main content. In Section A, we introduce important properties of matrix  $\mathbf{R}$  and prove Theorem 3.4. In Section B, we prove the main result, i.e., Theorem 3.3. In Section C, we introduce some extension of the proposed framework to broaden its applicability. In Section D, we will comment on real-world experiments and explain why  $\mathrm{MTP}_2$  assumption approximately holds for Crop Image dataset.
A Proof of Theorem 3.4
This section contains a detailed description of how we construct the matrix  $\mathbf{R}$  and prove Theorem 3.4 based on the properties of  $\mathbf{R}$ . The proof relies on the concept of a 'generalized path product', which involves a set of edges labeled as a 'bridge path'. The matrix  $\mathbf{R}$  has several notable properties that are essential in demonstrating Theorem 3.3.
A.1 Definitions and Important Properties
To begin, let us define the 'bridge path'  $\mathcal{B}_{ij}$  for nodes  $i$  and  $j$  that belong to different clusters, i.e.,  $\psi (i)\neq \psi (j)$  based on the bridge set  $\mathcal{B}$  in the thresholded graph  $G$ . The bridge path  $\mathcal{B}_{ij}$  is defined as the set of bridges in any path  $d_{ij}$  from node  $i$  to node  $j$ , where each element in  $\mathcal{B}_{ij}$  is a bridge in  $\mathcal{B}$  and the elements in  $\mathcal{B}_{ij}$  follow the order of  $d_{ij}$  while preserving the elements in  $\mathcal{B}$ . Symbolically, we can express this as:
$$
\mathcal {B} _ {i j} = d _ {i j} \cap \mathcal {B} = \left\{\left(u _ {1}, u _ {2}\right), \left(u _ {3}, u _ {4}\right), \dots , \left(u _ {2 T - 1}, u _ {2 T}\right) \right\}, \tag {14}
$$
where  $T$  denotes the number of clusters in any path from nodes  $i$  to  $j$ . It is important to note the following:
Each element in  $\mathcal{B}_{ij}$  is a bridge in the thresholded graph.
- The elements in  $\mathcal{B}_{ij}$  follow the orders of  $d_{ij}$ . This means that for any path from nodes  $i$  to  $j$ , the path will visit  $(u_{2i - 1}, u_{2i})$ , then visit  $(u_{2j - 1}, u_{2j})$  whenever we have  $i < j$ . Consequently, we have
$$
\psi (i) = \psi (u _ {1}), \quad \psi (u _ {2}) = \psi (u _ {3}), \dots , \psi (u _ {2 T - 2}) = \psi (u _ {2 T - 1}), \quad \psi (u _ {2 T}) = \psi (j). \tag {15}
$$
- For any path  $d_{ij}$ , the bridge path  $\mathcal{B}_{ij} = d_{ij} \cap \mathcal{B}$  is unique.
-  $\mathcal{B}_{ij}$  is defined as an empty set if there is no path from nodes  $i$  to  $j$ .
It is important to mention that if the thresholded graph is acyclic, the bridge path can be reduced to a traditional path. In this scenario, all edges in the path  $d_{ij}$  become bridges, and the bridge path  $\mathcal{B}_{ij}$  is equivalent to the path  $d_{ij}$ , which can be expressed as  $\mathcal{B}_{ij} = d_{ij}$ .
We define the matrix  $\mathbf{R} \in \mathbb{S}^p$  based on the definition of bridge-path, as shown below:
$$
R _ {i j} = \left\{ \begin{array}{l l} {\left[ \widehat {\boldsymbol {R}} _ {k} \right] _ {\pi (i), \pi (j)}} & \text {i f} i, j \in \mathcal {V} _ {k}, \\ T _ {i j} & \text {i f} (i, j) \in \mathcal {B}, \\ \sqrt {S _ {i i} S _ {j j}} \cdot g _ {i j} (\boldsymbol {R}) & \text {o t h e r w i s e}, \end{array} \right. \tag {16}
$$
where  $\widehat{\pmb{R}}_k = [\widehat{\Theta}_k]^{-1}$  and for each  $i,j$  in different clusters,  $g_{ij}$  is given as
$$
g _ {i j} (\boldsymbol {R}) = \prod_ {t = 0} ^ {2 T} R _ {u _ {t}, u _ {t + 1}} / \sqrt {S _ {u _ {t} , u _ {t}} S _ {u _ {t + 1} , u _ {t + 1}}}, \quad \text {w h e r e} u _ {0} \triangleq i, u _ {2 T + 1} \triangleq j, \tag {17}
$$
where  $u$  denotes the bridge path  $\mathcal{B}_{ij}$  and  $g_{ij}(\pmb {R}) = 0$  if  $\mathcal{B}_{ij} = \emptyset$