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ICML/2025/Enhancing Graph Invariant Learning from a Negative Inference Perspective/full.md

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+# Enhancing Graph Invariant Learning from a Negative Inference Perspective
+
+Kuo Yang  $^{12\dagger}$  Zhengyang Zhou  $^{12\dagger}$  Qihe Huang  $^{12}$  Wenjie Du  $^{12*}$  Limin Li  $^{12}$  Wu Jiang  $^{3}$  Yang Wang  $^{12*}$
+
+# Abstract
+
+The out-of-distribution (OOD) generalization challenge is a longstanding problem in graph learning. Through studying the fundamental cause of data distribution shift, i.e., the changes of environments, significant progress has been achieved in addressing this issue. However, we observe that existing works still fail to effectively address complex environment shifts. Existing practices place excessive attention on extracting causal subgraphs, inevitably treating spurious subgraphs as environment variables. While spurious subgraphs are controlled by environments, the space of environment changes encompass more than the scale of spurious subgraphs. Therefore, existing efforts have a limited inference space for environments, leading to failure under severe environment changes. To tackle this issue, we propose a negative inference graph OOD framework (NeGo) to broaden the inference space for environment factors. Inspired by the successful practice of prompt learning in capturing underlying semantics and causal associations in large language models, we design a negative prompt environment inference to extract underlying environment information. We further introduce the environment-enhanced invariant subgraph learning to effectively exploit inferred environment embedding, ensuring the robust extraction of causal subgraph in the environment shifts. Lastly, we conduct a comprehensive evaluation of NeGo on real-world datasets and synthetic datasets across domains. NeGo outperforms baselines on nearly all datasets, which verify the effectiveness of our framework.
+
+†Equal Contribution. <sup>1</sup>University of Science and Technology of China (USTC), Hefei, China <sup>2</sup>Suzhou Institute for Advanced Research, USTC, Suzhou, China <sup>3</sup>China Mobile Communications Group Co.,Ltd. Correspondence to: Yang Wang* <angyan@ustc.edu.cn>, Wenjie Du* <duwenjie@ustc.edu.cn>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+# 1. Introduction
+
+Graph Neural Networks (GNNs) have emerged as the predominant approach for encoding graph data (Kipf & Welling, 2016; Xu et al., 2018), delivering notable achievements in various research fields including molecular property prediction (Jumper et al., 2021; Yang et al., 2022), recommendation systems (Wu et al., 2022b; Gao et al., 2022), and traffic flow forecasting (Liang et al., 2018; Zhou et al., 2020). However, as real-world data is evolving with complex patterns, the challenge of data distribution shift has become a major obstacle for GNNs (Gui et al., 2022; Ji et al., 2022; Wang et al., 2023; Zhou et al., 2022b; Zou et al., 2023; Sun et al., 2022c; 2024). Therefore, various studies concentrate on improving the Out-Of-Distribution (OOD) generalization ability of graph learning models (Chen et al., 2024; 2022; Gui et al., 2024; Miao et al., 2022; Sui et al., 2022; Li et al., 2022; Wu et al., 2022c; Sun et al., 2022b).
+
+Recently, environment-centered invariant learning methods achieved impressive OOD generalization performance with the aim of inferring underlying environment factors in data (Chen et al., 2024; Gui et al., 2024; Xia et al., 2023; Yuan et al., 2023). Those efforts demonstrate that the changes of environment are the fundamental reason for the shift of data distribution (Grice & White, 1961; Liu et al., 2021; Peters et al., 2016). However, existing approaches still lack the ability to decouple causal subgraphs from complex environments. As shown in Fig. 1(a), we double the scale of spurious substructures in the SPURIOUS-MOTIF(0.5), and observe a significant decrease in the performance of current methods when they are re-conducted on this modified dataset. The reason lies in that current methods, even those claiming to model environments, focus much of their attention on extracting causal subgraphs (Chen et al., 2022; Wu et al., 2022a;c; Sun et al., 2021). This results in the model being able to extract causal subgraphs only in known environments, and failing in modeling unseen complex environments. Therefore, this poses a challenging research question: how to broaden the inference space of environments, enabling model to handle complex environment shifts.
+
+We argue that this limitation arises from the positive learning paradigm that focuses solely on extracting causal subgraphs as its primary objective. In contrast, negative inference paradigm, modeling the sample space except the invariant
+
+![](images/1953bdf9fde07c99f13f20a7e9276bd155ef73c3b5bfc4362cc9e8a7c23fea9f.jpg)  
+(a) Degradation in performance.
+
+![](images/4d864d26bd4f1c9061c109696490c463212bf7fb6702b3345028d81c8e60ed7a.jpg)  
+(b) Prediction failure in complex environments.  
+Figure 1. The motivation of our work. (a) We double the scale of spurious substructures in the SPURIOUS-MOTIF(0.5) (Ying et al., 2019), and observe a significant decrease in the performance of current methods when they are re-conducted on this modified dataset. (b) The OOD methods, which treat spurious subgraphs as the environments, fail to address the shift of complex environments.
+
+subgraph as environments, has the potential to broaden the perception scope of environment. As shown in Fig. 1(b), the positive inference can only infer the specific ladder, wheel, and tree as environment variables, while the negative inference approach can model all variable space except the cycle and house as environments. However, the inaccessibility of environment information pose challenges to implementing such negative inference. Specifically, (1) how to formulate the negative inference learning to achieve environment awareness, and (2) how to fully utilize environment information for facilitating causal invariant learning.
+
+In this work, we propose a novel Negative inference Graph OOD framework (NeGo). NeGo aims to achieve causal invariant learning against complex environment shifts by a negative inference. Firstly, we design a negative prompt learning framework for inferring underlying environment factors. Given a specific sample with label  $Y$ , we model all extra-class samples as the environment space for this sample. This design enables the model to capture a broader scale of environments, no more limiting to in-sample spurious subgraphs. Secondly, we introduce an environment-enhanced invariant learning strategy to effectively utilize inferred environment variables. Specifically, we design an interactive decoding scheme that utilizes an attention-based residual connection to encapsulate environment embedding into node representations. Different from traditional approaches that neglect the information of environment variables during subgraph extraction (Chen et al., 2024; Gui et al., 2024), our design incorporates the underlying environment patterns into the process of invariant subgraph learning. Lastly, we conduct a comprehensive evaluation of NeGo on real-world datasets across domains, and synthetic datasets. NeGo outperforms baselines on nearly all datasets. Our contributions can be summarized as follows:
+
+- We observe that existing environment-centered OOD practices encounter difficulties in handling complex environment shifts. Through a comprehensive investigation, we
+
+identify that limited environment space for positive inference is the main reason restricting the generalization capacity of existing OOD approaches.
+
+- We propose a novel invariant learning framework with negative inference  $\mathsf{N e G o}$ . To be specific, we design an innovative environment inference strategy via negative inference, which effectively broadens the inference space of environment factors. Moreover, we introduce an attention-based residual connection to offer our model with the ability to resist complex environment shifts.  
+- We conduct extensive experiments on both synthetic and real-world datasets with distribution shifts to evaluate the performance of  $\mathsf{N e G o}$ . The results from both visualization and quantitative analysis indicate that our framework successfully achieves accurate prediction in complex environmental scenarios.
+
+# 2. Background
+
+Preliminaries. A graph is denoted as  $G = (\mathcal{X},\mathcal{A})\in \mathcal{G}$  where  $\mathcal{G}$  is the observed graph dataset.  $\mathcal{A}\in \mathbb{R}^{N\times N}$  represents the adjacency matrix and  $\mathcal{X}\in \mathbb{R}^{N\times d}$  denotes node features, where  $N$  indicates the number of nodes and  $d$  is the feature dimension. Each graph is associated with a corresponding label  $Y$ . From the perspective of causal theory, the graph data can be partitioned into a spurious subgraph  $G_{S}$  and a causal subgraph  $G_{C}$ , where  $G_{C}$  directly determines its label  $Y$ . The spurious subgraph  $G_{S}$  is controlled by the spurious variable  $C$ , while the causal subgraphs  $G_{C}$  is controlled by the causal invariant factor  $C$ , as shown in Fig. 4. Based on the different interdependencies among  $C$ ,  $S$  and  $Y$ , structural causal models (SCMs) can be further classified into Full Informative Invariant Features (FIIF) and Partially Informative Invariant Features (PIIF) (Ahuja et al., 2021; Chen et al., 2022).
+
+Problem definition. Our work aims to address the limitations of existing approaches in handling complex data
+
+distribution shifts. We specifically focus on broadening the inference scope of environments, enabling the network to handle intricate scenarios of environment shifts. Additionally, our framework is required to effectively tackle both FIIF and PIIF challenges.
+
+Environment inference with negative prompt. Our negative prompter is proposed to achieve a broader inference scale of environments, which is inspired by the success of prompt learning in language models (Brown et al., 2020; Gao et al., 2020). Prompt learning is designed to capture underlying semantic knowledge in language data, which improves the generalization ability of models by introducing appropriate prompt tokens to guide the network learn desired answers (Rao et al., 2022; Sordoni et al., 2024; White et al., 2023; Sun et al., 2023). For example, in the semantic emotion classification task, the language model constructs a template such as "the emotion expressed by this sentence is [class]", where [class] is trained to learn real label. In a similar way, our framework can be viewed as constructing a set of text prompts such as "the underlying environments of current sample are [answer]", where [answer] can be guided to capture the real environment states.
+
+Different from random data augmentation techniques (Han et al., 2022; Li et al., 2021; Lu et al., 2024; Rong et al., 2019; Wang et al., 2021; You et al., 2020; Zhao et al., 2021) and distributionally robust optimization (DRO) methods (Staib & Jegelka, 2019; Wu et al., 2024; Zhu et al., 2021), our prompt-based approach not only broadens the scale of environment inference but also deepens the understanding of underlying data generation process. Existing methods always expand the inference boundary of the model by incorporating stochastic perturbations. However, the introduction of randomness prevents the model from capturing the underlying semantics and hinders its ability to deepen the understanding of generation process. In contrast, our prompt-based approach allows us to deeply extract the underlying casual correlation of variables, which is the reason why we introduce the technique of prompt learning.
+
+# 3. Related works
+
+# 3.1. OOD Generalization.
+
+Out-of-Distribution (OOD) generalization refers to the task of adapting a model that has been trained on a specific distribution to effectively process data from a potentially different distribution. Various approaches can be employed for OOD generalization, including data augmentation (Rong et al., 2019; Wang et al., 2021; You et al., 2020), domain adaptation (Wang & Deng, 2018), and causal invariant learning (Sui et al., 2022; Wu et al., 2022c). Among them, causal invariant learning demonstrates impressive performance in
+
+various fields, due to its powerful interpretability (Chen et al., 2022; Li et al., 2022; Miao et al., 2022; Wu et al., 2022c). Most existing invariant learning methods focus on extracting the causal subgraph to achieve invariant learning. This strategy limits the inference space of the environments to the dimension of spurious subgraphs, which hinders the ability of models to capture the complex environment states. In this work, we propose an invariant learning mechanism based on negative inference to address this limitation.
+
+# 3.2. Prompt Learning
+
+Prompt learning is proposed in NLP models to infer underlying semantic and potential causal associations in linguistic data. Many effective prompt methods have developed with the introduction of large language models, including some hand-crafted prompts (Brown et al., 2020), discrete prompts (Gao et al., 2020; Shin et al., 2020), and learnable prompts design (Li & Liang, 2021). In recent years, prompt learning has also been developed in the graph learning field (Sun et al., 2022a; Li et al., 2024; Sun et al., 2023; Zi et al., 2024). Our approach is the pioneering effort to apply prompt learning to address the graph OOD generalization issue.
+
+# 3.3. Comparisons to Existing Graph OOD Works
+
+Environment-centered studies (Chen et al., 2024; Gui et al., 2024; Li et al., 2022; Wu et al., 2022a; Yang et al., 2022) consider that the data distribution shifts stem from the changes of environments. Therefore, these practices enable the model to withstand data distribution shifts by inferring environment variables. Concretely, the networks are often trained with the objective of equipping models to effectively handle mixed environments scenarios. However, this design allows the networks to make narrow inference about the environments, and makes the networks unable to handle with distribution shifts in complex environments. We attribute this limitation of inference scale to the shortcomings of positive inference, which is proved both empirically and theoretically. To this end, we propose a negative inference mechanism to broaden the inference space for environments, without relying on the mixed environments hypothesis. Our approach, which represents a pioneering practice in utilizing negative inference, is distinct from all existing practices in this field.
+
+# 4. Graph OOD generalization via environment inference
+
+Existing environment-centered practices aim to enable the networks with the ability to resist data distribution shifts. However, our empirical observations indicate that these approaches are insufficient in handling complex environment shifts. To address this issue, we conduct a theoretical
+
+![](images/bd60bfd1294f031663d76a6e2ffeba8ef67f77dc807fe73b47e0c97dbb396622.jpg)  
+Figure 2. The architecture of NeGo. We implements an environment-enhanced graph learning framework in which the environment is extracted through a negative prompt mechanism. The training process is guided by both a positive loss and a negative loss, aiming to broaden the modeling space for the environment.
+
+analysis of these methods, and identify that their failures stem from the limited environment inference space by only positively treating spurious subgraphs as environments. In contrast, we propose a promising method based on negative inference.
+
+# 4.1. Limited Environment Cognitive Space for Positive Inference
+
+From the perspective of causal theory (Pearl, 2009; 2010), the variables of generating the graph data include causal subgraph  $G_{C}$  and spurious subgraph  $G_{S}$ , where  $G_{S}$  is controlled by environment variable  $E$ . As shown in Fig. 4,  $G_{C} \rightarrow Y$  demonstrates a stable casual relationship from  $G_{C}$  to  $Y$  in the data generation process. Consequently, the distribution shift between the training data and the test data can be attributed to the shifts of environment  $E$ , which can be formally expressed as  $\mathbb{P}_{train}(\mathcal{G},E) \neq \mathbb{P}_{test}(\mathcal{G},E)$ . Modeling environment variables becomes crucial for tackling OOD generalization issue (Chen et al., 2024; Gui et al., 2024; Xia et al., 2023; Yuan et al., 2023). With the observed training dataset  $\mathcal{G}$ , environment-centered approaches strive to learn the distribution of the environment factor  $E$ ,
+
+$$
+\mathbb {P} (E | \mathcal {G}) = \frac {\mathbb {P} (\mathcal {G} , E)}{\mathbb {P} (\mathcal {G})} = \frac {\mathbb {P} (\mathcal {G} | E) \mathbb {P} (E)}{\int_ {E} \mathbb {P} (\mathcal {G} | E) \mathbb {P} (E) d E}. \tag {1}
+$$
+
+The prior distribution  $\mathbb{P}(E)$  and the likelihood  $\mathbb{P}(\mathcal{G}|E) = \prod_{i=1}^{N} \mathbb{P}(G_i|E)$  make the numerator theoretically computable.
+
+However, due to the uncertainty in the scale of environments  $E$ , the denominator of Eq. 1 involving integration becomes intractable. To tackle this issue, we impose a distribution shift boundary based on following environment mixture assumption.
+
+Assumption 4.1. (Li et al., 2022) If  $K$  different environment labels can be extracted from the observed dataset  $\mathcal{G}$ ,
+
+they are formulated by  $K$  independent  $D$ -dimensional Gaussian distributions  $\mathcal{N}(\mu_i,I)$ , where  $\mu_{i}\in \mathbb{R}^{1\times D}$ . Therefore, environment variables can be modeled from a vector space perspective, allowing us to approximate the environment space by exploring the mixture space of vectors.
+
+Given the Assumption 4.1, we can model the environments codebook  $\pmb{\mu} = (\mu_1,\mu_2,\dots,\mu_K)\in \mathbb{R}^{K\times D}$ . This environment codebook serves as a proxy for the environment space, representing the entire environment space through the mixture of vectors.
+
+Therefore, the scale of environments is modeled as a mixing space of extracted environment variables. As a result, the new data  $G_{i}$  is associated with the environment state  $E_{i} \sim \mathcal{N}(e_{i} \cdot \boldsymbol{\mu}, I)$ , where  $e_{i} \in \mathbb{R}^{1 \times K}$  representing the mixing weight. This indicates that the latent variables  $e = (e_{1}, e_{2}, \dots, e_{N}) \in \mathbb{R}^{N \times K}$  be regarded as the proxy factor for the environment variable  $E$ , directly determining the observed data  $\mathcal{G}$  generation process. The posterior probability of the environments  $\mathbb{P}(E | \mathcal{G})$  is transformed into,
+
+$$
+\mathbb {P} (\boldsymbol {e} | \mathcal {G}) = \frac {\mathbb {P} (\mathcal {G} , \boldsymbol {e})}{\mathbb {P} (\mathcal {G})} = \frac {\prod_ {i = 1} ^ {N} \mathbb {P} \left(e _ {i}\right) \mathbb {P} \left(G _ {i} \mid e _ {i}\right)}{\sum_ {e} \mathbb {P} (\boldsymbol {e}) \prod_ {i = 1} ^ {N} \mathbb {P} \left(G _ {i} \mid e _ {i}\right)} = \frac {\prod_ {i = 1} ^ {N} \mathbb {P} \left(e _ {i}\right)}{\sum_ {e} \mathbb {P} (\boldsymbol {e})}. \tag {2}
+$$
+
+The finite space of  $e$  allows for the approximation strategy to be feasible. However, the limited scale of  $e$  may limit the capacity of model to effectively counter complex environment shifts, which is verified by our empirical results. We next delve into the reason contributing to this limitation. We first present a definition of the basis and base environments of the environment space, similar to the concepts of basis and base vectors in the vector space.
+
+Assumption 4.2. Let  $E_{b} = \{E_{1},\dots,E_{K}\}$  be the basis of environment space, and each element  $E_{i}$  within it is referred
+
+to as the base environment. The linear combination of base environments can completely describe the entire environment space.
+
+Actually, the environment mixing assumption fundamentally relies on the expectation that extracted environment codebook can cover the basis  $\mathbf{E}_b$ . However, we observe that such goal cannot be achieved by existing methods. Given a graph  $G$ , current environment-centered methods aim to decompose it into causal subgraph  $G_{C}$  and spurious subgraph  $G_{S}$ . The spurious subgraph  $G_{S}$  is inferred as the environment variable. Although  $G_{S}$  is controlled by environment factor  $(E \rightarrow G_{S})$ , the space of environment changes encompass more than the scale of spurious subgraphs. For example, consider the substructure  $G_{C}$  that is causally associated with one graph-level property  $l$ , but the variants of such  $G_{C}$  act as environment factors for other properties. Existing methods fail to capture such other properties with invariance.
+
+Proposition 4.3. Given an observed graph dataset  $\mathcal{G}$ , the inference process, considering  $G_{S}$  as the environment factor, fails to capture the basis  $\mathbf{E}_b$  that can represent the entire environment space.
