SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where  $q_{\phi}(\mathcal{G}_p|\mathcal{G}_{sub})$  is the variation approximation of  $p(\mathcal{G}_p|\mathcal{G}_{sub})$ . Equation 10 shows that the maximization of the mutual information  $I(\mathcal{G}_{sub};\mathcal{G}_p)$  can be attained by minimizing  $\mathcal{L}_{\mathrm{MI}}^2 (\mathcal{G}_{sub},\mathcal{G}_p)$ . We select a single-layer linear transformation as a modeling option for  $q_{\phi}$  to minimize the information loss of  $\mathcal{G}_{sub}$  when predicting  $\mathcal{G}_p$ .
2) Contrastive learning-based approach. Recent studies on contrastive learning [20, 30, 7] have proven that minimizing contrastive loss is equivalent to maximizing the mutual information between two variables. Hence, we additionally propose a variant of PGIB, i.e.,  $\mathrm{PGIB}_{\mathrm{cont}}$ , that minimizes the contrastive loss instead of minimizing the lower bound as defined in Equation 10. More precisely, we consider  $\mathcal{G}_p$  and  $\mathcal{G}_{sub}$  with the same label as a positive pair, and the contrastive loss is defined as follows:
$$
\mathcal {L} _ {\mathrm {M I}} ^ {2} = - \frac {1}{n} \sum_ {i = 1} ^ {n} \log \frac {\sum_ {j : \mathbf {z} _ {\mathcal {G} _ {p} ^ {j}} \in \mathbb {P} _ {y _ {i}}} \exp \left(g \left(\mathbf {z} _ {\mathcal {G} _ {s u b} ^ {i}} , \mathbf {z} _ {\mathcal {G} _ {p} ^ {j}}\right) / \tau\right)}{\sum_ {k : \mathbf {z} _ {\mathcal {G} _ {p} ^ {k}} \notin \mathbb {P} _ {y _ {i}}} \exp \left(g \left(\mathbf {z} _ {\mathcal {G} _ {s u b} ^ {i}} , \mathbf {z} _ {\mathcal {G} _ {p} ^ {k}}\right) / \tau\right)}. \tag {11}
$$
where  $\tau$  is the temperature hyperparameter,  $n$  denotes the number of graphs in a batch, and  $j$  and  $k$  indicate indices of positive and negative samples, respectively.  $\mathbb{P}_{y_i}$  is the set of prototypes that belong to class  $y_{i}$ . We effectively confer interpretability to  $\mathcal{G}_p$  by increasing its similarity with each  $\mathcal{G}_{sub}$ .
3.4 Prediction Layer
We obtain the set of similarity scores  $\mathbf{r} \in \mathbb{R}^M$ , whose  $m$ -th element  $r_m = g(\mathbf{z}_{\mathcal{G}_{sub}}, \mathbf{z}_{\mathcal{G}_p}^m)$  denotes the similarity score between  $\mathbf{z}_{\mathcal{G}_{sub}}$  and  $\mathbf{z}_{\mathcal{G}_p}$  as defined in Equation 7. Then, we compute the final predicted probability  $\pi \in \mathbb{R}^K$  by passing  $\mathbf{r}$  and  $\mathbf{z}_{\mathcal{G}_{sub}}$  through a linear layer with weights  $\omega$ , followed by the softmax function. Specifically,  $\omega[m, :]$  denotes weight assigned to  $r_m$ , i.e.,  $\omega(\mathbf{z}_{\mathcal{G}_p^m})$ . Finally, we calculate the cross-entropy classification loss, as follows:
$$
\mathcal {L} _ {\mathrm {c l s}} = - \frac {1}{N} \sum_ {i = 1} ^ {N} \sum_ {c = 1} ^ {K} \mathbb {I} \left(y _ {i} = c\right) \log \left(\pi_ {c}\right). \tag {12}
$$
3.5 Interpretability Stabilization
