SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In fact, with the reconstructed  $\widetilde{\mathbf{U}}$ , we can set  $\widetilde{\mathbf{Q}} = \mathbf{C}\widetilde{\mathbf{U}}\mathbf{R}$  as the regularization  $\mathbf{B}$ . It is, of course, sensible and feasible. However, a potential drawback is that the reconstruction of  $\widetilde{\mathbf{Q}}$  requires matrix multiplications of three matrices, which is time-consuming. Unexpectedly, we find that only constructing matrix  $\widetilde{\mathbf{U}}$  is enough to solve the SS problem since it models the relationships between 1-hop starved nodes and labeled ones.
vector  $\widetilde{\mathbf{A}}_{:j}$  of  $\widetilde{\mathbf{A}}$  for  $j\in \mathbb{C}\mathsf{M}_{+}$ . We define the weight contribution rates (WCR) of non-starved nodes  $\rho_{-}$  and the WCR of starved nodes  $\rho_{+}$  as:
$$
\rho_ {-} = \frac {\sum_ {i \notin \mathrm {R M} _ {+}} \widetilde {\mathbf {A}} _ {i j}}{\sum_ {i} \widetilde {\mathbf {A}} _ {i j}} = \frac {\sum_ {i \notin \mathrm {R M} _ {+}} \mathbf {A} _ {i j}}{\sum_ {i} \widetilde {\mathbf {A}} _ {i j}}, \quad \rho_ {+} = 1 - \rho_ {-} = 1 - \frac {\sum_ {i \notin \mathrm {R M} _ {+}} \mathbf {A} _ {i j}}{\sum_ {i} \widetilde {\mathbf {A}} _ {i j}} = \frac {\sum_ {i \in \mathrm {R M} _ {+}} \alpha \widetilde {\mathbf {U}} _ {i j}}{\sum_ {i} \widetilde {\mathbf {A}} _ {i j}}. \tag {7}
$$
From Eq. (7), we observe that the WCR of non-starved nodes  $\rho_{-}$  decays as the number of starved nodes increases (i.e.,  $\rho_{+}$  increases). This means that  $\rho_{-}$  becomes negligible if there are numerous starved nodes. As a result, the  $j$ -th column vector  $\widetilde{\mathbf{A}}_{:j}$  of  $\widetilde{\mathbf{A}}$  for  $j \in \mathbb{C}\mathbb{M}_{+}$  heavily relies on the  $j$ -th column vector  $\widetilde{\mathbf{U}}_{:j}$  of the reconstructed intersection matrix  $\widetilde{\mathbf{U}}$ . This outcome is not desirable since the regularization imposed should not dominate a significant portion of the final adjacency matrix. Recalling our initial objective of properly refining the original latent graph to replenish the missed supervision, we will design two simple strategies to relieve this issue.
Decrease  $\rho_{+}$ . On the one hand, we can decrease the WCR of starved nodes  $\rho_{+}$  by selecting only  $\tau (\tau < c)$  labeled nodes as the supplementary adjacent points for each 1-hop starved node. This results in a sparse intersection matrix  $\widehat{\mathbf{U}}$ . For this strategy, we present the following proposition:
Proposition 1. Suppose that we randomly select  $\tau$  out of  $c$  labeled nodes as the supplementary adjacent points for each 1-hop starved node. If there are  $r$  1-hop starved nodes, then for  $\forall j \in CM_{+}$ , we have  $\widehat{\rho}_{+} = \frac{\sum_{i \in R_{H}} \alpha \widehat{U}_{ij}}{\sum_{i} \widetilde{A}_{ij}} \leq \rho_{+}$ , where  $\widehat{\rho}_{+} = \rho_{+}$  with only a probability of  $\left(\frac{\tau}{c}\right)^r$ .