+
+Proposition 4.3 indicates that the mixing of  $\mu$  is unable to encompass the entire environment space. Existing environment-centered methods still have a narrow space of the environments, which leads to incapability of network to extract the causal graph from the complex environments. Therefore, the limitations of existing works are attributed to the narrow inference space of the model for environment variables. Detailed proof can be found in Appendix D.1.
+
+# 4.2. The Enhancement of Negative Inference
+
+Negative inference has a major advantage in effectively expanding the cognitive boundary of models. For example, the positive inference can only infer the specific ladder, wheel, and tree as environment variables, as shown in Fig. 1(b), while the negative inference approach can infer all variable space except the cycle and house as environments. While the ultimate objective is to extract invariant subgraphs, the negative inference mechanism prioritizes inferring the environment space, empowering the model with the capability to adapt to complex environment shifts. From the perspective of information theory, the training objective of negative inference can be formalized as,
+
+$$
+\begin{array}{l} \max  I (E; G _ {C} | \overline {{G ^ {Y}}}) = \max  I (E; \mathcal {G} - G _ {C} | Y) \tag {3} \\ = \max  I (E; \mathcal {G} | \overline {{G ^ {Y}}}) - I (E; G _ {C} | Y), \\ \end{array}
+$$
+
+where  $\overline{G^Y}$  denotes the extra-class samples of the graph  $G$  which is labeled as  $Y$ .
+
+Proposition 4.4. The learning objective of negative inference paradigm (Eq. 3) encompasses a broader cogni-
+
+tive space for environments, with its upper limit being the ground-truth environment distribution.
+
+Proposition 4.4 emphasizes that the negative inference paradigm enables a broader-scale environment inference space by cooperatively modeling both intra-class spurious subgraphs and extra-class samples. Detailed proof can be found in Appendix D.2.
+
+# 5. Graph Invariant Learning with Negative Inference
+
+In this section, we introduce a novel negative inference graph OOD framework NeGo to address the limitation of existing efforts in handling complex environments shifts. Specifically, NeGo is developed to design a negative inference learning task to capture underlying environments, and leverage inferred environment embeddings to enhance graph invariant learning.
+
+# 5.1. Negative Prompt Learning for Environment Inference
+
+Negative inference focuses on indirectly extracting invariant subgraph by investigating the information beyond the causal factors. This leads to the problem that the space of variables beyond causal information is infinite-dimensional. Given the insight from Theorem 4.4, we decouple the process of modeling the environment through negative inference into two components: the extraction of intra-class spurious subgraphs and the inference of extra-class samples. Modeling spurious subgraphs is relatively straightforward and extensively studied. The crucial challenge lies in achieving a comprehensive understanding of extra-class sample space.
+
+Formally, let the prior distribution of extra-class samples for  $G$  be denoted as  $\mathbb{P}(\overline{Y})$ , where  $G$  is with the label  $Y$ . We introduce a variational estimate of the environment variables denoted as  $\mathbb{Q}_{\phi}(E|G)$  (a.k.a.,  $f_{\phi}$ ), where  $\phi$  is the parameterized network. Denoting KL-divergence as  $\mathrm{KL}(\cdot ||\cdot)$ , the training of  $\mathbb{Q}_{\phi}$  is to implement the first term of Eq. 3, which can be formalized as following optimization,
+
+$$
+\min  _ {\phi} \mathbb {E} [ \mathrm {K L} (\mathbb {Q} _ {\phi} (E | G)) | | \mathbb {P} (\overline {{G ^ {Y}}}) ]. \tag {4}
+$$
+
+Inspired by the success of prompt learning in capturing underlying semantic and causal associations in large language models (Floridi & Chiriatti, 2020; Sordoni et al., 2024), we introduce a negative prompter to achieve this goal. Specifically, given a sample  $G$  belonging to class  $l$ , the negative prompter treats all extra-class samples as environments. Designing appropriate prompt tokens to guide effective learning is the primary question that needs to be addressed when employing the concept of prompt learning.
+
+Given the proven efficacy of learnable prompts in various
+
+practices, we design class-specific learnable prompt tokens  $\pmb{P} = [\pmb{v}^{(1)},\pmb{v}^{(2)},\dots,\pmb{v}^{(L)}]$ , where  $\pmb{v}^{(i)}\in \mathbb{R}^{1\times d}$  and  $L$  is the number of classes. The class-specific design manner aims to capture the extra-class sample space for each graph, in order to achieve the objective defined by Eq. 4. The negative prompter  $f_{\phi}(\cdot)$  is guided to learn the prompt answers  $\pmb{A}_N\in \mathbb{R}^{L\times d}$  by interacting the encoded graph embedding  $\pmb{Z}_{G}\in \mathbb{R}^{1\times d}$  and the learnable prompts  $\pmb{P}$ ,
+
+$$
+\boldsymbol {A} _ {N} = f _ {\phi} \left(\boldsymbol {Z} _ {G}, \boldsymbol {P}\right). \tag {5}
+$$
+
+The negative prompter  $f_{\phi}$  is parameterized by a cross-attention network in Transformer decoder (Vaswani et al., 2017), where  $Z_{G}$  is obtained by a GNN backbone encoder  $h_{\psi}(\cdot)$ . For a sample  $G$  with label  $l$ , such negative prompts answers  $A_{N}$  should satisfy the following two properties:
+
+- The prompts answers  $\mathbf{A}_N$  should produce a low match with graphs whose labels are  $l$ .  
+- The prompts answers  $\mathbf{A}_N$  should produce a high match with graphs whose labels are not  $l$ .
+
+With the explanation in the language models, our negative prompt mechanism involves designing prompt tokens to learn the desired [answer] of "the underlying environments of current sample are [answer]". These two properties guide  $f_{\phi}(\cdot)$  to learn a positive answer when interacting with each extra-class sample and a negative answer when interacting with each intra-class sample. Therefore, the training objective of our negative prompt mechanism can be formulated as,
+
+$$
+\begin{array}{l} \mathcal {L} _ {\text {n e g a}} = \mathbb {E} [ \mathrm {K L} (\mathbb {P} (\overline {{G ^ {Y}}}) | | \mathbb {Q} _ {\phi} (E | G)) ] \\ = - \mathbb {E} \left[ \log \mathbb {P} _ {\phi} (\bar {Y} | G, P) - \log \mathbb {P} _ {\phi} (Y | G, P) \right]. \tag {6} \\ \end{array}
+$$
+
+The environment variables we infer are class-specific, in contrast to the global environment factors constructed by previous methods. Our design is intuitively reasonable, as a specific subgraph may be perceived by one class as causal information, while its minor variations are perceived by other classes as environments. Moreover, it is worth noting that we never overlook the inference of the environments (spurious subgraphs) within intra-class samples. Given that the intra-class environments are always intertwined with causal factors, we incorporate the inference of intra-class environment variables into the discovery of the causal subgraph, which is provided in the next subsection.
+
+# 5.2. Environment-enhanced Graph Invariant Learning
+
+While inferring environment variables is a crucial step in understanding the data generation process, the ultimate goal of graph learning is to achieve casual invariant prediction. Thus, the next challenge to address is the disentanglement of
+
+the causal subgraph from environments. Existing methods often neglect the design of a graph-tailored environment exploitation algorithm, which can lead to the failure in extracting causal subgraphs when environment becomes complex (Gui et al., 2024).
+
+We propose an environment-enhanced invariant learning mechanism that leverages perceived latent environment embeddings to achieve the extraction of causal subgraphs with resistance to complex environment disturbances. Different from the negative prompter that investigates the extra-class sample space, we concentrate on the disentanglement of causal invariant substructures within the intra-class samples in this subsection.
+
+Let the marginal distribution of the causal subgraph  $G_{C}$  be  $\mathbb{P}(G_C)$ . We introduce a variational estimation of the subgraph extraction  $\mathbb{Q}_{\xi}(G_C|G,E)$  (a.k.a.,  $g_{\xi}$ ), where  $\xi$  is the parameterized networks. The model can make casual invariant predictions of the label distribution  $\mathbb{P}_{\theta}(Y|G_C)$  (a.k.a.,  $g_{\theta}$ ), only when the causal graph is accurately extracted from complex environments. The learning objective for environment-enhanced graph invariant learning  $\mathbb{P}_{\theta} \circ \mathbb{Q}_{\xi}(\cdot)$  is to implement the second term of Eq. 3, which can be formalized as following optimization,
+
+$$
+\min  _ {\xi , \theta} \mathbb {E} [ \mathrm {K L} (\mathbb {Q} _ {\xi} (G _ {C} | G, E) | | \mathbb {P} (G _ {C})) - \log \mathbb {P} _ {\theta} (Y | G _ {C}) ]. \tag {7}
+$$
+
+The environment embedding  $\mathbf{A}_N \in \mathbb{R}^{L \times d}$  is inferred at the graph level, but the extraction of substructures often requires node-level operations. Therefore, the primary focus of environment-enhanced invariant learning is to propagate the perceived environment embedding  $\mathbf{A}_N$  to individual nodes. We design an interaction-decoding module  $g_{\xi_1}(\cdot)$  to address this issue.
+
+Specifically,  $g_{\xi_1}(\cdot)$  consists of three families of learnable parameters, i.e.,  $W^{Q}, W^{K}, W^{V} \in \mathbb{R}^{d \times d}$ .  $g_{\xi_1}(\cdot)$  takes the node-level representation  $\mathbf{Z} \in \mathbb{R}^{N \times d}$  encoded by the GNN encoder  $h_{\psi}(\cdot)$  and the environment embedding  $\mathbf{A}_N \in \mathbb{R}^{L \times d}$  obtained by negative prompt as inputs. Three hidden state matrices are calculated by,
+
+$$
+\boldsymbol {Z} ^ {Q} = \boldsymbol {Z} W ^ {Q}, \boldsymbol {A} ^ {K} = \boldsymbol {A} _ {N} W ^ {K}, \boldsymbol {A} ^ {V} = \boldsymbol {A} _ {N} W ^ {V}. \tag {8}
+$$
+
+The node embedding with environment information obtained through residual connections is,
+
+$$
+\boldsymbol {Z} _ {E} = \operatorname {s o f t m a x} \left(\frac {\boldsymbol {Z} ^ {Q} (\boldsymbol {A} ^ {K}) ^ {T}}{\sqrt {d}}\right) \boldsymbol {A} ^ {V} + \boldsymbol {Z}. \tag {9}
+$$
+
+We exploit a subgraph extractor  $G_{C} = g_{\xi_{2}}(Z_{E})$  to realize invariant subgraph discovery. Then,  $G_{C}$  is encoded by  $h_{\psi}(\cdot)$  to obtain the causal representation for prediction. This representation is passed through an MLP layer  $g_{\theta}$  to model the distribution of  $Y$ . Therefore, the training objective of
+
+environment-enhanced invariant learning is,
+
+$$
+\begin{array}{l} \mathcal {L} _ {\text {p o s i}} = - \mathbb {E} [ \log \mathbb {P} _ {\xi , \theta} (Y | G _ {C}) ] \\ = - \mathbb {E} \left[ \log \mathbb {P} _ {\theta} \left(Y \mid G _ {C}\right) + \log \mathbb {P} _ {\xi_ {1}, \xi_ {2}} \left(G _ {C} \mid G, \boldsymbol {A} _ {N}\right) \right]. \tag {10} \\ \end{array}
+$$
+
+# 5.3. Optimization Objective and Theoretical Analysis
+
+Our NeGo achieves a graph learning framework with a wider space of environment inference. This is accomplished through two sequential approaches, first focusing on constructing the learning task for negative inference, and then leveraging the environment embeddings obtained from negative inference to enhance graph causal invariant learning. Thus, the training objective of our NeGo is,
+
+$$
+\mathcal {L} = \mathcal {L} _ {\text {n e g a}} + \lambda \cdot \mathcal {L} _ {\text {p o s i}}, \tag {11}
+$$
+
+where  $\lambda$  is a hyperparameter, which is set to 1 in the implementation. The training process of NeGo is provided in Alg. 1. Note that the two sub-challenges addressed by NeGo are not independent but closely interconnected. The environment negative inference mechanism assists the network in comprehending the distribution shift of data, while the causal invariant learning with environment enhancement empowers the network to accurately extract causal invariant subgraphs even in complex environments. Thus, the former serves as a foundation for the latter. This design reflects the principle that understanding data generation process is crucial to enhance the generalization of models. We also provide theoretical evidence supporting the ability of NeGo to effectively address both FIIF and PIIF under both cases of  $H(G_{C}|Y) < H(G_{S}|Y)$  and  $H(G_{C}|Y) > H(G_{S}|Y)$ . Detailed proof can be found in Appendix D.3.
+
+Theorem 5.1. Given the FIIF or PIIF assumptions under both cases when  $H(G_{C}|Y) < H(G_{S}|Y)$  and  $H(G_{C}|Y) > H(G_{S}|Y)$ , the causal subgraph  $G_{C}$  can be extracted by optimizing Eq. 11.
+
+# 6. Experiments
+
+We evaluate the effectiveness of NeGo by answering the following questions. Q1. Does our approach effectively address the issue unresolved in existing works? Q2. Is our framework sufficiently interpretable? Q3. Does each component in our NeGo effectively enhance the generalization capacity? Q4. Does our framework operate with high efficiency?
+
+# 6.1. Baselines
+
+We choose four representative OOD methods and seven graph-specific OOD approaches for comparison. The representative OOD frameworks we select consist of ERM, IRM (Arjovsky et al., 2019), V-Rex (Krueger et al., 2021), and
+
+# Algorithm 1 The training process of NeGo
+
+Input: training data  $\mathcal{G}$ , negative prompts  $P$ .
+
+Initial: the GNN encoder  $h_{\psi}$ , the negative prompter  $f_{\phi}$ , environment-enhanced invariant learning mechanism  $g_{\xi}$ , final predictor  $g_{\theta}$ , learnable prompt tokens  $P$ , the number of epochs  $K$ .
+
+for  $i = 1$  to  $K$  do
+
+$$
+\mathbf {Z} _ {G} = h _ {\psi} (G)
+$$
+
+$$
+\boldsymbol {A} _ {N} = f _ {\phi} (\boldsymbol {Z} _ {G}, \boldsymbol {P})
+$$
+
+$$
+\boldsymbol {Z} ^ {Q} = \boldsymbol {Z} \boldsymbol {W} ^ {Q}, \boldsymbol {A} ^ {K} = \boldsymbol {A} _ {N} \boldsymbol {W} ^ {K}, \boldsymbol {A} ^ {V} = \boldsymbol {A} _ {N} \boldsymbol {W} ^ {V}
+$$
+
+$$
+\boldsymbol {Z} _ {E} = \operatorname {s o f t m a x} \left(\frac {\boldsymbol {Z} ^ {Q} (\boldsymbol {A} ^ {K}) ^ {T}}{\sqrt {d}}\right) \boldsymbol {A} ^ {V}
+$$
+
+$$
+Y = g _ {\theta} \left(G _ {C}\right), G _ {C} = g _ {2} \left(\mathbf {Z} _ {E} + \mathbf {Z}\right)
+$$
+
+Optimizing:
+
+$$
+\mathcal {L} _ {n a g a} \quad = \quad \mathbb {E} [ \mathrm {K L} (\mathbb {P} (\overline {{G ^ {Y}}}) | | \mathbb {Q} _ {\phi} (E | G)) ] \quad =
+$$
+
+$$
+- \mathbb {E} [ \log \mathbb {P} _ {\phi} (\bar {Y} | G, P) - \log \mathbb {P} _ {\phi} (Y | G, P) ]
+$$
+
+$$
+\mathcal {L} _ {\text {p o s i}} = - \mathbb {E} [ \log \mathbb {P} _ {\xi , \theta} (Y | G _ {C}) ] = - \mathbb {E} [ \log \mathbb {P} _ {\theta} (Y | G _ {C}) +
+$$
+
+$$
+\log \left. \mathbb {P} _ {\xi_ {1}, \xi_ {2}} \left(G _ {C} | G, \boldsymbol {A} _ {N}\right) \right]
+$$
+
+$$
+\min  _ {\psi , \phi , \theta , \xi , \mathbf {P}} \mathcal {L} = \mathcal {L} _ {n e g a} + \mathcal {L} _ {p o s i}
+$$
+
+end for
+
+Return  $h_{\psi},f_{\phi},g_{\xi},g_{\theta}$  and  $\mathbf{P}$
+
+IB-IRM (Ahuja et al., 2021). The Empirical Risk Minimization (ERM) baseline is a vanilla GNN with ERM objective, which is trained using the same settings with (Gui et al., 2024). Graph OOD approaches include DIR (Wu et al., 2022c), GSAT (Miao et al., 2022), CAL (Sui et al., 2022), CIGA (Chen et al., 2022), GIL (Li et al., 2022), LECI (Gui et al., 2024) and GALA (Chen et al., 2024).
+
+# 6.2. Datasets
+
+We adopt two synthetic datasets with distribution shift and six real-world scenario shift datasets from both molecular and social science domains. Synthetic datasets include GOOD-Motif (Wu et al., 2022c) and GOOD-CMNIST (Gui et al., 2022). In molecular property prediction fields, we select the scaffold and size splits of GOOD-HIV dataset (Gui et al., 2022; Wu et al., 2018) and the assay and size splits of DrugOOD LBAP-core-ic50 dataset (Ji et al., 2022). We also choose two social sentiment graph datasets with distribution shifts, including GOOD-SST2 and GOOD-Twitter (Yuan et al., 2022). Detailed statistics on the number of graphs in those datasets are provided in Tab. 6.