Merging Prototypes. Since the number of prototypes for each class is determined before training, some of the learned prototypes may share similar semantics, which negatively affects the model interpretability for which the small size and low complexity are desirable [6, 25]. Inspired by [17], we propose a method to effectively merge the prototypes for graph-structured data, which, in turn, enhances the explanation of the reasoning process and improves performance on downstream tasks while reducing model complexity. The main idea is to merge prototypes based on the similarity between prototype pairs using the embeddings  $\mathbf{z}_{\mathcal{G}_{sub}}$ . This similarity utilizes all training subgraphs (i.e.,  $\bigcup_{\mathcal{G}\in \mathcal{X}}\mathcal{G}_{sub}$ , where  $\mathcal{X}$  is the training set) by measuring the disparity between  $g(\mathbf{z}_{\mathcal{G}_{sub}},\mathbf{z}_{\mathcal{G}_p^i})$  and  $g(\mathbf{z}_{\mathcal{G}_{sub}},\mathbf{z}_{\mathcal{G}_p^j})$  as follows:
$$
h \left(\mathbf {z} _ {\mathcal {G} _ {p} ^ {i}}, \mathbf {z} _ {\mathcal {G} _ {p} ^ {j}}\right) = \left[ \sum_ {\mathcal {G} \in \mathcal {X}} \left(g \left(\mathbf {z} _ {\mathcal {G} _ {s u b}}, \mathbf {z} _ {\mathcal {G} _ {p} ^ {i}}\right) - g \left(\mathbf {z} _ {\mathcal {G} _ {s u b}}, \mathbf {z} _ {\mathcal {G} _ {p} ^ {j}}\right)\right) ^ {2} \right] ^ {- 1}. \tag {13}
$$
Then, for every pair  $(\mathbf{z}_{\mathcal{G}_p^i}, \mathbf{z}_{\mathcal{G}_p^j})$  that falls within the highest  $\xi$  percent of similar pairs, the prototype  $\mathbf{z}_{\mathcal{G}_p^j}$  and its corresponding weights  $\omega(\mathbf{z}_{\mathcal{G}_p^j})$  are removed, and the weights  $\omega(\mathbf{z}_{\mathcal{G}_p^i})$  are updated to the sum of  $\omega(\mathbf{z}_{\mathcal{G}_p^i})$  and  $\omega(\mathbf{z}_{\mathcal{G}_p^j})$ . We combine the  $\xi$  percentage of the most similar prototype pairs based on the calculated similarity scores.
Prototype Projection. Since the learned prototypes are embedding vectors that cannot be directly interpreted, we project each prototype  $\mathbf{z}_{\mathcal{G}_p}$  onto the nearest latent training subgraph from the same class. This process establishes a conceptual equivalence between each prototype and a training subgraph, thereby enhancing interpretability of the prototypes. Specifically, we update prototype  $\mathbf{z}_{\mathcal{G}_p}$  of class  $k$  (i.e.,  $\mathbf{z}_{\mathcal{G}_p} \in \mathbb{P}_k$ ) by performing the following operation:
$$
\mathbf {z} _ {\mathcal {G} _ {p}} \leftarrow \underset {\tilde {\mathbf {z}} \in \mathbb {Z}} {\arg \min } \| \tilde {\mathbf {z}} - \mathbf {z} _ {\mathcal {G} _ {p}} \| _ {2}, \text {w h e r e} \mathbb {Z} = \left\{\tilde {\mathbf {z}}: \operatorname {R e a d o u t} \{f _ {g} (\tilde {\mathcal {G}}) \}, \tilde {\mathcal {G}} \in \operatorname {S u b g r a p h} \left(\mathcal {G} _ {i}\right) \forall i s. t. y _ {i} = k \right\}. \tag {14}
$$
In the Equation 14, we use Monte Carlo Tree Search (MCTS) [18] to explore training subgraphs  $\tilde{\mathcal{G}}$  during prototype projection.