Increase  $\rho_{-}$ . On the other hand, we can improve the WCR of non-starved nodes  $\rho_{-}$  by magnifying the weights of non-starved nodes. Specifically, we construct an additional regularization matrix  $\mathbf{Q} \in \mathbb{R}^{n \times c}$ , where its  $c$  columns correspond to the  $c$  labeled nodes and  $\mathbf{Q}_{ij} = 0$  for  $\forall i \in \mathbb{R}\mathbb{M}_{+}$ . For  $\forall i \notin \mathbb{R}\mathbb{M}_{+}$ , we establish connections between the node  $V_{i}$  and  $\tau(\tau < c)$  labeled nodes  $V_{j}$ , and assign the corresponding weights  $\mathbf{Q}_{ij}$  using a similar strategy as in constructing  $\widehat{\mathbf{U}}$ . By adding the additional regularization matrix  $\mathbf{Q}$ , the refined adjacency matrix  $\widehat{\mathbf{A}}$  can be expressed as:
$$
\widehat {\mathbf {A}} = \mathbf {A} + \alpha (\Gamma (\widehat {\mathbf {U}}, n) + \Gamma (\mathbf {Q}, n)). \tag {8}
$$
Similarly, for this strategy, we have the following proposition:
Proposition 2. Suppose that we randomly select  $\tau$  out of  $c$  labeled nodes as the supplementary adjacent points for each non-starved node. If there are  $r$  1-hop starved nodes, then for  $\forall j\in CM_{+}$ , we have  $\widehat{\rho}_{-} = \frac{\sum_{i\notin RM_{+}}\widehat{\mathbf{A}}_{ij}}{\sum_{i}\widehat{\mathbf{A}}_{ij}} = \frac{\sum_{i\notin RM_{+}}\widetilde{\mathbf{A}}_{ij} + \alpha\sum_{i\notin RM_{+}}\mathbf{Q}_{ij}}{\sum_{i}\widehat{\mathbf{A}}_{ij} + \alpha\sum_{i\notin RM_{+}}\mathbf{Q}_{ij}}\geq \rho_{-}$ , where  $\widehat{\rho}_{-} = \rho_{-}$  with only a probability of  $(1 - \frac{\tau}{c})^{n - r}$ .
3.4 End-to-End Training
Note that the proposed approach is model-agnostic and can be seamlessly integrated into existing LGI models. In Sec. 4, we will apply our design to state-of-the-art LGI methods [4, 10, 13, 46] and compare its performance with the original approach. Therefore, we follow these methods and implement the node encoder  $\mathcal{F}_{\Theta}$  using a 2-layer GNN. Then, we calculate the cross-entropy loss  $\mathcal{L}_{\mathrm{ce}}$  between the true labels  $\mathbf{Y}$  and the predictions  $\mathbf{Z}$ , as well as a graph regularization loss  $\mathcal{L}_{\mathrm{reg}}$  on  $\widehat{\mathbf{A}}$ :
$$
\min  \mathcal {L} = \mathcal {L} _ {\mathrm {c e}} + \gamma \mathcal {L} _ {\mathrm {r e g}} = \sum_ {i \in \mathcal {Y} _ {L}} \sum_ {j = 1} ^ {| \mathcal {C} |} \mathbf {Y} _ {i j} \ln \mathbf {Z} _ {i j} + \gamma \mathcal {L} _ {\mathrm {r e g}}, \tag {9}
$$
where  $\gamma$  is a balanced parameter and  $\mathbf{Z} = \mathcal{F}_{\Theta}(\mathbf{X},\widehat{\mathbf{A}}) = \mathrm{softmax}\left(\widehat{\mathbf{A}}\sigma \left(\widehat{\mathbf{A}}\mathbf{X}\Theta^{0}\right)\Theta^{1}\right)$ . According to different LGI methods, the graph regularization loss  $\mathcal{L}_{\mathrm{reg}}$  can be different, such as Dirichlet energy [4] and self-supervision [10] (see supplementary material for more details). For fairness, in the experiments, we will adopt the same graph regularization loss as the comparison methods.