+
+# 6.3. Result Comparison and Analysis
+
+We comprehensively evaluate the OOD performance of NeGo on both real-world and synthetic datasets to answer Q1. Tab. 1 and 2 present the performance of NeGo on chemical and sentiment graph datasets. Tab. 8 showcases the performance of our framework on two synthetic datasets. Compared to existing methods, our method achieves optimal performance on almost all datasets. Besides, the per
+
+Table 1. The ROC-AUC performance of NeGo on four real-world datasets in chemical research field. ID val and OOD val represent the results of OOD test set using the in-distribution and out-of-distribution validation sets, respectively (Gui et al., 2024). The best results are shown in bold and the second best results are underlined. * indicates statistical significance against the second-best results.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">GOOD-HIV-scaffold</td><td colspan="2">GOOD-HIV-size</td><td colspan="2">DrugOOD-assay</td><td colspan="2">DrugOOD-size</td></tr><tr><td>ID val</td><td>OOD val</td><td>ID val</td><td>OOD val</td><td>ID val</td><td>OOD val</td><td>ID val</td><td>OOD val</td></tr><tr><td>ERM</td><td>69.61±1.32</td><td>70.37±1.19</td><td>61.66±2.45</td><td>57.31±1.06</td><td>70.03±0.16</td><td>72.18±0.18</td><td>62.97±0.26</td><td>63.29±0.33</td></tr><tr><td>IRM</td><td>73.35±2.30</td><td>70.89±0.29</td><td>58.52±0.86</td><td>60.86±2.78</td><td>71.56±0.32</td><td>72.69±0.29</td><td>63.24±0.26</td><td>63.46±0.23</td></tr><tr><td>V-Rex</td><td>71.73±3.51</td><td>71.18±0.69</td><td>58.39±1.54</td><td>60.10±2.09</td><td>70.22±0.86</td><td>72.32±0.58</td><td>63.87±0.42</td><td>64.11±0.39</td></tr><tr><td>IB-IRM</td><td>67.56±2.31</td><td>66.25±0.93</td><td>57.45±0.74</td><td>56.65±1.22</td><td>69.34±0.48</td><td>71.32±0.76</td><td>64.03±0.61</td><td>64.59±0.70</td></tr><tr><td>DIR</td><td>65.84±1.71</td><td>68.59±3.70</td><td>59.69±1.59</td><td>60.85±0.52</td><td>67.29±0.73</td><td>69.70±0.65</td><td>63.85±0.65</td><td>64.73±0.54</td></tr><tr><td>GSAT</td><td>71.55±3.58</td><td>71.39±1.41</td><td>60.92±1.00</td><td>60.61±1.19</td><td>71.01±0.54</td><td>72.26±0.45</td><td>65.12±0.38</td><td>65.67±0.45</td></tr><tr><td>CAL</td><td>73.48±2.64</td><td>72.38±1.03</td><td>62.83±1.26</td><td>62.58±1.04</td><td>71.89±0.92</td><td>71.23±1.13</td><td>63.85±0.49</td><td>64.22±0.74</td></tr><tr><td>CIGA</td><td>66.25±2.89</td><td>71.47±1.29</td><td>58.24±3.78</td><td>62.56±1.76</td><td>67.68±1.14</td><td>70.54±0.59</td><td>64.14±0.66</td><td>64.83±0.79</td></tr><tr><td>GIL</td><td>70.89±1.60</td><td>70.23±1.23</td><td>61.74±1.76</td><td>61.29±1.34</td><td>70.45±0.89</td><td>70.73±1.36</td><td>64.91±0.51</td><td>65.43±0.64</td></tr><tr><td>LECI</td><td>74.04±0.65</td><td>74.43±1.69</td><td>64.83±2.59</td><td>65.44±1.78</td><td>72.67±0.46</td><td>73.45±0.17</td><td>65.93±0.43</td><td>66.49±0.60</td></tr><tr><td>GALA</td><td>73.85±1.10</td><td>74.02±1.34</td><td>63.99±1.54</td><td>64.45±2.26</td><td>72.83±0.73</td><td>73.23±0.29</td><td>65.23±0.72</td><td>65.84±0.52</td></tr><tr><td>NeGo</td><td>75.21±0.73*</td><td>75.87±1.02</td><td>65.23±1.74</td><td>65.92±1.82*</td><td>73.20±0.18*</td><td>73.94±0.25*</td><td>65.49±0.73</td><td>66.91±0.84*</td></tr></table>
+
+Table 2. The accuracy of NeGo on two sentiment graph datasets, where * indicates statistical significance against the second-best results.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">GOOD-SST2</td><td colspan="2">GOOD-Twitter</td></tr><tr><td>ID val</td><td>OOD val</td><td>ID val</td><td>OOD val</td></tr><tr><td>ERM</td><td>78.37±2.64</td><td>80.41±0.69</td><td>54.93±0.96</td><td>57.04±1.70</td></tr><tr><td>IRM</td><td>79.73±1.45</td><td>80.17±1.52</td><td>55.27±1.19</td><td>57.72±1.03</td></tr><tr><td>V-Rex</td><td>79.31±1.40</td><td>80.33±1.09</td><td>56.46±0.93</td><td>56.37±0.76</td></tr><tr><td>IB-IRM</td><td>78.93±1.23</td><td>80.22±0.55</td><td>54.23±1.21</td><td>56.73±1.02</td></tr><tr><td>DIR</td><td>77.65±0.71</td><td>81.50±0.55</td><td>55.32±1.85</td><td>56.81±0.91</td></tr><tr><td>GSAT</td><td>79.25±1.09</td><td>80.46±0.38</td><td>55.09±0.66</td><td>56.07±0.53</td></tr><tr><td>CAL</td><td>81.20±1.21</td><td>82.34±0.67</td><td>56.77±0.86</td><td>57.82±0.44</td></tr><tr><td>CIGA</td><td>80.37±1.46</td><td>82.93±0.75</td><td>57.51±1.36</td><td>57.19±1.15</td></tr><tr><td>GIL</td><td>81.43±1.02</td><td>83.31±0.50</td><td>58.21±1.24</td><td>57.82±1.18</td></tr><tr><td>LECI</td><td>82.93±0.22</td><td>83.44±0.27</td><td>59.35±1.44</td><td>59.64±0.15</td></tr><tr><td>GALA</td><td>82.60±0.66</td><td>82.98±0.42</td><td>59.03±0.65</td><td>60.45±1.36</td></tr><tr><td>NeGo</td><td>82.72±0.51</td><td>84.16±0.23*</td><td>60.82±0.22*</td><td>61.25±0.70*</td></tr></table>
+
+formance of environment-centered OOD methods, such as LECI and GALA, often achieves suboptimal or even optimal results on various datasets. This demonstrates the effectiveness of modeling environment factors in addressing data distribution shifts.
+
+Table 3. The performance comparison (ID val) of NeGo on new environment scenarios.  
+
+<table><tr><td>Model</td><td>GSAT</td><td>CAL</td><td>CIGA</td><td>LECI</td><td>GALA</td><td>NeGo</td></tr><tr><td>Acc.(%)</td><td>70.13</td><td>75.21</td><td>71.82</td><td>78.46</td><td>77.93</td><td>79.75</td></tr></table>
+
+To further validate the effectiveness of our NeGo in new environment scenarios (nonlinear environment mixing), we conduct additional experimental discussion. Considering that the environments in SPURIOUS-MOTIF (Ying et al., 2019) include five specific label-irrelevant base subgraphs (wheel, tree, ladder, star, and path), we increase the environment complexity by diversifying these base subgraphs. Specifically, we randomly add connections to the base subgraphs in each sample, totaling  $10\%$  of the original edge
+
+count. As shown in Tab. 3, we observe that the advantage of our NeGo on common OOD datasets is persists in complex environments, with a  $1.29\%$  performance improvement.
+
+Moreover, we evaluate the performance of our framework in the complex environments scenario illustrated in Fig. 1(a). NeGo achieves  $87.34\%$  and  $80.29\%$  on the original and adjusted dataset, respectively. There is only a minor decrease in performance, suggesting that our method effectively tackles the limitations encountered by existing methods in handling complex environments.
+
+To answer the Q2, we visually represent the causal subgraphs extracted by NeGo on the modified dataset in Fig. 1(a). As depicted in Fig. 3, our method consistently extracts the ground-truth subgraph. The visualized results further validate the effectiveness of our proposed negative inference method. By modeling extensive extra-class samples as environments, our approach offers undeniable advantages in handling complex environment shifts.
+
+# 6.4. Ablation Studies
+
+To answer Q3, we investigate each component of NeGo. Specifically, we conduct ablation studies to explore the effectiveness of negative prompter and interactive decoding component. Tab. 4 shows that the performance drops significantly when there is either no negative prompter or interactive decoding component. NeGo-NoPro refers to the framework that eliminates negative prompter and negative loss, which causes the most performance drop. Therefore, the negative inference mechanism plays a vital role in enhancing the capability of environment perception. This further validates the rationale of our motivation for incorporating negative inference. NeGo-NoEnv indicates that the casual subgraphs are extracted directly using node embedding without integrating inferred environment information.
+
+![](images/99596c138b3cc04789cb97a6d8f6531e344f6291046bed7e31ebeea5789d716d.jpg)  
+Figure 3. The causal subgraphs extracted by NeGo on the modified dataset in Fig. 1(a).
+
+The performance decline emphasizes the significance of environment utilizing strategies overlooked by existing works.
+
+Table 4. Ablation studies of NeGo.  
+
+<table><tr><td>Model</td><td>DrugOOD(assay)</td><td>GOOD(Twitter)</td></tr><tr><td>NeGo-NoPro</td><td>70.37</td><td>58.41</td></tr><tr><td>NeGo-NoEnv</td><td>71.71</td><td>59.17</td></tr><tr><td>NeGo</td><td>73.20</td><td>60.82</td></tr></table>
+
+# 6.5. Efficiency Analysis
+
+To address Q4, we explore the training efficiency of  $\mathsf{NeGo}$  from both theoretical and practical perspectives. The time complexity of  $\mathsf{NeGo}$  is  $\mathcal{O}(|\mathcal{V}| \times d^2 + |\mathcal{V}| \times d \times h + |\mathcal{E}| \times d)$ , where  $|\mathcal{V}|$  represents the number of nodes,  $|\mathcal{E}|$  denotes the number of edges,  $d$  is the feature dimension, and  $h$  represents the number of cross-attention heads. Our method has linear time complexity with high training efficiency.
+
+We empirically compare the training efficiency of NeGo with other baselines on DrugOOD-size dataset as shown in Tab. 5. Compared with some earlier invariant learning methods (DIR and GSAT), the minor increase in running time of our mothod brings out the substantial performance boost. Additionally, our approach demonstrates greater competitiveness in both training efficiency and performance compared to existing environment-centered methods.
+
+Table 5. The training efficiency of NeGo with other baselines on DrugOOD-size (s/epoch).  
+
+<table><tr><td>Models</td><td>GSAT</td><td>DIR</td><td>CIGA</td><td>LECI</td><td>GALA</td><td>NeGo</td></tr><tr><td>Time</td><td>51.6</td><td>52.6</td><td>54.2</td><td>59.1</td><td>62.3</td><td>58.7</td></tr></table>
+
+# 7. Conclusion
+
+In this work, we propose a negative inference graph OOD framework NeGo to handle complex environment shift in OOD scenarios. By inheriting the practices of prompt learning in large language models, we design a negative prompter to model the environment on a larger scale. We then introduce an environment-enhanced invariant learning strategy to eliminate spurious subgraphs from the data. This strategy effectively leverages the inferred environment variables to
+
+enhance the ability to remove irrelevant information. Extensive experiments on real-world datasets across domains and synthetic datasets validate the effectiveness of NeGo.
+
+# Impact Statement
+
+This paper investigates the advanced application of graph learning in chemistry and social science domains, utilizing publicly available datasets. Consequently, there are no ethical concerns in this work.
+
+# Acknowledgment
+
+This work was supported by the National Natural Science Foundation of China (No.12227901), Natural Science Foundation of Jiangsu Province (BK.20240460, BK.20240461), the grant from State Key Laboratory of Resources and Environmental Information System. The AI-driven experiments, simulations and model training were performed on the robotic AI-Scientist platform of Chinese Academy of Science.
+
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+
+# A. More Related works
+
+# A.1. OOD Generalization.
+
+Out-of-Distribution (OOD) generalization learning refers to the task of adapting a model that has been trained on a specific distribution to effectively process data from a potentially different distribution. This study holds significant importance because the issue of data distribution shifts is a common occurrence in the real world. External factors, such as changes in environmental conditions, technological advancements, or evolving user preferences, can lead to shifts in the data distribution. Various approaches can be employed for OOD generalization, including data augmentation (Rong et al., 2019; Wang et al., 2021; You et al., 2020), domain adaptation (Wang & Deng, 2018), and causal invariant learning (Sui et al., 2022; Wu et al., 2022c). Jia et al. (Jia et al., 2024) innovatively proposes a mixup-based environment modeling framework, IGM, to enhance graph invariant learning. IGM focuses on expanding the environment space through a mixing generation scheme, while our NeGo aims to mine environmental space as much as possible from the novel perspective of negative learning. Piao et al. creatively proposes a hierarchical environment inference paradigm to enhance graph invariant learning methods (Piao et al., 2024). This work focuses on generating sample-level hierarchical environments to expand the modeling of the environment space. Unlike this method, our NeGo focuses on class-level environment augmentation, collaborating with extra-class environment modeling and inter-class invariant learning to achieve global inference of environment space.
+
+Among them, causal invariant learning demonstrates impressive performance in various fields, due to its powerful interpretability (Chen et al., 2022; Li et al., 2022; Miao et al., 2022; Wu et al., 2022c). Our NeGo is aligned with this research line, as an environment-centered invariant learning method based on causal theory. However, in the field of graph learning, most existing invariant learning methods focus on extracting the causal graph to achieve invariant learning. This strategy limits the inference space of the environments to the dimension of spurious subgraphs, which hinders the ability of models to capture the complex environment states. In this work, we propose an invariant learning mechanism based on negative inference to address this limitation.
+
+# A.2. Prompt Learning
+
+Prompt learning is proposed in NLP models to infer underlying semantic and potential causal associations in linguistic data. Many effective prompt methods have developed with the introduction of large language models, including some hand-crafted prompts (Brown et al., 2020), discrete prompts (Gao et al., 2020; Shin et al., 2020), and learnable prompts design (Li & Liang, 2021). There have been various works on the interaction of computer vision and natural language processing fields, e.g., text-to-image retrieval text-to-image retrieval (Wang et al., 2019), visual question answering(Antol et al., 2015; Rao et al., 2022; Zhou et al., 2022a) and so on.
+
+In recent years, prompt learning has also been developed in the graph learning field (Sun et al., 2022a; Li et al., 2024; Sun et al., 2023; Zi et al., 2024). Our approach is the pioneering effort to apply prompt learning to address the challenge of graph OOD generalization issue.
+
+# A.3. Comparisons to Existing Graph OOD Works
+
+Environment-centered studies (Chen et al., 2024; Gui et al., 2024; Li et al., 2022; Wu et al., 2022a; Yang et al., 2022) consider that the data distribution shifts stem from the changes of environments. Therefore, these practices enable the model to withstand data distribution shifts by inferring environment variables. Concretely, the networks are often trained with the objective of equipping models to effectively handle mixed environments scenarios. However, this design allows the networks to make narrow inference about the environments, and makes the networks unable to handle with distribution shifts in complex environments. We attribute this limitation of inference scale to the shortcomings of positive inference, which is proved both empirically and theoretically. Therefore, we propose a negative inference mechanism to broaden the inference space for environments, without relying on the mixed environments hypothesis.
+
+Our approach, which represents a pioneering practice in utilizing negative inference, is distinct from all existing practices in this field. DIR (Wu et al., 2022c) aims to identify causal patterns that are stable across different distributions and filter out spurious patterns that are unstable. This work is a classic work in the early application of causal theory to address the challenge of graph OOD generalization. It focuses on obtaining invariant subgraphs with a positive inference manner. GIL (Li et al., 2022) aims to capture the invariant relationships between predictive graph structural information and labels in a mixture of latent environments through jointly optimizing three mutually promoting modules. This method relies on the mixing environment hypothesis and has limited inference space for environments. CIGA (Chen et al., 2022) build
+
+three Structural Causal Models (SCMs) to characterize the distribution shifts that could happen on graphs: one is to model the graph generation process, and the other two are to model two possible interactions between invariant and spurious features during the graph generation, i.e., FIIF and PIIF. This work provides a fresh perspective on existing research on out-of-distribution generalization based on causality. However, it still falls within the framework of positive inference, aiming to extract causal subgraphs. GALA (Chen et al., 2024) utilized proxy prediction mechanism to infer environment label. It is worth noting that the negative samples mentioned in this work are different from our negative inference, and their design is also to improve performance under the mixed environments hypothesis. Thus, it essentially follows a positive inferring process for environment variables. LECI (Gui et al., 2024) primarily focused on spurious substructures space to model the environment variables. Such environment inference strategy still relies on a positive inference with narrow cognitive space of the environments. Unlike existing graph OOD researches that centers on environment awareness, our work presents greater technical challenges, and targets scenarios that are more complex. G-Splice (Li et al.) focuses on linear mixed scenarios with spurious structures, whereas our work aims to address the effectiveness of models in nonlinear mixed scenarios. AIA (Sui et al., 2024) aims to model the environment factors in graph through the lens of data augmentation. Unlike AIA that focuses only on modifications within spurious subgraphs inside samples, our NeGo is more concerned with environment extraction in mixed scenarios at the class level. Thus, in scenarios where the spurious subgraphs within a sample are highly complex, we believe that our method is more effective.
+
+# B. Detailed Datasets
+
+- GOOD-Motif (Wu et al., 2022c) is a synthetic dataset designed for studying structure shifts. Each graph in the dataset is created by connecting a base graph and a motif, where the label is determined by the motif. This accessible ground-truth substructure brings a lot of convenience to the invariant subgraph learning with interpretability. This dataset includes five label-irrelevant base graphs (wheel, tree, ladder, star, and path) and three label-determining motifs (house, cycle, and crane) are used to generate the graphs in the dataset. In environment-centered invariant learning, such base graphs can be seen as environment factors and such motifs are be considered as the casual factors.  
+- GOOD-CMNIST (Gui et al., 2022) is a semi-synthetic dataset that has been purposefully created to evaluate node feature shifts. It comprises graphs constructed from hand-written digits extracted from the MNIST database, with the transformation applied using superpixel techniques (Monti et al., 2017).  
+- GOOD-HIV (Gui et al., 2022) is a compact and real-world molecular dataset that has been derived from (Wu et al., 2018). It comprises molecular graphs, where atoms represent nodes and chemical bonds represent edges. The primary task associated with this dataset is to predict a molecule's potential for inhibiting HIV replication. Its distribution shift scenario is developed into two, i.e., the scaffold, and the size of nodes in a molecular graph.  
+- DrugOOD(LBAP-core-ic50) (Ji et al., 2022) is utilized in the Ligand-based Affinity Prediction (LBAP) task, where the core noise level and IC50 measurement type serve as domain features. Its distribution shift scenario is developed into three, i.e., the scaffold, the size, and the assay.  
+- GOOD-SST2 (Yuan et al., 2022) is a real-world social sentiment dataset derived from natural language. This dataset represents each sentence as a graph, where individual words are treated as nodes, and their corresponding word embeddings serve as node features. The primary task in this dataset involves binary classification, aiming to predict the sentiment polarity of each sentence.  
+- GOOD-Twitter (Yuan et al., 2022) is a real-world natural language sentiment dataset that shares the same transformation process as the SST2 dataset. The classification task of this dataset involves predicting one of three sentiment polarities for each sentence. Similar to the GOOD-SST2 dataset, the sentence lengths are chosen as the domains.
+
+# C. Detailed Baselines
+
+- DIR (Wu et al., 2022c) is an early work using causal theory to address the distribution shifts issue in graph data. This work provides detailed theoretical proofs that demonstrate the feasibility of extracting invariant subgraph from graph data.  