Connectivity Loss. For an input graph  $\mathcal{G}$ , we construct a node assignment  $S_{\mathcal{G}} \in \mathbb{R}^{|\mathcal{V}_{\mathcal{G}}| \times 2}$  based on the probability values that are computed by Equation 5. Specifically,  $S_{\mathcal{G}}[j,0]$  and  $S_{\mathcal{G}}[j,1]$  denote the probability of node  $v_{j} \in \mathcal{V}_{\mathcal{G}}$  belonging to  $\mathcal{G}_{sub}$  and  $\bar{\mathcal{G}}_{sub}$ , respectively. Following [31], poor initialization of the matrix  $S$  may result in the proximity of its elements  $S[j,0]$  and  $S[j,1]$  for  $\forall v_{j} \in \mathcal{V}_{\mathcal{G}_{i}}$ , leading to an unstable connectivity of  $\mathcal{G}_{sub}$ . This instability can have adverse effects on the subgraph generation process. To enhance the interpretability of  $\bar{\mathcal{G}}_{sub}$  by inducing a compact topology, we utilize a batch-wise loss function as follows:
$$
\mathcal {L} _ {\text {c o n}} = \left\| \operatorname {N o r m} \left(S _ {\mathrm {B}} ^ {T} \mathbf {A} _ {\mathrm {B}} S _ {\mathrm {B}}\right) - I _ {2} \right\| _ {F} \tag {15}
$$
where  $S_{\mathrm{B}} \in \mathbb{R}^{\sum_{i=1}^{n} |\mathcal{V}_{\mathcal{G}_i}| \times 2}$  and  $\mathbf{A}_{\mathrm{B}} \in \mathbb{R}^{\sum_{i=1}^{n} |\mathcal{V}_{\mathcal{G}_i}| \times \sum_{i=1}^{n} |\mathcal{V}_{\mathcal{G}_i}|}$  are the node assignment and the adjacency matrix at the batch level, respectively.  $I_2$  is 2-by-2 identity matrix,  $\| \cdot \|_F$  is the Frobenius norm and  $\mathrm{Norm}(\cdot)$  is the row normalization. Minimizing  $\mathcal{L}_{\mathrm{con}}$  indicates that if  $v_j$  is in  $\mathcal{G}_{sub}$  its neighbors also have a high probability to be in  $\mathcal{G}_{sub}$ , while if  $v_i$  is in  $\mathcal{G}_{sub}$ , its neighbors have a low probability to be in  $\mathcal{G}_{sub}$ .
Final Objectives. Finally, we define the objective of our model as the sum of the losses as follows:
$$
\mathcal {L} _ {\text {t o t a l}} = \mathcal {L} _ {\text {c l s}} + \alpha_ {1} \mathcal {L} _ {\mathrm {M I}} ^ {1} + \alpha_ {2} \mathcal {L} _ {\mathrm {M I}} ^ {2} + \alpha_ {3} \mathcal {L} _ {\text {c o n}} \tag {16}
$$
where  $\alpha_{1},\alpha_{2}$  and  $\alpha_{3}$  are hyper-parameters that adjust the weights of the losses. A detailed ablation study for each loss term is provided in Appendix A.4.1 for further analysis.
Each dataset is split into training, validation, and test sets with a ratio of  $80\%$ ,  $10\%$ , and  $10\%$  respectively. All models are trained for 300 epochs using the Adam optimizer with a learning rate of 0.005. GIN [28] is used as the encoder for all models used in the experiment. We evaluate the performance based on accuracy, which is averaged over 10 independent runs with different random seeds. For simplicity, the hyperparameters  $\alpha_{1}$ ,  $\alpha_{2}$ , and  $\alpha_{3}$  in Equation 16 are set to 0.0001, 0.01 to 0.1 and 5, respectively. The prototype merge operation starts at epoch 100 and is performed every 50 epochs thereafter. We set the number of prototypes per class to 7 and combine  $30\%$  of the most similar prototype pairs.