Table 1: Test accuracy (\%) of the baselines (M) and our CUR extension versions (M_U and M_R) on various datasets with different labeling rates (marked in **bold**), where “OOM” indicates out of memory. The highest and second highest results are marked in red and blue, respectively.
Models / datasets	ogbn-arxiv	Cora390	Cora140	Citeseer370	Citeseer120	Pubmed
# of labeled / all nodes	90941/169343	390/2708	140/2708	370/3327	120/3327	60/19717
Labeling rate	53.70%	14.40%	5.17%	11.12%	3.61%	0.30%
GCN+KNN	55.15 ± 0.11	72.82 ± 0.39	67.94 ± 0.29	73.28 ± 0.23	69.68 ± 0.53	68.66 ± 0.05
GCN+KNN_U (ours)	55.82 ± 0.11	72.82 ± 0.21	68.18 ± 0.44	73.68 ± 0.10	69.74 ± 0.54	74.12 ± 0.32
GCN+KNN_R (ours)	55.86 ± 0.10	72.92 ± 0.28	68.12 ± 0.48	73.66 ± 0.14	69.90 ± 0.68	74.78 ± 0.17
GCN&KNN	OOM	72.16 ± 0.54	68.76 ± 1.20	77.28 ± 0.64	68.64 ± 1.14	OOM
GCN&KNN_U (ours)	OOM	73.04 ± 0.20	70.16 ± 0.91	78.40 ± 0.44	70.52 ± 1.04	OOM
GCN&KNN_R (ours)	OOM	73.20 ± 0.25	70.24 ± 0.97	78.48 ± 0.30	69.48 ± 0.77	OOM
IDGL [4]	OOM	74.00 ± 0.38	70.74 ± 0.50	71.30 ± 0.17	69.24 ± 0.19	OOM
IDGL_U (ours)	OOM	74.54 ± 0.52	70.82 ± 0.49	72.46 ± 0.14	69.32 ± 0.39	OOM
IDGL_R (ours)	OOM	74.48 ± 0.47	71.14 ± 0.22	72.56 ± 0.12	69.86 ± 0.50	OOM
LCGS [13]	OOM	72.02 ± 0.37	69.88 ± 0.66	73.84 ± 0.83	72.30 ± 0.33	OOM
LCGS_U (ours)	OOM	72.18 ± 0.31	70.04 ± 0.80	74.18 ± 0.43	72.38 ± 0.43	OOM
LCGS_R (ours)	OOM	72.22 ± 0.45	70.14 ± 0.64	74.20 ± 0.36	72.40 ± 0.42	OOM
GRCN [46]	OOM	73.34 ± 0.27	68.86 ± 0.25	73.62 ± 0.23	71.24 ± 0.19	69.24 ± 0.20
GRCN_U (ours)	OOM	74.10 ± 0.25	69.44 ± 0.34	73.88 ± 0.34	71.54 ± 0.31	72.80 ± 0.99
GRCN_R (ours)	OOM	74.14 ± 0.22	69.56 ± 0.22	74.22 ± 0.13	71.64 ± 0.41	72.82 ± 1.03
SLAPS [10]	55.46 ± 0.12	76.62 ± 0.83	74.26 ± 0.53	74.32 ± 0.56	70.66 ± 0.97	74.86 ± 0.79
SLAPS_U (ours)	55.68 ± 0.09	76.94 ± 0.42	74.56 ± 0.21	74.82 ± 0.27	71.68 ± 0.47	76.74 ± 0.59
SLAPS_R (ours)	56.11 ± 0.15	76.82 ± 0.19	75.00 ± 0.49	74.90 ± 0.42	72.36 ± 0.49	77.12 ± 0.77
Baselines. As mentioned earlier, the proposed regularization module can be easily integrated into most existing LGI methods. To evaluate its effectiveness, we select representative LGI methods as baselines, including IDGL [4], GRCN [46], SLAPS [10], and LCGS [13]. We also consider two additional baselines marked as GCN+KNN [10, 19] and GCN&KNN [4, 19]. GCN+KNN is a two-step method that first constructs a KNN graph based on feature similarities and then feeds the pre-constructed graph to a GCN for training. GCN&KNN is an end-to-end method that learns the latent graph and network parameters simultaneously. For methods that require a prior graph, we use the same KNN graph as in GCN+KNN. For GCN&KNN, we simply adopt the Dirichlet energy [4] as the graph regularization loss.