+- GSAT (Miao et al., 2022) employ information bottleneck theory to select causal subgraphs under only the FIIF assumption. The proposed stochastic attention mechanism in this paper is highly robust in extracting casual subgraphs,
+
+Table 6. Statistics on the number of graphs in the datasets.  
+
+<table><tr><td>Dataset</td><td>Training</td><td>ID validation</td><td>ID test</td><td>OOD validation</td><td>OOD test</td></tr><tr><td>GOOD-HIV-Scaffold</td><td>24682</td><td>4112</td><td>4112</td><td>4113</td><td>4108</td></tr><tr><td>GOOD-HIV-Size</td><td>26169</td><td>4112</td><td>4112</td><td>2773</td><td>3961</td></tr><tr><td>GOOD-SST2-Length</td><td>24744</td><td>5301</td><td>5301</td><td>17206</td><td>17490</td></tr><tr><td>GOOD-Twitter-Length</td><td>2590</td><td>554</td><td>554</td><td>1785</td><td>1457</td></tr><tr><td>GOOD-CMNIST-Color</td><td>42000</td><td>7000</td><td>7000</td><td>7000</td><td>7000</td></tr><tr><td>GOOD-Motif-Basis</td><td>18000</td><td>3000</td><td>3000</td><td>3000</td><td>3000</td></tr><tr><td>GOOD-Motif-Size</td><td>18000</td><td>3000</td><td>3000</td><td>3000</td><td>3000</td></tr><tr><td>DrugOOD-assay</td><td>34179</td><td>11314</td><td>11683</td><td>19028</td><td>19032</td></tr><tr><td>DrugOOD-size</td><td>36597</td><td>12153</td><td>12411</td><td>17660</td><td>16415</td></tr></table>
+
+Table 7. Comparison of existing methods on addressing OOD generalization issue.  
+
+<table><tr><td>Methods</td><td>SCMs</td><td>Inferred Environment Space</td></tr><tr><td>DIR</td><td>FIIF</td><td>Spurious subgraphs</td></tr><tr><td>GSAT</td><td>FIIF</td><td>Spurious subgraphs</td></tr><tr><td>CIGA</td><td>FIIF &amp; PIIF</td><td>Spurious subgraphs</td></tr><tr><td>GALA</td><td>FIIF &amp; PIIF</td><td>Spurious subgraphs</td></tr><tr><td>LECI</td><td>FIIF &amp; PIIF</td><td>Spurious subgraphs</td></tr><tr><td>NeGo</td><td>FIIF &amp; PIIF</td><td>Intra-class spurious subgraphs and extra-class sample space</td></tr></table>
+
+and has emerged as a backbone model in numerous methods. Actually, the subgraph extractor used in our work is also inspired by GSAT.
+
+- CAL (Sui et al., 2022) is guided by the backdoor adjustment principle derived from causal theory. It encourages the Graph Neural Networks (GNNs) to focus on exploiting causal features while disregarding shortcut connections.  
+- CIGA (Chen et al., 2022) is the first graph OOD method considering both Fully Informative Invariant Feature (FIIF) and Partially Informative Invariant Feature (PIIF) assumptions. This work presents an OOD algorithm for graphs that is provably generalizable under different types of distribution shifts.  
+- GIL (Li et al., 2022) is designed to capture invariant graph patterns in a mixture of underlying environments and handle the distribution shift issue. This work introduces a GNN-based subgraph generator to identify potentially invariant subgraphs from the complex interaction between invariant and variant patterns.  
+- LECI (Gui et al., 2024) comprehensively reviews existing OOD approaches and identifies the current causalsubgraph discovery challenges. This work jointly optimize label and environment causal independence to achieve powerful causal subgraphs learning.  
+- GALA (Chen et al., 2024) designs an additional assistant model to enhance model with more powerful OOD generalization ability without explicit environment labels. Theoretical proofs establish that GALA possesses robust out-of-distribution generalization capabilities under the FIIF and PIIF assumptions.
+
+![](images/753c2ea81715c87a73cecaefb8c1fff7d3f7faa2038f9342d5c573ff4c55d559.jpg)  
+(a). Traditional SCM
+
+![](images/4d4289c338175f2b59f6a993475de2d2bea0d711cdae791a2eb8d8f42c4c893b.jpg)  
+(b). FIIF SCM
+
+![](images/d7f432846500ec479ad252316b834c24a2258f1d267f8859b7ee292ac1b1abbd.jpg)  
+(c).PIIF SCM  
+Figure 4. Illustrations of three structural causal models (SCMs).
+
+# D. Detailed Proofs
+
+# D.1. Proof of Proposition 4.3.
+
+The basis  $E_{b}$  represents a set of fundamental components or features that can accurately represent the entire environment space. These components capture the essential variations, patterns, and characteristics present in the environment. However, if the inference process fails to capture this basis, it implies that the process is unable to fully understand and model the complexities of the environment. Thus, we next investigate that whether the environment variable inferred from  $G_{S}$  covers such base environments. We consider two SCMs hypotheses FIIF and PIIF as shown in Fig. 4.
+
+Under the FIIF assumption,  $Y \bot G_S|G_C$ , we have  $P(Y,G_S|G_C) = P(Y|G_C) \cdot P(G_S|G_C)$ . This conditional independence assumption leads to an equivalent expression:  $P(Y|G) = P(Y|G_S,G_C) = P(Y|G_C)$ . Therefore, the process of extracting the causal subgraph  $G_C$  is equivalent to the process of modeling the spurious correlations  $G_S$ . Traditional positive casual learning methods are capable of handling the FIIF assumption.
+
+Under the PIIF assumption,  $Y \not\angle G_S|G_C$ , we have  $P(Y,G_S|G_C) \neq P(Y|G_C) \cdot P(G_S|G_C)$ . Furthermore, we can obtain  $P(Y|G) = P(Y|G_S,G_C) \neq P(Y|G_C)$ . Thus, the process of extracting the causal subgraph  $G_C$  cannot be used to infer the labels of samples. More formally, using mutual information theory, we derive the following,
+
+$$
+I (Y; G _ {S} | G _ {C}) = H (Y | G _ {C}) - H (Y | G _ {S}, G _ {C}) > 0, \tag {12}
+$$
+
+$$
+H \left(Y \mid G _ {C}\right) > H \left(Y \mid G _ {S}, G _ {C}\right). \tag {13}
+$$
+
+This indicates that, given the causal subgraph  $G_{C}$ , the uncertainty of  $Y$  is higher than when both the spurious subgraph  $G_{S}$  and the causal subgraph  $G_{C}$  are given. It suggests that the spurious subgraph  $G_{S}$  contains additional information of  $Y$ .
+
+Therefore, the causal subgraph  $\hat{G}_C$  learned by the model with the positive learning manner contains components of the spurious subgraph, i.e.,  $G_S \cap \hat{G}_C \neq \emptyset$ . At this point, if we can obtain the basis for the environment space, the model should be able to infer the spurious subgraph  $G_S$  and treat it as part of the environment  $E$ . The extracted causal subgraph  $\hat{G}_C$  should be able to effectively remove the spurious subgraph, i.e.,  $G_S \cap \hat{G}_C = \emptyset$ . This clearly contradicts the PIIF assumption, indicating that the model currently lacks the capability to obtain a basis for the environmental space. Therefore, simply inferring the causal subgraph with a positive manner is not sufficient to address the PIIF assumption. Since  $E \to G_S$ , modeling the spurious subgraph  $G_S$  requires modeling and understanding its root  $E$ . Existing methods that simply model  $G - G_C$  also lack the capability to address the PIIF assumption.
+
+# D.2. Proof of Proposition 4.4.
+
+The optimization of Eq. 3 enables a broader scale environment inference space by cooperatively modeling intra-class spurious subgraphs and extra-class samples. Given that  $\max - I(E; G_C|Y) = \max I(E; G_S|Y)$ , maximizing  $I(E; G_S|Y)$  implements the inference process for intra-class spurious subgraphs. Consider  $I(E; \mathcal{G}|\bar{Y}) = \sum_{y_i \in \bar{Y}} I(E; G^{(i)})$ , maximizing
+
+$I(E;\mathcal{G}|\overline{G^Y})$  implements the modeling of extra-class sample space. The optimization procedure of max  $I(E;\mathcal{G}|\overline{G^Y})$  indicates that all other extra-class samples  $\{G^{(i)}|y_i\in \bar{Y}\}$  are modeled as environment variables when making environment inference on samples with label  $Y$ . Therefore, the optimization process for Eq. 3 encompasses a broader cognitive space for environments, with its upper limit being the ground-truth environment distribution.
+
+# D.3. Proof of Theorem 5.1.
+
+Given that PIIF shifts in the absence of environment labels are more challenging (Chen et al., 2024), our work focuses on the ability of NeGo on the PIIF assumption, namely PIIF implies that the causal variable  $G_{C}$  indirectly influences the spurious variable  $G_{S}$  through the mediator  $Y$ . In the following analysis, we analyze the two specific scenarios under PIIF assumption, i.e.,  $H(G_{C}|Y) < H(G_{S}|Y)$  and  $H(G_{C}|Y) > H(G_{S}|Y)$ . NeGo aims to comprehensively capture the underlying environment space by inferring the extra-class sample space and the intra-class spurious subgraphs. The learning objective of extracting casual subgraph  $G_{C}$  can be rewritten as follows,
+
+$$
+\begin{array}{l} \arg \max _ {\hat {G} _ {C}} \left(I \left(\hat {G} _ {C} ^ {e _ {i}}, \hat {G} _ {C} ^ {e _ {j}} | C\right) - I \left(\hat {G} _ {C}, \bar {G} | Y\right)\right) \\ \forall e _ {i}, e _ {j} \in E \\ = \underset {\forall e _ {i}, e _ {j} \in E} {\arg \max } _ {\hat {G} _ {C}} (- I (\hat {G} _ {C}, \bar {G} | Y) + I (\hat {G} _ {C} ^ {e _ {i}}, \hat {G} _ {C} ^ {e _ {j}} | Y)), \tag {14} \\ \end{array}
+$$
+
+where  $\hat{G}_C^{e_i}$  denotes the extracted causal subgraph under any environmental scenario  $e_i$ . The first term represents the constraint of negative inference, meaning that NeGo models all extra-class samples as environmental space. The second term represents the constraint of positive causal inference, meaning that the causal subgraph extracted under any environmental condition remains consistent, and is most useful for label prediction. Next, we will demonstrate that NeGo can address the two scenarios of the PIIF assumption.
+
+For the case of  $I(G_{C};Y) > H(G_{C}) - H(G_{S})$ , we can get following derivation,
+
+$$
+H \left(G _ {S} | Y\right) > H \left(G _ {C} | Y\right), \tag {15}
+$$
+
+$$
+H \left(G _ {S}\right) - I \left(G _ {S}; Y\right) > H \left(G _ {C}\right) - I \left(G _ {C}; Y\right), \tag {16}
+$$
+
+$$
+H \left(G _ {S}\right) - H \left(G _ {C}\right) + I \left(G _ {C}; Y\right) > I \left(G _ {S}; Y\right) > 0, \tag {17}
+$$
+
+$$
+I \left(G _ {C}; Y\right) > H \left(G _ {C}\right) - H \left(G _ {S}\right). \tag {18}
+$$
+
+We can get that inferring  $G_{C}$  from  $Y$  is more effective and seamless compared to simply separating causal and spurious substructures based on entropy differences. Thus, our positive inference approach,  $\arg \max_{\forall e_i,e_j\in E}I(\hat{G}_C^{e_i},\hat{G}_C^{e_j}|Y)$ , is sufficient to achieve the decoupling of  $G_{C}$  from the label  $Y$ .
+
+For the case of  $I(G_{C};Y) < H(G_{C}) - H(G_{S})$ , we get  $I(G_{C};Y) < H(G_{C}) - H(G_{S})$ . This means that we need to consider entropy differences in the data composition to assess the differences between causal and spurious relationships. In other words, positive inference  $\operatorname*{arg\max}_{\forall e_i,e_j\in E}I(\hat{G}_C^{e_i},\hat{G}_C^{e_j}|Y)$  alone may result in  $\hat{G}_C$  containing spurious subgraph information,
+
+meaning  $G_{S} \in \hat{G}_{C}$ . Fortunately, our negative inference strategy can further refines  $\hat{G}_{C}$  by considering entropy differences  $H(G_{C}) - H(G_{S})$  to better distinguish between causal and spurious relationships. Specifically, our  $G_{C}$  is also subject to this constraint through a negative inference approach to learn  $\hat{G}_{S}$ ,
+
+$$
+G _ {C} \in G - \arg \max  \left(I (Y | \hat {G} _ {S}) - I (\hat {G} _ {S} | \bar {Y})\right). \tag {19}
+$$
+
+# E. More Experiment Results
+
+# E.1. Implementation Details
+
+We implement our Nego and parts of baselines with PyTorch 1.10.1 on a server with NVIDIA A100-PCIE-40GB. All experiments are repeated with 10 different random seeds of [1,2,3,4,5,6,7,8,9,10]. The reported results include the mean and standard deviation obtained from these 10 runs. During the training stage, we employ the Adam optimizer. We set the maximum number of training epochs to 200. The batch size of training is set as 32 except for GOOD-CMNIST, which uses a batch size of 64. For GOOD-Motif, GOOD-CMNIST and GOODSST2, the learning rate is set to  $5 \times 10^{-4}$ . For GOOD-HIV, GOOD-Twitter, and DrugOOD, we exploit a learning rate of  $10^{-4}$ . Additionally, we utilize a weight decay of  $10^{-4}$  to help with regularization and prevent overfitting. The experiment setup of all baselines is same as (Gui et al., 2024).
+
+# E.2. Performance on the Synthetic Datasets
+
+As shown in Tab. 8, our method achieves optimal performance on almost all datasets. Besides, the performance of environment-centered OOD methods, such as LECI and GALA, often achieves suboptimal or even optimal results on various datasets. This demonstrate the effectiveness of modeling environment factors in addressing data distribution shifts.
+
+# E.3. Hyperparameter Sensitivity Analysis
+
+To investigate the sensitivity of  $\lambda$  to performance in Eq. 11, we conducted experiments on both GOOD-HIV-scaffold and DrugOOD-assay. As illustrated by the results from Fig. 5(b), the performance of NeGo improves significantly when  $\lambda \approx 1$ . We also find that neither smaller nor larger values of  $\lambda$  lead to improvements in performance. Therefore, only by jointly utilizing both positive learning and negative inference can better performance be achieved.
+
+Table 8. The accuracy of NeGo on two synthetic datasets, where GOOD-Motif has a structure shift and GOOD-CMNIST has a feature shift. * indicates statistical significance against the second-best results.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">GOOD-Motif</td><td colspan="2">GOOD-CMNIST</td></tr><tr><td>basis</td><td>size</td><td>color</td><td>covariate</td></tr><tr><td>ERM</td><td>60.93±11.11</td><td>56.63±7.12</td><td>26.64±2.37</td><td>57.56±9.59</td></tr><tr><td>IRM</td><td>64.94±4.85</td><td>54.52±3.27</td><td>29.63±2.06</td><td>58.11±5.14</td></tr><tr><td>V-Rex</td><td>61.59±6.58</td><td>55.85±9.42</td><td>27.13±2.90</td><td>48.78±7.81</td></tr><tr><td>IB-IRM</td><td>63.45±5.42</td><td>52.76±4.67</td><td>28.95±1.98</td><td>50.56±6.62</td></tr><tr><td>DIR</td><td>34.39±2.02</td><td>43.11±2.78</td><td>22.53±2.56</td><td>44.67±0.00</td></tr><tr><td>GSAT</td><td>62.27±8.79</td><td>50.03±5.71</td><td>35.02±2.78</td><td>68.22±7.23</td></tr><tr><td>CAL</td><td>59.45±3.34</td><td>51.27±2.50</td><td>28.87±1.80</td><td>52.59±2.76</td></tr><tr><td>CIGA</td><td>37.81±2.42</td><td>51.87±5.15</td><td>25.06±3.07</td><td>56.78±2.99</td></tr><tr><td>GIL</td><td>68.48±2.46</td><td>63.61±2.75</td><td>47.32±2.27</td><td>57.61±2.98</td></tr><tr><td>LECI</td><td>84.56±2.22</td><td>71.43±1.96</td><td>51.80±2.71</td><td>83.20±5.89</td></tr><tr><td>GALA</td><td>80.95±1.31</td><td>70.45±1.30</td><td>52.68±2.40</td><td>81.23±3.29</td></tr><tr><td>NeGo</td><td>83.96±1.90</td><td>72.65±1.47*</td><td>53.28±1.79</td><td>82.43±1.73</td></tr></table>
+
+# E.4. Case Studies
+
+We also explore whether incorporating prompt learning can enhance the model's performance, rather than our overall negative prompt framework. To this end, we develop a variant of our NeGo framework, referred to as PoGo, which incorporates the positive prompt practice. We evaluate the effectiveness (ROC-AUC) of PoGo on four distribution shift datasets. We present the final performance by averaging the results from two runs conducted on an NVIDIA A100-PCIE-40GB with different random seeds. As shown in Fig. 5(a), the performance of PoGo is competitive with recent successful practices like LECI and GALA, demonstrating that the design of positive prompt can still obtain excellent generalization. However, our framework of negative prompt shows superior performance.
+
+We further investigate the reason of such performance of positive prompt practice PoGo. We modify PoGo by masking the  $\mathcal{L}_{posi}$  (the original Negative Loss  $\mathcal{L}_{naga}$ ), obtaining PoGo (w/o.  $\mathcal{L}_{posi}$ ). With all other configurations remaining the same, we observe a significant decrease in the performance of PoGo (w/o.  $\mathcal{L}_{posi}$ ). Our analysis is as follows: although both  $\mathcal{L}_{posi}$  and  $\mathcal{L}_{pred}$  are positive losses in PoGo, we argue they serve different purposes and convey distinct information.  $\mathcal{L}_{prompt}$ , as a guidance strategy for the positive prompt, guides the prompt module to learn more potential environment semantics, while  $\mathcal{L}_{pred}$  enhances prediction accuracy. Without prompt guidance  $\mathcal{L}_{prompt}$ , the advantage of prompt learning is not released. Therefore, we argue that positive prompt may also enhance the model to capture a broader scale of environments. A more in-depth investigation will be left for our future work.
+
+![](images/49a209e6567396af6c6c378c7a2503cb9259688691722c36758d0eec7ed79178.jpg)  
+Figure 5. Left: the performance comparison of PoGo, which is a variant of our NeGo framework by incorporating the positive prompt practice. Right: hyperparameter sensitivity analysis.