4.2 Graph Classification
Datasets and Baselines. We use the MUTAG [16], PROTEINS [1], NCI1 [24], and DD[5] datasets. These are datasets related to molecules or bioinformatics, and are widely used for evaluations on graph classification. We consider three GNN baselines, including GCN [10], GIN [28], GAT [22]. In addition, we compare PGIB with several state-of-the-art built-in models that integrate explanation functionality internally, including a prototype-based method ProtGNN [35], and IB-based models such as GIB [31], VGIB [32], and GSAT [13]. Further details about the baselines and datasets are provided in Appendix A.5 and A.6, respectively.
Experiment Results. Experimental results for graph classification are presented in the Table 1. In the table, PGIB and  $\mathrm{PGIB}_{\mathrm{cont}}$  represent our proposed methods. PGIB utilizes a Variational IB-based approach, while  $\mathrm{PGIB}_{\mathrm{cont}}$  employs a Contrastive learning-based approach to maximize  $I(\mathcal{G}_{\mathrm{sub}}; \mathcal{G}_p)$  (Section 3.3.2). We have the following observations: 1) All variants of PGIB outperform the baselines including both the prototype-based and IB-based methods on all datasets. Notably, PGIBs incorporate the crucial information of the key subgraph, which significantly contributes to enhancing the classification performance.  $\mathrm{PGIB}_{\mathrm{cont}}$  achieves a significant improvement of up to  $5.6\%$  compared to the runner-up baseline. 2) We observe that  $\mathrm{PGIB}_{\mathrm{cont}}$  performs relatively better than PGIB. We attribute this to the nature of the contrastive loss, which is generally shown to be effective in classifying instances between different classes, allowing the prototypes learned based on the contrastive loss to be more distinguishable from one another.
Table 1: Evaluation on graph classification (accuracy).
Dataset	Methods
GCN	GIN	GAT	ProtGNN	GIB	VGIB	GSAT	PGIB	\( PGIB_{cont} \)
MUTAG	74.50±7.89	80.50±7.89	73.50±7.43	80.50±9.07	79.00±6.24	81.00±6.63	80.88±8.94	85.00±7.07	85.50±5.22
PROTEINS	72.83±4.23	70.30±4.84	71.35±4.85	73.83±4.22	75.25±5.92	73.66±3.32	69.64±4.71	77.14±2.19	77.50±2.42
NCI1	73.16±3.49	75.04±2.08	66.05±1.03	74.13±2.10	64.65±6.78	63.75±3.37	68.13±2.64	77.65±2.20	78.25±2.13
DD	72.53±4.51	72.04±3.62	70.81±4.33	69.15±4.33	72.61±8.26	72.77±5.63	71.93±2.74	73.36±1.80	73.70±2.14
4.3 Graph Interpretation
In this section, we evaluate the process of extracting subgraphs that possess the most similar properties of the original graph. We present qualitative results including subgraph visualizations, and conduct quantitative experiments to measure how accurately explanations capture the important components that contribute to the model's predictions.
Datasets and Baselines. We use four molecular properties from the ZINC [8] dataset, which consists of 250,000 molecules, for graph interpretation. QED measures the likelihood of a molecule being a drug and DRD2 indicates the probability of a molecule being active on dopamine type 2 receptors. HLM-CLint and RLM represent estimates of in vitro human and rat liver microsomal metabolic stability (mL/min/g as base 10 logarithm). We compare  $\mathrm{PGIB}_{\mathrm{cont}}$  with several representative interpretation models, including GNNexplainer [29], PGexplainer [12], GIB [31], and VGIB [32]. Note that as PGIB shows similar results with  $\mathrm{PGIB}_{\mathrm{cont}}$ , we only report the results of  $\mathrm{PGIB}_{\mathrm{cont}}$  for simplicity. Further details on baselines and datasets are described in Appendix A.5 and A.6, respectively.