Datasets. Following the common settings of existing LGI methods [4, 10, 13, 46], we conduct experiments on four well-known benchmarks: Cora, Citeseer, Pubmed [19, 34], and ogbn-arxiv [14]. For detailed dataset statistics, please refer to the supplementary material. For all datasets, we only provide the original node features for training. To test the performance under different labeling rates, for the Cora and Citeseer datasets, we add half of the validation samples to the training sets, resulting in Cora390 and Citeseer370, where the suffix number represents the total number of labeled nodes.
Implementation. We compare the above baselines with their corresponding CUR extension versions. Specifically, we consider two CUR extensions termed M_U and M_R (M refers to the baseline), where the former adopts the sparse intersection matrix  $\widehat{\mathbf{U}}$  as the regularization, and the latter combines  $\widehat{\mathbf{U}}$  and  $\mathbf{Q}$  together (see Sec. 3.3 for details). In experiments, we practically select the  $\tau$  closest labeled nodes as supplementary adjacent points for each row of  $\widehat{\mathbf{U}}$  and  $\mathbf{Q}$ . For postprocessing operations on the graph, such as symmetrization and normalization [4, 10], we follow the baselines and adopt the same operations for fairness. We select the values of  $\tau$  and  $\alpha$  from the sets  $\{10, 15, 20, 25, 30, 50\}$  and  $\{0.01, 0.1, 1.0, 10, 50, 100\}$ , respectively. For other hyperparameters such as learning rate and weight decay, we follow the baselines and use the same settings. For each method, we record the best testing performance and report the average accuracy of five independent experiments, along with the corresponding standard deviation.
Table 1 presents the comparison results on all used datasets. It is evident that our proposed CUR extensions consistently outperform the corresponding baselines, demonstrating the effectiveness of eliminating starved nodes. When considering the labeling rates of different datasets listed in the third row of Table 1, we observe that the lower the labeling rate, the greater the performance
(a) GCN+KNN
(b) GRCN
Figure 2: Training loss (left vertical axis) and testing accuracy (right vertical axis) curves of GCN+KNN, GRCN, SLAPS, and their corresponding CUR extensions on the Pubmed dataset.
(c) SLAPS
Table 2: Number of  $k$ -hop starved nodes ( $k \in \{1,2\}$ ) on various datasets when selecting different number of neighbors ( $\tau$ ) in graph sparsification operation.
Number of neighbors (τ)	10	20
Datasets	Cora140	CiteSeer120	Pubmed	Cora140	CiteSeer120	Pubmed
1-hop starved nodes	1,625	2,268	19,076	993	1,577	18,478
2-hop starved nodes	100	299	16,756	0	0	12,843
improvement achieved by our methods. Notably, on the Pubmed dataset with an extremely low labeling rate of  $0.30\%$ , the accuracy of our proposed methods (M_R) increase by  $6.12\%$ ,  $3.58\%$ , and  $2.26\%$  compared to the basic models (M) of GCN+KNN, GRCN, and SLAPS, respectively. This is because the lower the labeling rate, the more starved nodes exist in the dataset (see Fig. 1 for an example). Our proposed methods aim to restore the destroyed affinities between starved nodes and labeled nodes, enabling us to leverage the semantic supervision missed by baselines, thereby achieving superior performance with extremely limited supervision.