+
+![](images/7ce498a8f0672023e41610095c34cda161d79c24814f1d8bebebcaeb4de0cbd2.jpg)
+
+# F. Future Works
+
+Our design can effectively solve the existing challenges, but there still exist a limitation. The negative prompter in our approach learns class-specific environment embeddings by considering all extra-class samples as environment variables. This results in our method relying on the class information of the dataset. With a larger number of classes, the model is better equipped to capture and recognize complex underlying environment factors. When the dataset is limited to a binary classification task, environment factors always present within the in-class samples. In this case, our negative prompter may have reduced capability to expand the environment inference space. The reason for this limitation is that the model is sensitive to the characteristics of dataset. Actually, we can realize that environment variables are often shareable across datasets. Therefore, it is a promising research direction to study cross-task graph OOD work to capture broader environmental information. In the future, we aim to investigate transferable multi-task graph OOD generalization learning, which is not discussed in existing works.

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+# Enhancing Ligand Validity and Affinity in Structure-Based Drug Design with Multi-Reward Optimization
+
+Seungbeom Lee  $^{*1}$  Munsun Jo  $^{*2}$  Jungseul Ok  $^{13}$  Dongwoo Kim  $^{13}$
+
+# Abstract
+
+Deep learning-based structure-based drug design aims to generate ligand molecules with desirable properties for protein targets. While existing models have demonstrated competitive performance in generating ligand molecules, they primarily focus on learning the chemical distribution of training datasets, often lacking effective steerability to ensure the desired chemical quality of generated molecules. To address this issue, we propose a multi-reward optimization framework that finetunes generative models for attributes, such as binding affinity, validity, and drug-likeness, together. Specifically, we derive direct preference optimization for a Bayesian flow network, used as a backbone for molecule generation, and integrate a reward normalization scheme to adopt multiple objectives. Experimental results show that our method generates more realistic ligands than baseline models while achieving higher binding affinity, expanding the Pareto front empirically observed in previous studies.
+
+# 1. Introduction
+
+Designing ligand molecules that simultaneously optimize multiple objectives – such as binding affinity, synthetic accessibility, and strain energy – is a central challenge in structure-based drug design (SBDD). Conventional deep-learning approaches typically address only one or two objectives at a time, most commonly focusing on high binding affinity and basic chemical feasibility, e.g., through simple validity checks via the valency rule. While these methods have advanced the generative capabilities of molecular mod
+
+*Equal contribution Graduate School of Artificial Intelligence, POSTECH, South Korea KT Corporation, South Korea Department of Computer Science & Engineering, POSTECH, South Korea. Correspondence to: Dongwoo Kim <dongwoo.kim@postech.ac.kr>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+![](images/581d00bfb1c22687e42c38cee12e19254de8d6320c36eae6291251f5ead11b88.jpg)  
+(a) Vina Dock vs. SA  
+Figure 1. Comparison of the median Vina docking score (lower is better) against Synthetic accessibility (SA, higher is better) and Strain Energy (SE, lower is better). The dashed blue line represents a Pareto front obtained from previous works.
+
+![](images/cbb2cb95fc840954b412fbd9d65c1ef51208cae986cc621e282f780d28da64ba.jpg)  
+(b) Vina Dock vs. SE
+
+els (Huang et al., 2024; Qu et al., 2024; Guan et al., 2023), they often neglect other crucial factors like ease of conformation stability such as strain energy or synthesis. In practice, drug discovery demands a careful balance of many properties, making single-objective or narrowly focused solutions insufficient.
+
+In current deep generative modeling for SBDD, standard strategies generally concentrate on minimizing the discrepancy between a generated ligand and a training set of protein-ligand complexes. Although this can lead to strong binding affinities, improving multiple objectives beyond binding, e.g., synthetic accessibility, usually requires careful tuning of loss terms or elaborate constraints. As a result, purely likelihood-based approaches can struggle to expand the Pareto front, which captures trade-offs among distinct objectives.
+
+Figure 1 illustrates the trade-offs among Vina docking score measuring binding affinity to target protein, synthetic accessibility (SA), and strain energy (SE) using results from nine different generative models, along with a reference dataset included in the test set. Although recently proposed methods such as MolCRAFT (Qu et al., 2024), IPDiff (Huang et al., 2024), and AliDiff (Gu et al., 2024), outperform the reference data in terms of Vina docking score, they underperform on SA and SE. Ideally, a design process would push
+
+the Pareto frontier toward both lower docking scores and higher SA (or lower SE). In contrast, these findings show that existing models remain overly focused on improving affinity, thus calling into question the practical utility of their generated molecules.
+
+Our Approach Building on recent developments in preference-based reinforcement learning, we introduce a multi-reward Direct Preference Optimization (DPO) framework adapted for Bayesian Flow Networks (BFNs), which recently show promising performance on ligand generation tasks (Qu et al., 2024). DPO refines a pretrained generative model using feedback from external software or metrics, enabling fine-grained control over multiple properties (Rafailov et al., 2024; Kim et al., 2024). In our setting, we treat each property, such as binding affinity, synthetic accessibility, and strain energy, as a separate reward. We then apply softmax-based normalization to ensure commensurability of metrics with different ranges and adopt an uncertainty-regularized ensemble strategy to penalize high variance in multi-reward predictions. This multi-reward scheme expands the trade-off frontier between objectives, thereby improving molecular candidates across multiple dimensions.
+
+# 2. Related Work
+
+Preference Alignment and Direct Preference Optimization (DPO) Preference alignment has emerged as a powerful paradigm for adjusting pretrained models to better reflect desired behaviors. Originally adopted in large language models (LLMs) to align with human feedback (Ouyang et al., 2022), DPO (Rafailov et al., 2024) provides a more direct means of incorporating user or expert preferences compared to traditional reinforcement learning. Subsequently, DPO-based methods have been extended to text-to-image generative models (Wallace et al., 2023), showing that preference alignment is not limited to language domains. In multi-objective settings, Kim et al. (2024) propose a confidence-based or variance-penalized approach to combine heterogeneous reward signals, further enhancing stability and interpretability.
+
+Structure-Based Drug Design Structure-based drug design (SBDD) focuses on generating or modifying small-molecule ligands to exhibit strong binding affinity for a target protein while maintaining favorable pharmacological properties. Classical computational approaches rely on de novo design or docking-based virtual screening (Schneider & Fechner, 2005) to identify candidate molecules, followed by iterative refinement and validation through experimental assays.
+
+Recent deep learning methods cast SBDD as a conditional
+
+generative modeling task, leveraging protein-ligand complex data to learn a distribution over valid 3D conformations. Autoregressive approaches place atoms or fragments within protein pockets in a sequential manner (Luo et al., 2021; Peng et al., 2022), while diffusion models generate candidate molecules by denoising from random noise, learning continuous atom positions and discrete atom types jointly (Guan et al., 2023; Huang et al., 2024; Guan et al., 2024; Zhou et al., 2024). Most recently, Qu et al. (2024) introduced MolCRAFT, which uses Bayesian Flow Networks (BFNs) (Graves et al., 2023) to sample in parameter space rather than directly in data space, leading to improved binding affinity and more stable 3D molecular structures. To further improve the quality of binding affinity, Gu et al. (2024) introduced AliDiff, which fine-tunes a diffusion-based generative model via DPO.
+
+On the other hand, Harris et al. (2023) raises questions about the evaluation practices with the deep generative models for SBDD. They find that many generated molecules from recent generative models violate physical constraints and call for an expanded evaluation of SBDD. A similar observation is made for deep learning based binding affinity prediction in Buttenschoen et al. (2024). Our approach builds upon this foundation by adopting a multi-reward perspective that jointly optimizes for binding affinity and other molecular properties (e.g., synthetic accessibility, strain energy), thereby expanding the Pareto frontier in multi-objective SBDD.
+
+# 3. Preliminaries
+
+We first introduce the problem formulation of structure-based drug design with notations used throughout this manuscript. We then provide an overview of Bayesian Flow Networks (BFNs)(Graves et al., 2023) and MolCRAFT(Qu et al., 2024), where the latter leverages BFNs to generate target molecules.
+
+# 3.1. Problem Formulation
+
+In structure-based drug design (SBDD), the goal is to generate ligand molecules conditioned on a given protein target. We denote a protein target by  $\mathcal{P} = \{(\mathbf{x}_{\mathcal{P}}^{(i)},\mathbf{v}_{\mathcal{P}}^{(i)})\}_{i = 1}^{N_{\mathcal{P}}}$  where  $\mathbf{x}_{\mathcal{P}}^{(i)}\in \mathbb{R}^3$  is the three-dimensional coordinate of the  $i$ -th atom, and  $\mathbf{v}_{\mathcal{P}}^{(i)}\in \mathbb{R}^{D_{\mathcal{P}}}$  is a one-hot vector encoding the atom type. A molecule is analogously represented as  $\mathcal{M} = \{(\mathbf{x}_{\mathcal{M}}^{(i)},\mathbf{v}_{\mathcal{M}}^{(i)})\}_{i = 1}^{N_{\mathcal{M}}}$ , where  $\mathbf{x}_{\mathcal{M}}^{(i)}\in \mathbb{R}^3$  and  $\mathbf{v}_{\mathcal{M}}^{(i)}\in \mathbb{R}^{D_{\mathcal{M}}}$  denote the three-dimensional coordinate and the atom-type vector, respectively. For convenience, we denote each molecule by  $\mathbf{m} = [\mathbf{x},\mathbf{v}]$  where  $[\cdot ,\cdot ]$  is the concatenation of  $\mathbf{x}\in \mathbb{R}^{N_{\mathcal{M}}\times 3}$  and  $\mathbf{v}\in \mathbb{R}^{N_{\mathcal{M}}\times D_{\mathcal{M}}}$  given target protein as  $\mathbf{p}$ .
+
+SBDD aims to discover a molecule  $\mathbf{m}$  given protein and its binding site  $\mathbf{p}$ . During training, a set of protein-reference molecule pairs is provided. As a primary performance metric, external software such as AutoDock Vina (Eberhardt et al., 2021) is employed to estimate the binding affinity between the protein and generated molecules. Along with affinity scores, additional molecular properties (e.g., validity, synthesizeability, and drug-likeness) can be evaluated to assess the quality of the generated compounds further.
+
+# 3.2. Bayesian Flow Networks
+
+Bayesian flow network (BFN) (Graves et al., 2023) is a class of generative models that integrate core ideas from both diffusion models (Ho et al., 2020) and Bayesian inference. Unlike traditional diffusion models that operate directly in the data space, BFNs perform their forward and reverse "diffusion" in the parameter space of the data distribution. This process ensures the generative process is continuous and differentiable regardless of whether data is discrete or continuous, making BFNs particularly suitable for molecule generation, where atom coordinates are continuous, and atom types are discrete. MolCRAFT (Qu et al., 2024) adopts the BFN as a backbone for conditional molecule generative model given proteins. Below, we outline the details of BFNs in the context of MolCRAFT.
+
+At a high level, BFNs can be conceptualized as an iterative message exchange between two components: a sender and a receiver. In each round, the sender transmits a noisy sample to the receiver, akin to the forward diffusion process in a standard diffusion model. The receiver then attempts to infer the sender's parameters from the noisy input, effectively performing a reverse "denoising" step analogous to the reverse diffusion phase. This iterative cycle continues until reaching the final estimate of the data distribution in parameter space.
+
+Sender distribution Formally, given a molecule  $\mathbf{m} = [\mathbf{x},\mathbf{v}]$ , the sender injects continuous noise for both coordinates and types independently according to a predefined noise schedule. We denote the resulting noise-injected molecule by  $\tilde{\mathbf{m}} = [\tilde{\mathbf{x}},\tilde{\mathbf{v}}]$ . The sender distribution is then factorized as:
+
+$$
+p _ {S} (\tilde {\mathbf {m}} | \mathbf {m}, t) = \prod_ {i} ^ {N _ {\mathcal {M}}} p (\tilde {\mathbf {x}} ^ {(i)} | \mathbf {x} ^ {(i)}, \alpha_ {t} ^ {\mathbf {x}}) \prod_ {i} ^ {N _ {\mathcal {M}}} p (\tilde {\mathbf {v}} ^ {(i)} | \mathbf {v} ^ {(i)}, \alpha_ {t} ^ {\mathbf {v}}), \tag {1}
+$$
+
+where  $\alpha_{t}$  is an accuracy parameter such that the sender samples are entirely uninformative about  $\mathbf{x}$  when  $\alpha = 0$  and become more informative as  $\alpha$  increases. One can use different schedules for the coordinate  $\alpha_{t}^{\mathbf{x}}$  and atom type  $\alpha_{t}^{\mathbf{y}}$ .
+
+Receiver distribution At time  $t$ , the receiver uses parameters  $\theta_{t-1}$  from the previous step and the noisy sample  $\tilde{\mathbf{m}}$  from the sender to reconstruct a denoised  $\mathbf{m}$ . Let  $\Phi(\theta_{t-1}, \mathbf{p}, t)$  be an SE(3)-equivariant neural network that outputs  $\hat{\mathbf{m}}$  from parameters  $\theta_{t-1}$ , the protein pocket  $\mathbf{p}$ , and time  $t$ . Given that the distribution and noise schedule of the sender distribution is known, the receiver distribution is defined as
+
+$$
+p _ {R} \left(\tilde {\mathbf {m}} \mid \boldsymbol {\theta} _ {t - 1}, \mathbf {p}, t\right) = p _ {S} \left(\tilde {\mathbf {m}} \mid \boldsymbol {\Phi} \left(\boldsymbol {\theta} _ {t - 1}, \mathbf {p}, t\right), \alpha_ {t}\right). \tag {2}
+$$
+
+Hence the receiver distribution becomes close to the sender distribution as  $\Phi$  accurately reconstructs the original molecule  $\mathbf{m}$ .
+
+Bayesian update To ensure that distribution parameters  $\theta_{t}$  can be updated in a tractable way, BFNs employ Bayesian inference. Given the previous parameter  $\theta_{t - 1}$  and the noisy sample  $\tilde{\mathbf{m}}_t$ , the updated parameter  $\theta_{t}$  can be obtained via a Bayesian update function  $h(\theta_{t - 1},\tilde{\mathbf{m}}_t,\alpha_t) = \theta_t$ . The update function infers the posterior after observing noisy sample  $\tilde{\mathbf{m}}_t$  from the sender distribution  $p_S(\tilde{\mathbf{m}}_t|\mathbf{m},\alpha_t)$  and an input distribution  $p_I(\mathbf{m}|\boldsymbol {\theta}_{t - 1})$ , often selected as a conjugate prior of the sender distribution.
+
+This induces the Bayesian update distribution  $p_U$ , which is a push-forward distribution of  $p_S$  under transformation  $h$ , i.e.,  $p_U(\pmb{\theta}') \coloneqq p_S(\{\tilde{\mathbf{m}} \mid h(\tilde{\mathbf{m}}) = \pmb{\theta}'\})$ , where we omit the conditions for brevity. By marginalizing out  $\pmb{\theta}_{1:t-1}$ , one obtains the Bayesian flow distribution:
+
+$$
+p _ {F} \left(\boldsymbol {\theta} _ {t} \mid \mathbf {m}, \mathbf {p}, t\right) = p _ {U} \left(\boldsymbol {\theta} _ {t} \mid \boldsymbol {\theta} _ {0}, \mathbf {m}, \mathbf {p}, \alpha_ {0: t}\right), \tag {3}
+$$
+
+where  $\theta_0$  is the initial parameter used to define a simple distribution such as the standard normal for continuous or uniform for discrete. A key advantage of BFNs is that, with suitable noise injections and the input distribution,  $\theta_t$  can be calculated recursively without explicit simulation.
+
+Training objective Training proceeds by minimizing the Kullback-Leibler divergence between the sender distribution  $p_S$  and the receiver distribution  $p_R$ :
+
+$$
+L (\mathbf {m}, \mathbf {p}) = \mathbb {E} _ {t \sim U (1, T), \tilde {\mathbf {m}} _ {t} \sim p _ {S}, \boldsymbol {\theta} _ {t - 1} \sim p _ {F}} \left[ D _ {\mathrm {K L}} \left(p _ {S} \| p _ {R}\right) \right]. \tag {4}
+$$
+
+For the continuous-time extension of BFNs and additional technical details, we refer readers to Graves et al. (2023).
+
+# 3.3. Direct Preference Optimization
+
+Reinforcement learning with human feedback (RLHF) has been proposed to align the large language models with human preference (Ouyang et al., 2022). Let  $\mathcal{D} = \{(c, x^w, x^l)\}$  be a preference dataset, where  $x^w$  and  $x^l$  are winning and losing responses given prompt  $c$ . In standard
+
+RLHF pipelines, one first trains a reward model to capture these preferences and then uses policy optimization methods (e.g., PPO) to fine-tune the model such that it produces higher-scoring outputs.
+
+Direct Preference Optimization (DPO) offers an alternative that bypasses the need for an explicit reward-model approximation in the training loop (Rafailov et al., 2024). Let  $p_{\theta}(\mathbf{x})$  be a model to be fine-tuned. Given a reference model  $p_{\mathrm{ref}}$ , DPO is derived from the optimal solution to the RLHF objective and can be expressed as
+
+$$
+\begin{array}{l} L _ {\mathrm {D P O}} (\theta) = \\ - \mathbb {E} _ {\mathbf {c}, \boldsymbol {x} ^ {w}, \boldsymbol {x} ^ {l}} \left[ \log \sigma \left(\beta \log \frac {p _ {\theta} \left(\boldsymbol {x} ^ {w}\right)}{p _ {\text {r e f}} \left(\boldsymbol {x} ^ {w}\right)} - \beta \log \frac {p _ {\theta} \left(\boldsymbol {x} ^ {l}\right)}{p _ {\text {r e f}} \left(\boldsymbol {x} ^ {l}\right)}\right) \right], \tag {5} \\ \end{array}
+$$
+
+where we omit the condition  $\mathbf{c}$  for brevity. Minimizing this loss encourages the fine-tuned model  $p_{\theta}$  to assign higher relative likelihood to the winning samples than to the losing samples, while remaining close to  $p_{\mathrm{ref}}$ . Recently, Diffusion-DPO extends DPO to diffusion-based generative models by expanding the reward model over the entire diffusion trajectory, enabling preference-driven fine-tuning in multi-step generative processes (Wallace et al., 2023).
+
+# 4. Multi-Reward Optimization for BFNs
+
+In this section, we present our Direct Preference Optimization for Bayesian Flow Networks (BFN-DPO), an end-to-end framework for controllable molecule generation. Our method draws on recent advances in DPO and adapts these ideas to BFNs. We then extend DPO-BFN to handle multiple reward signals via a softmax-based normalization strategy and an uncertainty-regularized ensemble, allowing fair comparisons among rewards with different numerical scales.