Qualitative Analysis. Figure 3(a)-(d) present the visualization of subgraphs in Mutag dataset. According to [29, 4], the  $\mathrm{NO}_2$  functional group is known to be a cause of mutagenicity, while the carbon ring is a substructure that is not related to mutagenicity. In the figure, the bold edges connect the nodes that the models consider important. The  $\mathrm{NO}_2$  group in Mutag is correctly identified by PGIB, while VGIB, ProtGNN, and GNNexplainer fail to recognize all  $\mathrm{NO}_2$  groups or include other unnecessary substructures. Figure 3(e)-(h) present the visualization of subgraphs in BA-2Motifs dataset. We observe that PGIB accurately recognizes motif graphs containing the label information such as a house or a five-node cycle, but other models have difficulty in detecting the complete motifs.
Quantitative Analysis. Although visualizing the explanations generated by models plays a crucial role in assessing various explanation models, relying solely on qualitative evaluations may not always be reliable due to their subjective nature. Therefore, we also perform quantitative experiments using the Fidelity metric [14, 34].
The Fidelity metric quantifies the extent to which explanations accurately capture the important components that contribute to the model's predictions. Specifically, let  $y_{i}$  and  $\hat{y}_i$  denote the ground-truth and predicted values for the  $i$ -th input graph, respectively. Moreover,  $k$  denotes the sparsity score of the selected subgraph in which nodes whose importance scores obtained by Equation 5 are among the top-  $(k\times 100)\%$  within the original graph, and its prediction is denoted by  $\hat{y}_i^k$ . Additionally,  $\hat{y}_i^{1 - k}$  denotes the prediction based on the remaining subgraph. The Fidelity scores are computed as follows:
$$
\mathcal {F} _ {-} = \frac {1}{N} \sum_ {i = 1} ^ {N} \mathbb {I} \left(y _ {i} = \hat {y} _ {i}\right) - \mathbb {I} \left(y _ {i} = \hat {y} _ {i} ^ {k}\right), \quad \mathcal {F} _ {+} = \frac {1}{N} \sum_ {i = 1} ^ {N} \mathbb {I} \left(y _ {i} = \hat {y} _ {i}\right) - \mathbb {I} \left(y _ {i} = \hat {y} _ {i} ^ {1 - k}\right), \tag {17}
$$
where  $\mathbb{I}(y_i = \hat{y}_i)$  is the binary indicator which returns 1 if  $y_{i} = \hat{y}_{i}$ , and 0 otherwise. In other words, they measure how well the predictions made solely based on the extracted subgraph (i.e.,  $\mathcal{F}_{-}$ ) and the remaining subgraph (i.e.,  $\mathcal{F}_{+}$ ) mimic the predictions made based on the entire graph, respectively. Hence, a low value of  $\mathcal{F}_{-}$  and a high value of  $\mathcal{F}_{-}$  indicate better explainability of the model.
Table 2 shows the fidelity scores on four datasets at the sparsity score of  $k = 0.5$ . Our proposed model outperforms both post-hoc and built-in state-of-the-art explanation models in all datasets. Furthermore,
Table 2: Evaluation on graph interpretation (Fidelity scores).
Method	\( \mathcal{F}_{-} \downarrow \)	\( \mathcal{F}_{+} \uparrow \)
RLM	HLM-CLint	QED	DRD2	RLM	HLM-CLint	QED	DRD2
GNNexplainer	0.478	0.616	0.498	0.433	0.694	0.778	0.602	0.740
PGexplainer	0.502	0.620	0.560	0.540	0.632	0.692	0.598	0.686
GIB	0.483	0.643	0.525	0.428	0.654	0.781	0.601	0.724
VGIB	0.463	0.579	0.487	0.424	0.765	0.792	0.627	0.756
PGIBcont	0.441	0.593	0.459	0.406	0.747	0.772	0.613	0.771
PGIBcont + merge	0.415	0.543	0.447	0.379	0.765	0.796	0.635	0.781
merging prototypes achieves significant improvements in terms of interpretability. This implies
(a) PGIBcont
(b) VGIB
(c) ProtGNN
(d) GNNexplainer
(e) PGIBcont
(f) VGIB
(g) ProtGNN
(h) GNNexplainer
Figure 3: Explanation visualizations on Mutag (a-d) and BA-2Motifs (e-h)
(a) RLM-CLint
(b) HLM-CLint
Figure 4: Comparisons of fidelity scores over sparsity scores  $k$ . Figure 5:  $\mathcal{G}_p$  visualization over  $\alpha_2$ .