Fig. 2 displays the training loss and testing accuracy curves of GCN+KNN, GRCN, SLAPS, and their CUR extensions on the Pubmed dataset. We omit the curves for the first 200 epochs of SLAPS because, during this stage, SLAPS focuses solely on learning the latent graph through an additional self-supervision task without involving the classification task. From Fig. 2, we observe that our proposed CUR extensions achieve higher testing accuracy compared to their corresponding baselines. In comparison with the CUR extensions of GCN+KNN, the CUR extensions of GRCN and SLAPS exhibit relatively stable accuracy results as the training epoch increases. This stability can be attributed to GRCN and SLAPS jointly learning the latent graph and network parameters in an end-to-end manner, whereas GCN+KNN pre-constructs a KNN graph and maintains a fixed graph during training without optimization. For GRCN, our proposed CUR extensions demonstrate lower training loss and higher testing accuracy, further indicating their improved generalization by removing starved nodes. Additionally, we observe relatively unstable training loss for the CUR extensions of SLAPS. The potential reason is that SLAPS introduces an additional graph regularization loss through a self-supervision task, while its CUR extensions aim to reconstruct the destroyed connections in the latent graph. As a result, the training process for the self-supervision task can exhibit some instability.
We would like to explore the question that why our proposed methods yield a slight improvement on the Cora and CiteSeer datasets, while achieving a substantial improvement on the Pubmed dataset. As shown in Table 2, when  $\tau = 10$ , there are 100 and 299 2-hop starved nodes on the Cora140 and CiteSeer120 datasets, respectively. However, when  $\tau = 20$ , there are no 2-hop starved nodes on the Cora140 and CiteSeer120 datasets, and the number of 1-hop starved nodes also sharply decreases. In this scenario, since the comparison baselines all utilize a 2-layer GNN, they are unaffected by 2-hop starved nodes and only minimally affected by 1-hop starved nodes. Consequently, our methods result in only slight improvements over the baselines on the Cora140 and CiteSeer120 datasets. On the other hand, when  $\tau = 20$ , there are still 12,843 2-hop starved nodes present on the Pubmed dataset. Since we effectively eliminate these starved nodes without requiring additional graph convolutional layers, our methods can provide notable benefits on this dataset with an extremely low labeling rate.
Table 3: Test accuracy (%) of our proposed CUR extensions for GCN+CNN, GRCN, and SLAPS when eliminating different number of staved nodes on the Pubmed dataset.
Staved nodes (%)	100	80	60	40	20	10	0
GCN+KNN_U	68.66 ± 0.05	69.42 ± 0.12	70.20 ± 0.22	71.52 ± 0.15	72.40 ± 0.27	73.04 ± 0.29	74.12 ± 0.32
GCN+KNN_R	69.12 ± 0.33	69.46 ± 0.22	70.46 ± 0.19	71.90 ± 0.15	73.24 ± 0.05	74.00 ± 0.14	74.78 ± 0.17
GRCN_U	-69.24 ± 0.20	-72.14 ± 0.35	-72.48 ± 0.43	-73.04 ± 0.52	-72.82 ± 0.75	-72.96 ± 0.87	-72.80 ± 0.99
GRCN_R	70.20 ± 0.06	72.34 ± 0.30	72.54 ± 0.48	72.98 ± 0.37	72.80 ± 0.70	72.80 ± 0.74	72.82 ± 1.03
SLAPS_U	-74.86 ± 0.79	-76.26 ± 0.62	-76.48 ± 0.61	-76.36 ± 0.39	-76.48 ± 0.67	-76.32 ± 0.71	-76.74 ± 0.59
SLAPS_R	75.64 ± 0.45	76.44 ± 1.27	76.52 ± 0.18	76.50 ± 1.22	76.38 ± 0.45	76.70 ± 0.59	77.12 ± 0.77
Table 4: Parameter sensitivity of  $\tau$  when applying our proposed method to GCN+CNN, GRCN, and SLAPS on the Pubmed dataset. The baseline results indicate the accuracy of the original methods.