+
+# 4.1. Direct Preference Optimization on BFNs
+
+We consider a preference dataset  $\mathcal{D} = \{(\mathbf{p},\mathbf{m}^w,\mathbf{m}^l)\}$ , where each triplet consists of a protein target  $\mathbf{p}$ , a "winning" molecule  $\mathbf{m}^w$ , and a "losing" molecule  $\mathbf{m}^l$ . These preferences are determined by pre-computed reward values (e.g., binding affinity). We denote by  $p_{\mathrm{ref}}(\cdot)$  and  $p_{\phi}(\cdot)$  the reference and fine-tuned BFN models, respectively. Drawing on Diffusion-DPO (Wallace et al., 2023), we define the BFN-DPO objective for a single preference pair  $(\mathbf{m}^w,\mathbf{m}^l)$  as
+
+$$
+\begin{array}{l} \mathcal {L} _ {\mathrm {B F N - D P O}} (\phi) = - \mathbb {E} \left[ \log \sigma \left(\beta \log \frac {p _ {\phi} \left(\hat {\mathbf {m}} _ {1 : T} ^ {w} \mid \boldsymbol {\theta} _ {0 : T - 1} ^ {w}\right)}{p _ {\text {r e f}} \left(\hat {\mathbf {m}} _ {1 : T} ^ {w} \mid \boldsymbol {\theta} _ {0 : T - 1} ^ {w}\right)} \right. \right. \\ \left. - \beta \log \frac {p _ {\phi} \left(\hat {\mathbf {m}} _ {1 : T} ^ {l} \mid \boldsymbol {\theta} _ {0 : T - 1} ^ {l}\right)}{p _ {\text {r e f}} \left(\hat {\mathbf {m}} _ {1 : T} ^ {l} \mid \boldsymbol {\theta} _ {0 : T - 1} ^ {l}\right)} \right], \tag {6} \\ \end{array}
+$$
+
+where  $\hat{\mathbf{m}}_t$  is a noisy sample drawn at time step  $t$  (the "receiver" output), governed by the BFN's sender/receiver distributions, and  $\theta_t$  are parameters sampled from the Bayesian flow distribution  $p_F$ , which performs Bayesian updates in parameter space.
+
+Unlike Diffusion-DPO, the reconstruction at time  $t$  depends on the parameter  $\theta_{t-1}$ . Expanding the ratio into a sum over time steps with Jensen's inequality, one obtains an upper bound involving differences of KL terms at each time step. By approximating the receiver distribution in expectation with the sender distribution, we can obtain the following loss:
+
+$$
+\mathcal {L} _ {\mathrm {B F N - D P O}} (\phi) = - \mathbb {E} \left[ \mathcal {L} _ {t} ^ {\mathbf {x}} + \mathcal {L} _ {t} ^ {\mathbf {v}} \right], \tag {7}
+$$
+
+which is separated into two parts: atom loss  $\mathcal{L}_t^{\mathbf{x}}$  and type loss  $\mathcal{L}_t^{\mathbf{v}}$ . A complete derivation is provided in Appendix A.
+
+Let  $\mathbf{x}^w$  and  $\mathbf{x}^l$  be the clean coordinates for the winning and losing molecules at time  $t = 0$ . Then the atom loss has the form of
+
+$$
+\mathcal {L} _ {t} ^ {\mathbf {x}} = - \mathbb {E} \left[ \log \sigma \left(- \beta T \frac {\alpha_ {t} ^ {\mathbf {x}}}{2} \left[ \Delta_ {\phi} ^ {w} - \Delta_ {\phi} ^ {l} \right]\right) \right], \tag {8}
+$$
+
+with
+
+$$
+\Delta_ {\phi} ^ {w} = \left\| \mathbf {x} ^ {w} - \Phi_ {\phi} ^ {\mathbf {x}} (\pmb {\theta} _ {t - 1} ^ {\mathbf {x}, w}) \right\| ^ {2} - \left\| \mathbf {x} ^ {w} - \Phi_ {\mathrm {r e f}} ^ {\mathbf {x}} (\pmb {\theta} _ {t - 1} ^ {\mathbf {x}, w}) \right\| ^ {2},
+$$
+
+and analogously for  $\Delta_{\phi}^{l}$ . Here,  $\Phi_{\phi}^{\mathbf{x}}(\cdot)$  and  $\Phi_{\mathrm{ref}}^{\mathbf{x}}(\cdot)$  generate coordinate predictions under the fine-tuned and reference BFNs, and  $\alpha_{t}^{\mathbf{x}}$  is a concentration parameter for the coordinates at step  $t$ .
+
+For the discrete atom types, we have a similar construction based on KL divergences. Each "winning" and "losing" noisy type  $\hat{\mathbf{v}}_t^w$  and  $\hat{\mathbf{v}}_t^l$  is compared to the true one-hot type vector  $\mathbf{v}^w$  or  $\mathbf{v}^l$ . The derivation yields:
+
+$$
+\mathcal {L} _ {t} ^ {\mathbf {v}} = - \mathbb {E} \left[ \log \sigma \left(- \beta T \left[ \left(\Delta_ {\phi} ^ {w}\right) _ {\mathbf {v}} - \left(\Delta_ {\phi} ^ {l}\right) _ {\mathbf {v}} \right]\right) \right], \tag {9}
+$$
+
+where
+
+$$
+\begin{array}{l} \left(\Delta_ {\phi} ^ {w}\right) _ {\mathbf {v}} = \\ \ln p _ {S} \left(\hat {\mathbf {v}} _ {t} ^ {w} | \mathbf {v} ^ {w}, \alpha_ {t} ^ {\mathbf {v}}\right) - \ln p _ {R} \left(\hat {\mathbf {v}} _ {t} ^ {w} \mid \Phi_ {\phi} ^ {\mathbf {v}} \left(\boldsymbol {\theta} _ {t - 1} ^ {\mathbf {v}, w}\right), \alpha_ {t} ^ {\mathbf {v}}\right) \\ - \left[ \ln p _ {S} \left(\hat {\mathbf {v}} _ {t} ^ {w} | \mathbf {v} ^ {w}, \alpha_ {t} ^ {\mathbf {v}}\right) - \ln p _ {R} \left(\hat {\mathbf {v}} _ {t} ^ {w} \mid \Phi_ {\text {r e f}} ^ {\mathbf {v}} \left(\boldsymbol {\theta} _ {t - 1} ^ {\mathbf {v}, w}\right), \alpha_ {t} ^ {\mathbf {v}}\right) \right], \\ \end{array}
+$$
+
+and similarly for  $(\Delta_{\phi}^{l})_{\mathbf{v}}$ . In practice, these log-likelihood terms could be implemented as cross-entropy or KL divergences for categorical predictions.
+
+By minimizing  $\mathcal{L}_{\mathrm{BFN - DPO}}(\phi)$ , the model  $p_{\phi}$  is driven to generate noisy samples  $\hat{\mathbf{m}}_t^w$  that better align with automated preferences than  $\hat{\mathbf{m}}_t^l$ , all while remaining close to  $p_{\mathrm{ref}}$ .
+
+# 4.2. Multi-Reward Normalization
+
+While one can fine-tune a Bayesian Flow Network (BFN) solely using a single reward signal (e.g., validity), doing so may degrade other desirable properties like binding affinity. To address this limitation, we adopt a multi-reward BFN-DPO scheme inspired by Kim et al. (2024), where multiple reward signals (e.g., binding affinity, validity) are simultaneously optimized.
+
+Suppose we have  $K$  reward functions that evaluate a molecule-protein pair  $(\mathbf{p},\mathbf{m})$ . Since these rewards may have different numerical ranges and may be preferred in different directions (e.g., negative for binding affinity, positive for strain energy), direct averaging can be misleading. To handle scale discrepancies, we normalize each reward via a softmax over a mini-batch of size  $B$ :
+
+$$
+\hat {r} _ {i} ^ {(j)} = \frac {\exp \left(f _ {j} \left(r _ {i} ^ {(j)}\right)\right)}{\sum_ {b = 1} ^ {B} \exp \left(f _ {j} \left(r _ {b} ^ {(j)}\right)\right)}, \tag {10}
+$$
+
+where  $r_i^{(j)}$  is the  $j$ -th reward for the  $i$ -th molecule, and  $f_j(\cdot)$  is a simple transformation based on the  $j$ -th reward, so that higher values consistently indicate better rewards across different metrics. This maps each reward to the (0,1) interval, ensuring that distinct reward signals become comparable despite inherently different numerical ranges.
+
+After normalization, one still must combine the m rewards into a single score. Following Kim et al. (2024), we reduce the influence of high-variance reward profiles by subtracting a variance-based penalty. Specifically, let  $\mu_{\hat{r}_i}$  be the mean of the normalized rewards for molecule  $i$ . The final multi-reward score for that molecule is
+
+$$
+\bar {r} _ {i} = \mu_ {\hat {r} _ {i}} - \gamma \frac {1}{K} \sum_ {j = 1} ^ {K} \left(\hat {r} _ {i} ^ {(j)} - \mu_ {\hat {r} _ {i}}\right) ^ {2} \tag {11}
+$$
+
+where  $\gamma$  is a hyperparameter controlling how strongly to penalize high reward variance. This strategy promotes molecules that achieve balanced performance across all reward dimensions, rather than excelling at some while failing at others. More details on the hyperparameters can be found in Appendix B.
+
+We further stabilize training by adopting the  $\mathrm{E}^2\mathrm{PO}$ , a variant of DPO (Gu et al., 2024), to avoid excessive focus on winning samples when their advantage is already large. Concretely, we replace the standard BFN-DPO loss  $\mathcal{L}_{\mathrm{BFN - DPO}}$  with
+
+$$
+\begin{array}{l} \mathcal {L} _ {\mathrm {B F N - E} ^ {2} \mathrm {P O}} = - \mathbb {E} \left[ \sigma (\bar {r} _ {w} - \bar {r} _ {l}) \mathcal {L} _ {\mathrm {B F N - D P O}} \right. \\ + \left. \left(1 - \sigma \left(\bar {r} _ {w} - \bar {r} _ {l}\right)\right) (2 - \mathcal {L} _ {\mathrm {B F N - D P O}}) \right], \tag {12} \\ \end{array}
+$$
+
+where  $\bar{r}_w$  and  $\bar{r}_l$  denote the final multi-reward scores for the winning and losing molecules, respectively. If the difference
+
+$\bar{r}_w - \bar{r}_l$  becomes too large (close to 1), the second term is nullified; thus, over-optimizing already "high-scoring" winners is discouraged. In effect,  $\mathrm{E}^2\mathrm{PO}$  enables more balanced training and helps prevent mode collapse onto a narrow set of winning samples.
+
+# 5. Experiment
+
+In this section, we provide the result of the multi-object optimization approach with empirical experiments. Due to the space limit, missing implementation details are provided in Appendix B.
+
+# 5.1. Experimental Setting
+
+Datasets We fine-tune a pretrained model on the Cross-Docked dataset (Francoeur et al., 2020). The CrossDocked dataset comprises 100K training pairs and 100 target proteins for testing. For each protein, win-lose pairs are selected based on multi-reward scores described in Section 4.2 from molecules generated by the pretrained model. These pairs serve as input for fine-tuning via our proposed BFN-DPO procedure. We generate 100 new molecules per target protein in the test set for evaluation.
+
+Baselines We compare our method to the following baselines: AR (Luo et al., 2021) and Poket2Mol (Peng et al., 2022) generate molecules autoregressive, and FLAG (Zhang & Liu, 2023) generates at the fragment level. TargetDiff (Guan et al., 2023), DecompDiff (Guan et al., 2024), DecompOpt (Zhou et al., 2024), and IPDiff (Huang et al., 2024) are diffusion-based generative models. IPDiff incorporates protein-ligand interactions as a priori in the forward process. MolCRAFT (Qu et al., 2024) employs BFNs, where noise and denoise processes are performed in a fully continuous parameter space. AliDiff (Gu et al., 2024) applies a single reward optimization to improve the binding affinity based on IPDiff.
+
+Evaluation metrics We evaluate the generated molecules from three different perspectives: binding affinity, conformation validity, and molecular properties. For binding affinity, we use Vina Score, Vina Min, and Vina Dock measured by AutoDock Vina (Eberhardt et al., 2021) following the setup used in Ragoza et al. (2022). The Vina Score measures the binding affinity of the generated molecule as is. Vina Min measures the affinity after optimizing the molecular structure without changing the docking position. Vina Dock measures the binding affinity after re-docking the molecule; hence, reconsider the molecule's position and orientation. Low scores indicate stronger protein-ligand binding affinity. We report average and median scores for all Vina metrics. For conformation validity, we report strain energy (SE) measuring molecule conformation energy and protein-ligand
+
+Table 1. Performance comparison across different reward methods. We mark the best, and the second-best performances in bold and underline, respectively.  $(\downarrow) / (\uparrow)$  indicate whether a smaller/larger number is better, respectively.  
+
+<table><tr><td rowspan="3">Method</td><td colspan="6">Binding Affinity</td><td colspan="4">Conformation Stability</td><td colspan="2">Molecular Properties</td><td rowspan="3">Avg. size</td></tr><tr><td colspan="2">Vina Score (↓)</td><td colspan="2">Vina Min (↓)</td><td colspan="2">Vina Dock (↓)</td><td colspan="3">SE (↓)</td><td>Clash (↓)</td><td>SA (↑)</td><td>QED (↑)</td></tr><tr><td>Avg.</td><td>Med.</td><td>Avg.</td><td>Med.</td><td>Avg.</td><td>Med.</td><td>25%</td><td>50%</td><td>75%</td><td>Avg.</td><td>Avg.</td><td>Avg.</td></tr><tr><td>Reference</td><td>-6.36</td><td>-6.46</td><td>-6.71</td><td>-6.49</td><td>-7.45</td><td>-7.26</td><td>1.98</td><td>3.98</td><td>6.23</td><td>5.51</td><td>0.73</td><td>0.48</td><td>22.8</td></tr><tr><td>AR</td><td>-5.75</td><td>-5.64</td><td>-6.18</td><td>-5.88</td><td>-6.75</td><td>-6.62</td><td>2.79</td><td>7.39</td><td>16.34</td><td>4.46</td><td>0.64</td><td>0.51</td><td>17.7</td></tr><tr><td>Pocket2Mol</td><td>-5.14</td><td>-4.70</td><td>-6.42</td><td>-5.82</td><td>-7.15</td><td>-6.79</td><td>0.22</td><td>1.50</td><td>4.82</td><td>6.24</td><td>0.74</td><td>0.57</td><td>17.7</td></tr><tr><td>FLAG</td><td>16.48</td><td>4.53</td><td>1.21</td><td>-4.04</td><td>-5.63</td><td>-6.61</td><td>1.32</td><td>4.96</td><td>16.84</td><td>40.83</td><td>0.70</td><td>0.49</td><td>21.5</td></tr><tr><td>TargetDiff</td><td>-5.47</td><td>-6.30</td><td>-6.64</td><td>-6.83</td><td>-7.80</td><td>-7.91</td><td>7.48</td><td>16.85</td><td>33.70</td><td>10.84</td><td>0.58</td><td>0.48</td><td>24.2</td></tr><tr><td>DecompDiff</td><td>-5.19</td><td>-5.27</td><td>-6.03</td><td>-6.00</td><td>-7.03</td><td>-7.16</td><td>4.17</td><td>10.81</td><td>22.77</td><td>8.16</td><td>0.66</td><td>0.51</td><td>21.2</td></tr><tr><td>DecompOpt</td><td>-5.67</td><td>-6.04</td><td>-7.04</td><td>-7.09</td><td>-8.39</td><td>-8.43</td><td>12.34</td><td>21.37</td><td>37.69</td><td>14.56</td><td>0.61</td><td>0.45</td><td>29.4</td></tr><tr><td>IPDiff</td><td>-6.70</td><td>-7.53</td><td>-7.63</td><td>-7.71</td><td>-8.41</td><td>-8.39</td><td>10.30</td><td>27.05</td><td>65.49</td><td>8.79</td><td>0.61</td><td>0.52</td><td>24.3</td></tr><tr><td>AliDiff</td><td>-6.97</td><td>-7.75</td><td>-7.87</td><td>-7.94</td><td>-8.65</td><td>-8.64</td><td>10.09</td><td>27.60</td><td>67.00</td><td>8.76</td><td>0.56</td><td>0.50</td><td>24.4</td></tr><tr><td>MolCRAFT</td><td>-6.59</td><td>-7.04</td><td>-7.27</td><td>-7.26</td><td>-7.92</td><td>-8.01</td><td>3.10</td><td>7.92</td><td>14.87</td><td>7.09</td><td>0.69</td><td>0.50</td><td>22.7</td></tr><tr><td>Ours</td><td>-7.18</td><td>-7.38</td><td>-7.89</td><td>-7.77</td><td>-8.62</td><td>-8.64</td><td>2.05</td><td>5.56</td><td>11.62</td><td>6.69</td><td>0.74</td><td>0.55</td><td>22.8</td></tr></table>
+
+clash measuring the number of potential overlaps between ligand atoms and protein (Harris et al., 2023). For molecular properties, we use synthetic accessibility (SA) (Ertl & Schuffenhauer, 2009) and quantitative estimation of drug-likeness (QED) (Bickerton et al., 2012).
+
+Multi-reward optimization We choose MolCRAFT (Qu et al., 2024) as a backbone for the multi-reward optimization. In the following analysis, we report the results of the fine-tuned model with Vina Score, SE, and QED. Although it is possible to optimize all seven evaluation metrics together with BFN-DPO, we find that this generally degrades the model performance. A more detailed analysis of reward selection is provided in Section 5.4.
+
+# 5.2. Results
+
+Table 1 shows the overall performance of ligand molecules generated by our method and baselines. Our approach achieves four best and four second-best results in eight metrics, indicating robust and well-rounded performance in SBDD. While autoregressive models ensure stable conformations and molecular properties but struggle with binding affinity, and diffusion-based models achieve high binding affinity at the cost of conformation stability and synthetic accessibility, MolCRAFT with BFNs balances both effectively. Qu et al. (2024) observes that binding affinity scores can be artificially boosted by generating larger molecules, which present more surface area for binding. The average size of our generated molecules remains comparable to those produced by MolCRAFT, indicating that no affinity hacking occurs. Note that smaller molecules often excel in SE and SA, which helps explain why AR and Pocket2Mol, both of which generate relatively smaller ligands, report particularly high SE and SA scores.
+
+![](images/d5f283801eb8110cf2d8d27b05d7def0b42c4c47329ff4db0c90161dd26edc26.jpg)  
+(a) Vina Dock vs. QED
+
+![](images/08b4b65f5810e293948b609952519643d8284a7ce8efe57fecd29acb1a50ca4d.jpg)  
+(b) SE vs. SA  
+Figure 2. Comparison between the median Vina docking score and QED, and SE and SA.
+
+Figure 1 illustrates how different models balance their average Vina Dock scores against SA and SE. In both subplots, previous approaches occupy a curve where improving one metric typically comes at the expense of the other. In contrast, our multi-reward method moves beyond this frontier by achieving stronger binding affinity while simultaneously preserving higher SA and lower SE. This outcome highlights the effectiveness of our framework in expanding the Pareto front for multi-objective structure-based drug design.