(d) DRD2
(a)  $\alpha_{2} = 0.0001$
(b)  $\alpha_{2} = 1$
that decreasing the number of prototypes can eliminate uninformative substructures and emphasize key substructures, which increases the interpretability of the extracted subgraphs. Figure 4 visualizes the comparison of fidelity scores over various sparsity scores of subgraphs. To ensure a fair comparison, the fidelity scores are compared under the same subgraph sparsity, as the difference between the predictions of the original graph and subgraph strongly depends on the level of sparsity. We observe that  $\mathrm{PGIB}_{\mathrm{cont}}$  achieves the best performance in most sparsity environments on the four datasets.
4.4 Hyperparameter Analysis
In Figure 6, we conduct a sensitivity analysis on the hyperparameters  $\alpha_{1}$  and  $\alpha_{2}$  of the final loss (Equation 16) relevant to mutual information. Note that  $\alpha_{1}$  and  $\alpha_{2}$  are related to minimizing  $I(\mathcal{G};\mathcal{G}_{sub})$  and maximizing  $I(\mathcal{G}_{sub};\mathcal{G}_p)$ , respectively. 1) Figure 6(a) shows a significant decrease in performance when  $\alpha_{1}$  becomes large, i.e., when the model focuses on compressing the subgraphs. This is because too much compression of
Figure 6: Impact of  $\alpha_{1}$  and  $\alpha_{2}$  on PROTEINS dataset.
(a)  $\alpha_{1}$
(b)  $\alpha_{2}$
subgraphs results in the loss of important information, ultimately having a negative impact on the downstream performance. However, when  $\alpha_{1} = 0$  (i.e.,  $\mathcal{G} = \mathcal{G}_{sub}$ ; no compression at all), uninformative information would be included in  $\mathcal{G}_{sub}$ , which incurs a performance degradation. 2) Figure 6(b) visualizes the change in performance depending on  $\alpha_{2}$ . A small value of  $\alpha_{2}$  prevents sufficient transmission of information from  $\mathcal{G}_{sub}$  to  $\mathcal{G}_{p}$ , whereas excessive value of  $\alpha_{2}$  allows the influence of  $\mathcal{G}_{sub}$  to dominate the prototypes  $\mathcal{G}_{p}$ , both of which lead to a performance deterioration. For example, in Mutag dataset, a low value of  $\alpha_{2}$  ultimately results in  $\mathcal{G}_{p}$  not obtaining label-relevant information (i.e.,  $\mathrm{NO}_2$ ) that is captured by  $\mathcal{G}_{sub}$  (See Figure 5(a)). On the other hand, a high value of  $\alpha_{2}$  hinders the formation of diverse prototypes (See Figure 5(b)).
5 Conclusion and Future work
We propose a novel framework of explainable GNNs, called interpretable Prototype-based Graph Information Bottleneck (PGIB), that integrates prototype learning into the information bottleneck framework. The main idea of PGIB is to learn prototypes that capture subgraphs containing key structures relevant to the label information, and to merge the semantically similar prototypes for better model interpretability and model complexity. Experimental results show that PGIB achieves improvements not only in the performance on downstream tasks, but also provides more precise
explanation of the reasoning process. For future work, we plan to further extend the applicability of PGIB by integrating domain knowledge into prototype learning by imposing constraints on subgraphs. We expect that this approach would enable the learning of prototypes that align with the domain knowledge as they obtain domain-specific information from the subgraphs.
This work was supported by Institute of Information & Communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT) (RS-2023-00216011, No.2022-000077), and the National Research Foundation of Korea(NRF) funded by Ministry of Science and ICT (NRF-2022M3J6A1063021).
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