The value of τ	baseline (0)	10	15	20	25	30	50
GCN+KNN_U	68.66 ± 0.05	71.86 ± 0.26	72.92 ± 0.17	73.50 ± 0.31	73.88 ± 0.28	74.12 ± 0.32	73.76 ± 0.29
GCN+KNN_R	68.66 ± 0.05	73.10 ± 0.09	73.90 ± 0.14	74.02 ± 0.10	74.56 ± 0.12	74.78 ± 0.17	74.62 ± 0.12
GRCN_U	-69.24 ± 0.20	-71.92 ± 0.87	-72.56 ± 0.77	-72.64 ± 1.03	-72.80 ± 0.99	-72.56 ± 1.02	-71.80 ± 1.13
GRCN_R	-69.24 ± 0.20	-72.12 ± 0.86	-72.54 ± 0.77	-72.80 ± 1.05	-72.82 ± 1.03	-72.44 ± 1.07	-71.86 ± 1.11
SLAPS_U	-74.86 ± 0.79	-75.98 ± 1.12	-76.20 ± 0.87	-75.58 ± 0.90	-76.28 ± 0.61	-76.74 ± 0.59	-75.98 ± 0.89
SLAPS_R	-74.86 ± 0.79	-76.00 ± 1.17	-76.48 ± 0.69	-76.40 ± 0.48	-77.12 ± 0.77	-76.58 ± 0.33	-76.50 ± 0.59
Table 5: Parameter sensitivity of  $\alpha$  when applying our proposed method to GCN+CNN, GRCN, and SLAPS on the Pubmed dataset. The baseline results indicate the accuracy of the original methods.
The value of α	baseline (0)	0.01	0.1	1.0	10	50	100
GCN+KNN_U	68.66 ± 0.05	68.04 ± 0.05	67.96 ± 0.08	68.14 ± 0.10	69.92 ± 0.12	72.70 ± 0.06	74.12 ± 0.32
GCN+KNN_R	68.66 ± 0.05	67.96 ± 0.10	67.90 ± 0.11	68.18 ± 0.07	70.04 ± 0.16	73.22 ± 0.07	74.78 ± 0.17
GRCN_U	-69.24 ± 0.20	-69.24 ± 0.10	-69.72 ± 0.19	-71.92 ± 0.24	-72.80 ± 0.99	-67.22 ± 3.93	-59.44 ± 6.74
GRCN_R	-69.24 ± 0.20	-69.24 ± 0.10	-69.66 ± 0.22	-71.94 ± 0.22	-72.82 ± 1.03	-68.98 ± 4.05	-60.34 ± 6.50
SLAPS_U	-74.86 ± 0.79	-74.88 ± 0.90	-74.48 ± 0.72	-74.94 ± 0.81	-76.26 ± 0.80	-76.74 ± 0.59	-76.08 ± 0.97
SLAPS_R	-74.86 ± 0.79	-74.62 ± 1.51	-74.32 ± 0.83	-74.76 ± 0.71	-76.22 ± 0.55	-77.12 ± 0.77	-76.74 ± 0.68
In this subsection, we aim to explore and answer the following questions.
How many starved nodes should be eliminated? The results shown in Table 1 indicate that eliminating all starved nodes contributes to the performance improvement of the baselines. It is important to understand the relationship between the number of starved nodes removed and the corresponding performance improvement of the baselines. To investigate this, we randomly remain  $10\%$ ,  $20\%$ ,  $40\%$ ,  $60\%$ ,  $80\%$  starved nodes and evaluate the performance of GCN+CNN, GRCN, and SLAPS accordingly. The results on the Pubmed dataset are summarized in Table 3. We find that, for GCN+KNN, a smaller number of starved nodes leads to higher testing accuracy. For GRCN and SLAPS, however, we need to check the number of starved nodes to obtain optimal performance.
How  $\tau$  affects the performance. Table 4 shows that the selection of  $\tau$  slightly differentiates the performance of our proposed methods. In general, a larger value of  $\tau$  leads to a higher improvement in performance. However, when  $\tau$  is set too large, such as 50, the performance starts to degrade. This degradation occurs due to the introduction of incorrect labels when  $\tau$  exceeds a certain threshold.