+
+We further illustrate the comparison between Vina Dock and QED in Figure 2a and SE and SA in Figure 2b. A similar trend can be observed from the Vina Dock and QED, but no clear trade-off between SE and SA is observable. Instead, we observe a strong correlation between SE and SA.
+
+# 5.3. Analysis
+
+Validity of generated molecules Although the conformation stability and molecular property metrics show the validity of generated molecules by our method, we con
+
+![](images/5083ae5c6233c664fc66fd0b5c802dbcf5bc1d57063d4b00f652b67476b1c188.jpg)  
+(a) The percentage of valid molecules.
+
+![](images/76ea5479abc62a77e84de1a42b356aa643a29688b9b602272f0d01aa4b920af7.jpg)  
+(b) Cumulative chart showing the remaining percent of valid molecules after each validity test.
+
+![](images/fa0d985d487b18c6e28e4a93f209f7d76680cadb45f29596dfbf14894e3c6483.jpg)  
+Figure 3. Validity check of the generated molecules from different models through PoseBuster (Buttenschoen et al., 2024).  
+Figure 4. Comparison of strain energy (SE) distributions for molecules generated by different methods.
+
+duct a more comprehensive validity check through the 20 validity tests proposed by PoseBuster (Buttenschoen et al., 2024). The test considers both intra-molecular and intermolecular validity with the protein. Figure 3a shows the percentage of molecules that pass all validity tests on the generated 10K molecules for the test proteins. Our multi-reward method generates significantly more valid ligands compared to other generative models. AliDiff, a fine-tuned IPDiff (Huang et al., 2024) with Vina Dock as a single reward, is known to achieve the best binding performance so far. However, the model fails to generate valid molecules in many cases. Figure 3b shows which test each model fails to satisfy the validity over 20 validity tests. Notably, many models, including IPDiff and AliDiff, fail to satisfy proper bond angles between atoms of generated molecules. Some models fail to keep a proper distance from the protein.
+
+Strain Energy (SE) Recent studies (Buttenschoen et al., 2024; Harris et al., 2023) caution that deep learning-based SBDD models may artificially inflate binding scores by producing ligands with unrealistically high SE. To examine
+
+Table 2. Comparison between different optimization approaches. GS is short for gradient surgery. A model fine-tuned with multi-rewards shows better binding affinity scores than the model with gradient surgery.  
+
+<table><tr><td rowspan="2"></td><td colspan="2">Binding Affinity</td><td colspan="2">Conformation Stability</td><td colspan="2">Molecular Properties</td></tr><tr><td>Vina Score Med. (↓)</td><td>SE Med. (↓)</td><td>Clash Avg. (↓)</td><td>SA Avg. (↑)</td><td>QED Avg. (↑)</td><td></td></tr><tr><td>Reference</td><td>-7.46</td><td>3.96</td><td>5.51</td><td>0.73</td><td>0.48</td><td></td></tr><tr><td>Pretrained</td><td>-7.04</td><td>7.62</td><td>7.09</td><td>0.69</td><td>0.50</td><td></td></tr><tr><td>3 GS</td><td>-6.80</td><td>9.69</td><td>7.18</td><td>0.66</td><td>0.50</td><td></td></tr><tr><td>6 GS</td><td>-6.87</td><td>8.62</td><td>7.11</td><td>0.69</td><td>0.52</td><td></td></tr><tr><td>Multi-Stage</td><td>-7.16</td><td>5.15</td><td>7.54</td><td>0.74</td><td>0.54</td><td></td></tr><tr><td>Ours</td><td>-7.38</td><td>5.56</td><td>6.69</td><td>0.74</td><td>0.55</td><td></td></tr></table>
+
+this issue, Figure 4 presents the SE distributions for different methods. Lower SE values correspond to more physically plausible, stable 3D conformations. Our method clearly generates molecules with lower SE distributions, indicating structurally sound ligands. These improvements do not come at the expense of binding affinity.
+
+Synthetic Accessibility (SA) Figure 5 presents the median SA scores of molecules generated by AliDiff, IPDiff, and MolCRAFT with reference molecules across target proteins in the test set. The compared models are recent state-of-the-art models for binding affinity. The results show that our model consistently generates molecules with high SA across all targets. We also report the proportion of proteins for which each model generated molecules with the highest median SA score. Our model obtains the best SA scores for  $44\%$  of proteins.
+
+# 5.4. Ablation Studies
+
+Different optimization approaches Multi-reward optimization can be framed as a multi-task learning problem
+
+![](images/85b44a12c7c427ca476a7f1eccbe0fe186dac729c9a0d7711559a5156eb9e20b.jpg)  
+Figure 5. The median Synthetic Accessibility (SA) scores of the molecules generated for each of the 100 target proteins in the test set. We compare our model to the top three models from the highest median binding affinity (AliDiff, IPDiff, MolCRAFT), as well as the reference. The values in parentheses represent the proportion of the best SA score for each protein
+
+Table 3. Comparison based on the number of rewards. A model fine-tuned with three rewards shows better binding affinity scores than the model with all seven rewards.  
+
+<table><tr><td rowspan="2" colspan="2"># Rewards</td><td colspan="3">Binding Affinity</td><td colspan="2">Conformation Stability</td><td colspan="2">Drug-like Properties</td></tr><tr><td>Vina Score Avg. (↓)</td><td>Vina Min Avg. (↓)</td><td>Vina Dock Avg. (↓)</td><td>SE Med. (↓)</td><td>Clash Avg. (↓)</td><td>SA Avg. (↑)</td><td>QED Avg. (↑)</td></tr><tr><td>No</td><td>Pretrained</td><td>-6.59</td><td>-7.27</td><td>-7.92</td><td>7.62</td><td>7.09</td><td>0.69</td><td>0.50</td></tr><tr><td rowspan="7">Single</td><td>Score</td><td>-6.63</td><td>-7.36</td><td>-8.01</td><td>6.85</td><td>7.25</td><td>0.71</td><td>0.53</td></tr><tr><td>Min</td><td>-6.61</td><td>-7.43</td><td>-8.02</td><td>6.88</td><td>7.36</td><td>0.71</td><td>0.53</td></tr><tr><td>Dock</td><td>-6.63</td><td>-7.40</td><td>-8.1</td><td>6.99</td><td>7.34</td><td>0.71</td><td>0.53</td></tr><tr><td>SE</td><td>-6.51</td><td>-7.25</td><td>-7.79</td><td>6.79</td><td>7.40</td><td>0.70</td><td>0.50</td></tr><tr><td>Clash</td><td>-6.44</td><td>-7.09</td><td>-7.58</td><td>9.87</td><td>7.16</td><td>0.66</td><td>0.48</td></tr><tr><td>QED</td><td>-6.45</td><td>-7.31</td><td>-7.80</td><td>6.13</td><td>7.38</td><td>0.72</td><td>0.54</td></tr><tr><td>SA</td><td>-6.50</td><td>-7.29</td><td>-7.80</td><td>7.33</td><td>7.47</td><td>0.71</td><td>0.53</td></tr><tr><td>Multi (Ours)</td><td>Dock+SE+QED</td><td>-7.18</td><td>-7.89</td><td>-8.62</td><td>5.56</td><td>6.69</td><td>0.74</td><td>0.55</td></tr><tr><td>All</td><td>7 rewards</td><td>-6.50</td><td>-7.28</td><td>-7.86</td><td>7.22</td><td>7.28</td><td>0.53</td><td>0.70</td></tr></table>
+
+when each reward is treated as a separate task. To explore this perspective, we adopt the widely used multi-task optimization technique known as gradient surgery (Yu et al., 2020). Specifically, we apply the gradient surgery to DPO losses of Vina score, SE, and QED (3GS). Since the DPO loss consists of atom and type losses as in Equation (7), we further apply the gradient surgery to these losses separately, resulting in six losses in total (6GS). In addition, we examine a multi-stage optimization procedure, wherein the model is iteratively fine-tuned using newly generated molecules from each preceding iteration. We report the result of the two-stage approach.
+
+Table 2 compares the performance of each approach to our uncertainty-based reward optimization. We only report the Vina Score among all Vina metrics, but they share similar trends. Neither the gradient surgery nor the multi-stage
+
+approach outperforms our uncertainty-based reward optimization. The gradient surgery methods perform worse than the reference model in all metrics except QED.
+
+Multi-reward selection Table 3 show the performances with different choices of metrics as rewards. Trying to optimize all seven rewards at once underperforms in several categories, implying the complex loss landscape with all rewards. In general, optimizing Vina metrics shows a strong correlation, i.e., optimizing one Vina metric can improve the performance of the other two Vina metrics. The optimization over QED improves the performance of SA, leading to our final choice of rewards. Note that Claw cannot be improved even by the corresponding single reward. We leave the improvement of Claw for future work.
+
+# 6. Conclusion
+
+In this work, we propose a multi-reward optimization framework to enhance deep learning-based structure-based drug design. Our method addresses the limitations of previous approaches by fine-tuning generative models to optimize multiple attributes, including binding affinity, validity, and drug-likeness. By utilizing direct preference optimization within a Bayesian flow network and integrating a reward normalization scheme, we enable the generation of more realistic ligands with improved binding affinity. Experimental results demonstrate that our approach outperforms the baseline models, offering a more robust solution for drug discovery tasks.
+
+# Acknowledgements
+
+This work was partly supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (RS-2019-II191906: Artificial Intelligence Graduate School Program (POSTECH), RS-2024-00457882: Artificial Intelligence Research Hub Project) and Korea Evaluation Institute of Industrial Technology (KEIT) grant funded by the Korea government (MOTIE, Korea) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2024-00337955, RS-2023-00217286)
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
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+
+# A. Complete Derivation of BFN-DPO
+
+Given a preference molecular dataset as  $\mathcal{D} = \{(\mathbf{p},\mathbf{m}^w,\mathbf{m}^l)\}$  where  $\mathbf{p}$  is the target proteins and  $\mathbf{m}^w, \mathbf{m}^l$  are winning-losing pair of molecules based on pre-computed rewards. With the similar argument used in Diffusion-DPO, one can define an objective for BFN over the entire timesteps as:
+
+$$
+\begin{array}{l} L _ {\mathrm {B F N - D P O}} (\phi) = - \mathbb {E} _ {\mathbf {m} ^ {w}, \mathbf {m} ^ {l}, \mathbf {p} \sim \mathcal {D}} \log \sigma \left(\beta \mathbb {E} _ {p _ {\phi} \left(\mathbf {m} _ {1: T} ^ {w} \mid \mathbf {m} ^ {w}\right) p _ {\phi} \left(\mathbf {m} _ {1: T} ^ {l} \mid \mathbf {m} ^ {l}\right)} \left[ \log \frac {p _ {\phi} \left(\mathbf {m} _ {1 : T} ^ {w}\right)}{p _ {\text {r e f}} \left(\mathbf {m} _ {1 : T} ^ {w}\right)} - \log \frac {p _ {\phi} \left(\mathbf {m} _ {1 : T} ^ {l}\right)}{p _ {\text {r e f}} \left(\mathbf {m} _ {1 : T} ^ {l}\right)} \right]\right) (13) \\ = - \mathbb {E} _ {\mathbf {m} ^ {w}, \mathbf {m} ^ {l}, \mathbf {p} \sim \mathcal {D}} \log \sigma \left(\beta \mathbb {E} _ { \begin{array}{c} p _ {\phi} (\mathbf {m} _ {1: T} ^ {w} | \mathbf {m} ^ {w}, \theta_ {0: T - 1} ^ {w}) p _ {\phi} (\theta_ {0: T - 1} ^ {w} | \mathbf {m} ^ {w}), \\ p _ {\phi} (\mathbf {m} _ {1: T} ^ {l} | \mathbf {m} ^ {l}, \theta_ {0: T - 1} ^ {l}) p _ {\phi} (\theta_ {0: T - 1} ^ {l} | \mathbf {m} ^ {l}) \end{array} } \left[ \log \frac {p _ {\phi} (\mathbf {m} _ {1 : T} ^ {w})}{p _ {\text {r e f}} (\mathbf {m} _ {1 : T} ^ {w})} - \log \frac {p _ {\phi} (\mathbf {m} _ {1 : T} ^ {l})}{p _ {\text {r e f}} (\mathbf {m} _ {1 : T} ^ {l})} \right]\right) (14) \\ = -\mathbb{E}_{\mathbf{m}^{w},\mathbf{m}^{l},\mathbf{p}\sim \mathcal{D}}\log \sigma \Bigg(\beta \mathbb{E}_{\substack{p_{\phi}(\mathbf{m}_{1:T}^{w}|\mathbf{m}^{w},\theta_{0:T - 1}^{w})\\ p_{\phi}(\theta_{0:T - 1}^{w}|\mathbf{m}^{w})\\ p_{\phi}(\mathbf{m}_{1:T}^{l}|\mathbf{m}^{l},\theta_{0:T - 1}^{l})\\ p_{\phi}(\theta_{0:T - 1}^{l}|\mathbf{m}^{l})}}\Bigg[\log \frac{p_{\phi}(\mathbf{m}_{1:T}^{w}|\theta_{0:T - 1}^{w})p_{\phi}(\theta_{0:T - 1}^{w}|\mathbf{m}^{w})}{p_{\mathrm{ref}}(\mathbf{m}_{1:T}^{w}|\theta_{0:T - 1}^{w})p_{\phi}(\theta_{1:T - 1}^{w}|\mathbf{m}^{w}) \\ \left. - \log \frac {p _ {\phi} \left(\mathbf {m} _ {1 : T} ^ {l}\right) p _ {\phi} \left(\theta_ {1 : T - 1} ^ {l} \mid \mathbf {m} ^ {l}\right)}{p _ {\text {r e f}} \left(\mathbf {m} _ {1 : T} ^ {l}\right) p _ {\phi} \left(\theta_ {0 : T - 1} ^ {l} \mid \mathbf {m} ^ {l}\right)} \right]\left. \right) (15) \\ = - \mathbb {E} _ {\mathbf {m} ^ {w}, \mathbf {m} ^ {l}, \mathbf {p} \sim \mathcal {D}} \log \sigma \left(\beta \mathbb {E} _ { \begin{array}{c} p _ {S} \left(\mathbf {m} _ {t} ^ {w} \mid \mathbf {m} ^ {w}, \alpha_ {t}\right) p _ {F} \left(\theta_ {t - 1} ^ {w} \mid \mathbf {m} ^ {w}\right) \\ p _ {S} \left(\mathbf {m} _ {t} ^ {l} \mid \mathbf {m} ^ {l}, \alpha_ {t}\right) p _ {F} \left(\theta_ {t - 1} ^ {l} \mid \mathbf {m} ^ {l}\right) \end{array} } \left[ \sum_ {t = 1} ^ {T} \left(\log \frac {p _ {\phi} \left(\mathbf {m} _ {t} ^ {w} \mid \theta_ {t - 1} ^ {w}\right)}{p _ {\text {r e f}} \left(\mathbf {m} _ {t} \mid \theta_ {t - 1} ^ {w}\right)} - \log \frac {p _ {\phi} \left(\mathbf {m} _ {t} ^ {l} \mid \theta_ {t - 1} ^ {l}\right)}{p _ {\text {r e f}} \left(\mathbf {m} _ {t} \mid \theta_ {t - 1} ^ {l}\right)}\right) \right]\right) + C, \\ \end{array}
+$$
+
+where we approximate the intractable  $p_{\phi}(\theta|\mathbf{m}^w)$  via Bayesian flow distribution  $p_F$  and  $p_{\phi}(\mathbf{m}_{1:T}^w \mid \mathbf{m}^w, \theta_{0:T-1}^w)$  via the sender distribution. We omit the conditioning on the proteins  $\mathbf{p}$  for brevity.