How  $\alpha$  affects the performance. Table 5 presents the sensitivity of parameter  $\alpha$  on the Pubmed dataset. It is observed that a relatively larger value of  $\alpha$ , such as 10 or 50, leads to a significant improvement in performance. This finding further emphasizes the effectiveness of our proposed regularization methods in enhancing the performance of existing LGI models.
Latent Graph Inference. Given only the node features of data, latent graph inference (LGI) aims to simultaneously learn the underlying graph structure and discriminative node representations from the features of data [10, 24, 39]. For example, Jiang et al. [15] propose to infer the graph structure by combining graph learning and graph convolution in a unified framework. Yang et al. [45] model the topology refinement as a label propagation process. Jin et al. [16] explore some intrinsic properties of the latent graph and propose a robust LGI framework to defend adversarial attacks on graphs. Though effective, these methods require a prior graph to guide the graph inference process. Controversially,
some methods have been proposed to directly infer an optimal graph from the data. For example, Franceschiet et al. [11] regard the LGI problem as a bilevel program task and learn a discrete probability distribution on the edges of the latent graph. Norcliffe-Brown et al. [29] focus on visual question answering task and propose to learn an adjacency matrix from image objects so that each edge is conditioned on the questions. Fatemi et al. [10] propose a self-supervision guided LGI method, called SLAPS, which yields supplementary supervision from node features through an additional self-supervision task. Our method is totally different from SLAPS as we provide supplementary supervision directly from the true labels. More importantly, our method is model-agnostic and can be easily integrated into most existing LGI methods.
CUR Matrix Decomposition. The CUR decomposition [2, 3] of a matrix  $\mathbf{Q} \in \mathbb{R}^{n \times m}$  aims to find a column matrix  $\mathbf{C} \in \mathbb{R}^{n \times c}$  with a subset of  $c < m$  columns of  $\mathbf{Q}$ , and a row matrix  $\mathbf{R} \in \mathbb{R}^{r \times m}$  with a subset of  $r < n$  rows of  $\mathbf{Q}$ , as well as an intersection matrix  $\mathbf{U} \in \mathbb{R}^{c \times r}$  such that the matrix multiplication of CUR approximates  $\mathbf{Q}$ . Unlike the SVD decomposition of  $\mathbf{Q}$ , the CUR decomposition obtains actual columns and rows of  $\mathbf{Q}$ , which makes it useful in various applications [20, 28]. In our method, instead of seeking an optimal matrix approximation, we employ CUR decomposition to identify and eliminate the starved nodes. The extracted intersection matrix  $\mathbf{U}$  is then reconstructed and served as a regularization term to provide supplementary supervision for better latent graph inference. To the best of our knowledge, we are the first to introduce CUR matrix decomposition into the field of graph neural networks.
In this paper, we analyze the common problem of supervision starvation (SS) in existing latent graph inference (LGI) methods. Our analysis reveals that this problem arises due to the graph sparsification operation, which destroys numerous important connections between pivotal nodes and labeled ones. Building upon this observation, we propose to recover the corrupted connections and replenish the missed supervision for improved graph inference. To this end, we begin by defining  $k$ -hop starved nodes and transform the SS problem into a more manageable task of reducing starved nodes. Then, we present two simple yet effective solutions to identify the starved nodes, including a more efficient method inspired by CUR matrix decomposition. Subsequently, we eliminate the starved nodes by constructing and incorporating a regularization graph. In addition, we propose two straightforward strategies to tackle the potential issue known as weight contribution rate decay. Extensive experiments conducted on representative benchmarks demonstrate that our proposed methods consistently enhance the performance of state-of-the-art LGI models, particularly under extremely limited supervision.
We are very grateful to Bahare Fatemi for her valuable discussion of our work. We thank the anonymous NeurIPS reviewers for providing us with constructive suggestions to improve our paper. This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-23-1-0290.
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