+
+By applying Jensen's inequality and dropping constant, we have
+
+$$
+L _ {\mathrm {B F N - D P O}} (\phi) = - \mathbb {E} _ {\substack {\mathbf {m} ^ {w}, \mathbf {m} ^ {l}, \mathbf {p} \sim \mathcal {D}, t \sim U (1, T), \\ p _ {F} \left(\theta_ {i} ^ {w} \mid \mathbf {m} ^ {w}\right), p _ {F} \left(\theta_ {i} ^ {l} \mid \mathbf {m} ^ {l}\right)}} \log \sigma \left(\beta T \mathbb {E} _ {\substack {p _ {S} \left(\mathbf {m} _ {i} ^ {w}\right) \\ p _ {S} \left(\mathbf {m} _ {i} ^ {l}\right)}} \left[ \left(\log \frac {p _ {\phi} \left(\mathbf {m} _ {i} ^ {w} \mid \theta_ {i - 1} ^ {w}\right)}{p _ {\text {ref}} \left(\mathbf {m} _ {i} ^ {w} \mid \theta_ {i - 1} ^ {w}\right)} - \log \frac {p _ {\phi} \left(\mathbf {m} _ {i} ^ {l} \mid \theta_ {i - 1} ^ {l}\right)}{p _ {\text {ref}} \left(\mathbf {m} _ {i} ^ {l} \mid \theta_ {i - 1} ^ {l}\right)}\right) \right]\right) \tag{17}
+$$
+
+Expanding the equation further gives us
+
+$$
+\begin{array}{l} \mathcal {L} _ {\mathrm {B F N - D P O}} (\phi) = - \mathbb {E} \Big [ \log \sigma \Big (- \beta T \Big (\mathbb {D} _ {\mathrm {K L}} \big (p _ {S} (\mathbf {m} _ {t} ^ {w} \mid \mathbf {m} ^ {w}) \mid p _ {\phi} \left(\mathbf {m} _ {t} ^ {w} \mid \theta_ {t - 1} ^ {w}\right) - \mathbb {D} _ {\mathrm {K L}} \big (p _ {S} (\mathbf {m} _ {t} ^ {w} \mid \mathbf {m} ^ {w}) \mid p _ {\text {r e f}} (\mathbf {m} _ {t} ^ {w} \mid \theta_ {t - 1} ^ {w}) \Big) \\ \left.\left. - \mathbb {D} _ {\mathrm {K L}} \left(p _ {S} \left(\mathbf {m} _ {t} ^ {l} \mid \mathbf {m} ^ {l}\right) \mid p _ {\phi} \left(\mathbf {m} _ {t} ^ {l} \mid \theta_ {t - 1} ^ {l}\right)\right) + \mathbb {D} _ {\mathrm {K L}} \left(p _ {S} \left(\mathbf {m} _ {t} ^ {l} \mid \mathbf {m} ^ {l}\right) \mid p _ {\text {r e f}} \left(\mathbf {m} _ {t} ^ {l} \mid \theta_ {t - 1} ^ {l}\right)\right)\right)\right)\left. \right]. \tag {18} \\ \end{array}
+$$
+
+The latent parameters  $\theta_{t} = [\theta_{t}^{\mathbf{x}},\theta_{t}^{\mathbf{v}}]$ , from which the noisy samples  $\mathbf{m}_t = [\mathbf{x}_t,\mathbf{v}_t]$  are drawn with the noise factors  $\alpha_{t} = [\alpha_{t}^{\mathbf{x}},\alpha_{t}^{\mathbf{v}}]$ , are assumed to follow distinct continuous distributions for each atom coordinates  $\mathbf{x}$  and atom types  $\mathbf{v}$ . With the neural network  $\Phi$  representing the predicted atom coordinates and types as  $[\Phi^{\mathbf{x}},\Phi^{\mathbf{v}}]$ , we can reformulate each of the above KL divergence terms as:
+
+$$
+\begin{array}{l} \mathbb {D} _ {\mathrm {K L}} \left(p _ {S} (\mathbf {m} _ {t} \mid \mathbf {x}) \mid p (\mathbf {m} _ {t} \mid \theta_ {t - 1})\right) \\ = \mathbb {D} _ {\mathrm {K L}} ^ {\mathbf {x}} \left(p _ {S} \left(\mathbf {x} _ {t} \mid \mathbf {x}\right) \mid p \left(\mathbf {x} _ {t} \mid \theta_ {t - 1} ^ {\mathbf {x}}\right)\right) + \mathbb {D} _ {\mathrm {K L}} ^ {\mathbf {v}} \left(p _ {S} \left(\mathbf {v} _ {t} \mid \mathbf {v}\right) \mid p \left(\mathbf {v} _ {t - 1} \mid \theta_ {t - 1} ^ {\mathbf {v}}\right)\right) (19) \\ = \frac {\alpha_ {t} ^ {\mathbf {x}}}{2} \| \mathbf {x} - \Phi^ {\mathbf {x}} (\theta) \| ^ {2} + \left(\log p _ {S} \left(\mathbf {v} _ {t} \mid \mathbf {v}, \alpha_ {t} ^ {\mathbf {v}}\right) - \log p _ {R} \left(\mathbf {v} _ {t} \mid \Phi^ {\mathbf {v}} \left(\theta_ {t - 1}\right), \alpha_ {t} ^ {\mathbf {v}}\right)\right). (20) \\ \end{array}
+$$
+
+As a result, we can show that the objective can be represented as a summation of the atom coordinates and atom type preference losses as follows:
+
+$$
+\mathcal {L} _ {\mathrm {B F N - D P O}} (\phi) = - \mathbb {E} _ {(\mathbf {p}, \mathbf {m} ^ {w}, \mathbf {m} ^ {l}) \sim \mathcal {D}, t \sim [ 0, T ], \theta_ {t} ^ {w} \sim p _ {F} (\cdot | \mathbf {m} _ {t} ^ {w}, \mathbf {p}; t), \theta_ {t} ^ {l} \sim p _ {F} (\cdot | \mathbf {m} _ {t} ^ {l}, \mathbf {p}; t)} \left[ \mathcal {L} _ {t - 1} ^ {\mathbf {x}} + \mathcal {L} _ {t - 1} ^ {\mathbf {v}} \right]. \tag {21}
+$$
+
+The atom coordinates loss is designed to directly train the model output  $\Phi_0^{\mathbf{x}}(\theta_t^{\mathbf{x}})$  to approximate the clean coordinate values  $\mathbf{x}_0$ . The concentration parameter  $\alpha$  for each time  $t$  is multiplied during training:
+
+$$
+\begin{array}{l} \mathcal{L}^{\mathbf{x}} = \operatorname *{-}\mathbb{E}_{\substack{(\mathbf{p},\mathbf{x}^{w},\mathbf{x}^{l})\sim \mathcal{D},t\sim [0,T],\\ \theta_{t}^{\mathbf{x},w}\sim p_{F}(\cdot |\mathbf{x}_{t}^{w},\mathbf{p};t),\theta_{t}^{\mathbf{x},l}\sim p_{F}(\cdot |\mathbf{x}_{t}^{l},\mathbf{p};t)}}\left[\log \sigma \Big(-\beta T\Big(\frac{\alpha_{t}^{\mathbf{x}}}{2}\Big)\Big(\| \mathbf{x}^{w} - \Phi_{\phi}^{\mathbf{x}}(\theta_{t - 1}^{\mathbf{x},w})\|^{2}- \| \mathbf{x}^{w} - \Phi_{\mathrm{ref}}^{\mathbf{x}}(\theta_{t - 1}^{\mathbf{x},w})\|^{2}\right) \\ \left.\left. - \| \mathbf {x} ^ {l} - \Phi_ {\phi} ^ {\mathbf {x}} \left(\theta_ {t - 1} ^ {\mathbf {x}, l}\right) \| ^ {2} + \| \mathbf {x} ^ {l} - \Phi_ {\mathrm {r e f}} ^ {\mathbf {x}} \left(\theta_ {t - 1} ^ {\mathbf {x}, l}\right) \| ^ {2}\right)\right)\left. \right]. \tag {22} \\ \end{array}
+$$
+
+Similarly, the atom types loss is designed to directly train the model output  $\Phi^{\mathbf{v}}(\theta_t^{\mathbf{v}})$  to approximate the clean atom types  $\mathbf{v}$ , which is the  $N_{\mathcal{M}} \times K$  matrix with each row as  $K$ -dimensional one-hot vector:
+
+$$
+\begin{array}{l} \mathcal {L} ^ {\mathbf {v}} = \underset { \begin{array}{c} (\mathbf {p}, \mathbf {v} ^ {w}, \mathbf {v} ^ {l}) \sim \mathcal {D}, t \sim [ 0, T ], \\ \theta_ {t - 1} ^ {\mathbf {v}, w} \sim p _ {F} (\cdot | \mathbf {v} ^ {w}, \mathbf {p}, t), \theta_ {t - 1} ^ {\mathbf {v}, l} \sim p _ {F} (\cdot | \mathbf {v} ^ {l}, \mathbf {p}, t) \end{array} } {- \mathbb {E}} \left[ \log \sigma \Big (- \beta T \left(\left(\ln p _ {S} (\cdot | \mathbf {v} ^ {w}, \alpha_ {t} ^ {\mathbf {v}}) - \ln p _ {R} (\cdot | \Phi_ {\phi} ^ {\mathbf {v}} (\theta_ {t - 1} ^ {\mathbf {v}, w}))\right) \right. \right. \\ - \left(\ln p _ {S} (\cdot | \mathbf {v} ^ {w}, \alpha_ {t} ^ {\mathbf {v}}) - \ln p _ {R} (\cdot | \Phi_ {\text {r e f}} ^ {\mathbf {v}} (\theta_ {t - 1} ^ {\mathbf {v}, w}))\right) \\ \left. - \left(\ln p _ {S} \left(\cdot | \mathbf {v} ^ {l}, \alpha_ {t} ^ {\mathbf {v}}\right) - \ln p _ {R} \left(\cdot | \Phi_ {\phi} ^ {\mathbf {v}} \left(\theta_ {t - 1} ^ {\mathbf {v}, l}\right)\right)\right) \right. \\ \left. \left. + \left(\ln p _ {S} \left(\cdot | \mathbf {v} ^ {l}, \alpha_ {t} ^ {\mathbf {v}}\right) - \ln p _ {R} \left(\cdot | \Phi_ {\text {r e f}} ^ {\mathbf {v}} \left(\theta_ {t - 1} ^ {\mathbf {v}, l}\right)\right)\right)\right) \right]. \tag {23} \\ \end{array}
+$$
+
+# B. Implementaton Details
+
+# B.1. SE-(3) Equivariant Network
+
+To model the interaction between ligand and protein pocket atoms, we employ an SE(3)-equivariant network, PosNet3D (Guan et al., 2022), as the foundational architecture  $\Phi$  in the receiver distribution. A protein-ligand graph is initially constructed by performing a  $k$ -nearest neighbor search on the atomic coordinates, represented as  $G = \langle V, E \rangle$ . At each layer, the atomic hidden states  $\mathbf{h}^l$  and coordinates  $\mathbf{x}^l$  are iteratively updated as follows:
+
+$$
+\mathbf {h} _ {i} ^ {l + 1} = \mathbf {h} _ {i} ^ {l} + \sum_ {j \in N _ {G} (i)} \phi_ {h} \left(d _ {i j}, \mathbf {h} _ {i} ^ {l}, \mathbf {h} _ {j} ^ {l}, e _ {i j}, t\right)
+$$
+
+$$
+\Delta \mathbf {x} _ {i} = \sum_ {j \in N _ {G} (i)} \left(\mathbf {x} _ {j} ^ {l} - \mathbf {x} _ {i} ^ {l}\right) \phi_ {x} \left(d _ {i j}, \mathbf {h} _ {i} ^ {l + 1}, \mathbf {h} _ {j} ^ {l + 1}, e _ {i j}, t\right)
+$$
+
+$$
+\mathbf {x} _ {i} ^ {l + 1} = \mathbf {x} _ {i} ^ {l} + \Delta \mathbf {x} _ {i} \cdot \mathbf {1} _ {m o l}
+$$
+
+where  $N_G(i)$  represents the neighborhood of atom  $i$  in graph  $G$ , while  $\mathbf{h}_i, \mathbf{x}_i$  and  $\mathbf{h}_j, \mathbf{x}_j$  denote the hidden states and coordinates of atoms  $i$  and  $j$ , respectively. The term  $d_{ij}^l$  represents the Euclidean distance between atoms  $i$  and  $j$ , and  $e_{ij}$  indicates whether the interaction is between protein atoms, ligand atoms, or a combination of both. The indicator function  $\mathbf{1}_{mol}$  ensures that only ligand atom positions are updated. The functions  $\phi_h$  and  $\phi_x$  act as attention mechanisms, utilizing  $\mathbf{h}_i^l$  as a query and  $[\mathbf{h}_i^l, \mathbf{h}_j^l, e_{ij}]$  as keys and values.
+
+For the initial layer, the atomic positions and hidden states are initialized as  $\mathbf{x}^0 = [\pmb{\mu},\mathbf{x}_{\mathcal{P}}]$  and  $\mathbf{h}^0 = \mathrm{linear}(\theta^v,\mathbf{v}_P,t)$ . In the final layer, the network produces an estimation  $\hat{\mathbf{x}} = \Phi^{\mathbf{x}}$ . Additionally, for the discrete variable  $\mathbf{v}^{(d)}$ , the function  $\Phi$  applies a softmax operation to yield the probability distribution:
+
+$$
+\hat {\mathbf {v}} ^ {(d)} = \operatorname {s o f t m a x} \left(\left(\boldsymbol {\Phi} ^ {\mathbf {v}}\right) ^ {(d)}\right).
+$$
+
+# B.2. Data Featurization
+
+Each protein atom is characterized by multiple features to effectively capture its structural and biochemical properties. These features include a one-hot encoded element identifier (comprising H, C, N, O, S, and Se) to specify the atomic type, a one-hot encoded amino acid type indicator of 20 dimensions to classify the amino acid identity, a binary flag to distinguish backbone atoms, and a one-hot encoded arm/scaffold region indicator that designates whether the atom is part of an arm or scaffold region based on its spatial proximity to the arm prior center.
+
+Similarly, ligand atoms are represented using a one-hot encoded element type indicator (spanning C, N, O, F, P, S, and Cl) and a one-hot encoded arm/scaffold indicator to differentiate between aromatic and non-aromatic atoms.
+
+For efficient message passing within the protein-ligand complex, two dynamically constructed graphs are employed: a k-nearest neighbors (k-NN) graph linking ligand and protein atoms and a fully connected graph for ligand atoms. The edge features in the k-NN graph are derived from the outer products of the distance embeddings, which are computed via radial basis function expansion, and a four-dimensional one-hot encoded edge type vector that classifies different types of atomic interactions. In contrast, the ligand graph represents ligand bonds through a one-hot encoded bond type vector, which categorizes bond types as non-bonded, single, double, triple, or aromatic.
+
+Following the existing work (Guan et al., 2023), proteins and ligands are represented through atomic coordinates alongside one-hot encoded atomic type vectors. Protein atoms are characterized by a one-hot encoded vector distinguishing among 20 amino acid types, while ligand atoms are encoded using a one-hot vector that identifies various elements, including H, C, N, O, F, P, S, and Cl. Additionally, a binary flag is incorporated to distinguish between protein and ligand atoms. To further refine these representations, two separate single-layer multi-layer perceptrons (MLPs) are applied, each mapping the input features into a 128-dimensional latent space, ensuring a compact yet informative feature embedding for subsequent computational processing.
+
+# B.3. Preference Pair Generation
+
+For each protein binding site, we initially generate ten molecular candidates from the pretrained model. The reward for each synthesized molecule is then evaluated based on a user-defined reward function specific to the corresponding binding site. To construct preference data, we identify a lower-performing sample with a comparatively lower reward and establish a preference ranking. The detailed procedure for reward computation and preference selection is provided in Section 4.2.
+
+# B.4. Model Hyperparameters
+
+For the SE(3)-equivariant network, we utilize k-nearest neighbors (kNN) graphs with a 32-nearest neighbor search to construct the graph. The model consists of nine layers, each with a hidden dimension of 128, employing a 16-head attention mechanism. ReLU activation functions are used in conjunction with Layer Normalization (Ba et al., 2016) to enhance stability and performance.
+
+Regarding the noise schedules, we set the parameters as follows:  $\beta_{1} = 1.5$  for atom types and  $\sigma_{1} = 0.03$  for atom coordinates. The model is trained using a discrete-time loss function over 1000 training steps.
+
+For pretraining, we adopt the same architecture as MolCRAFT, which is based on Bayesian Flow Networks (Graves et al., 2023).
+
+For finetuning, we employ the Adam optimizer with a learning rate of 0.005 and a batch size of 32. Additionally, an exponential moving average (EMA) of model parameters is maintained with a decay factor of 0.999. Training converges within two epochs on a single RTX 4090 GPU, requiring approximately six hours for completion. For sampling, we generate 100 sample steps using a noise-reduced sampling strategy.
+
+To ensure a higher value corresponds to a better reward across different metrics, we use a transformation function in Equation (10) as normalizing each rewards.
+
+$$
+\hat {r} _ {i} ^ {(j)} = \frac {\exp \big (f _ {j} (r _ {i} ^ {(j)}) \big)}{\sum_ {b = 1} ^ {B} \exp \big (f _ {j} (r _ {b} ^ {(j)}) \big)}
+$$
+
+$$
+f _ {j} (r) = \left\{ \begin{array}{l l} - r, & \text {i f j c o r r e s p o n d s t o t h e V i n a D o c k}, \\ \frac {1 0}{r}, & \text {i f j c o r r e s p o n d s t o s t r a i n e n e r g y}, \\ r, & \text {i f j c o r r e s p o n d s t o Q E D}. \end{array} \right.
+$$
+
+where  $r_i^{(j)}$  is the  $j$ -th reward for the  $i$ -th molecule, and  $f_j(\cdot)$  is a simple transformation function to ensure each reward is reasonable. This maps each reward to the  $(0,1)$  interval, ensuring that distinct reward signals become comparable despite inherently different numerical ranges. We set  $\gamma = 0.4$  in Equation (11)
+
+# B.5. Computation Time
+
+We report the time required for one epoch of pretraining and fine-tuning our model, comparing it to the AliDiff baseline trained with DPO. To measure the computation time, we use the official source code provided by the author and run the code on a single NVIDIA A6000 GPU.
+
+Table 4. Time needed for training per one epoch (in seconds)  
+
+<table><tr><td>Model</td><td>Pretraining</td><td>Finetuning only</td></tr><tr><td>AliDiff (DPO with diffusion model)</td><td>7461.73</td><td>22,884</td></tr><tr><td>Ours (DPO with BFN model)</td><td>5695.40</td><td>16,354</td></tr></table>
+
+This shows applying DPO to a pre-trained model takes 16,354 seconds with our method per one epoch, compared to 22,884 seconds for AliDiff, representing a  $28.5\%$  reduction. Our approach does not significantly increase overall training time; however, obtaining rewards from an external tool can be time-consuming. To address this, we preprocess and store rewards in advance, but this requires an offline DPO framework. A promising future direction for SBDD is reducing reliance on external tools, enabling a more iterative and efficient online training loop.

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@@ -0,0 +1,598 @@
+# Yuchen Guan\*1 Runxi Cheng\*1 Kang Liu 1 Chun Yuan
+
+# Abstract
+
+Knowledge distillation typically minimizes the Kullback-Leibler (KL) divergence between teacher and student logits. However, optimizing the KL divergence can be challenging for the student and often leads to sub-optimal solutions. We further show that gradients induced by KL divergence scale with the magnitude of the teacher logits, thereby diminishing updates on low-probability channels. This imbalance weakens the transfer of inter-class information and in turn limits the performance improvements achievable by the student. To mitigate this issue, we propose a plug-and-play auxiliary ranking loss based on Kendall's  $\tau$  coefficient that can be seamlessly integrated into any logit-based distillation framework. It supplies inter-class relational information while rebalancing gradients toward low-probability channels. We demonstrate that the proposed ranking loss is largely invariant to channel scaling and optimizes an objective aligned with that of KL divergence, making it a natural complement rather than a replacement. Extensive experiments on CIFAR-100, ImageNet, and COCO datasets, as well as various CNN and ViT teacher-student architecture combinations, demonstrate that our plug-and-play ranking loss consistently boosts the performance of multiple distillation baselines. Code is available at https://github.com/OvernighTea/RankingLoss-KD
+
+# 1. Introduction
+
+The recent advancements in deep neural networks (DNN) have significantly enhanced performance in the field of computer vision. However, the heightened computational and
+
+*Equal contribution  ${}^{1}$  Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen, China. Correspondence to: Kang Liu  $<$  liuk22@mails.tsinghua.edu.cn>, Chun Yuan  $<$  yuanc@sz.tsinghua.edu.cn>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+![](images/1f1b2439fa7fcf0724b0b82d4fac0fa76c4a31d85f723cb116387c8f1e4d12c5.jpg)  
+Figure 1. Suboptimal Case of the KL Divergence.
+
+storage costs associated with complex networks limit their applicability. To address this, knowledge distillation (KD) has been proposed to obtain performant lightweight models. Typically, knowledge distillation involves a well-trained heavy teacher model and an untrained lightweight student model. The same data is fed to both the teacher and the student, with the teacher's outputs serving as soft labels for training the student. By constraining the student to produce predictions that match the soft labels, the knowledge in the teacher model is transferred to the lightweight student model.
+
+Most logit-based KD methods adhere to the paradigm introduced by Hinton (Hinton et al., 2015), employing the Kullback-Leibler (KL) divergence (Kullback & Leibler, 1951) to align the logits output by the student and teacher models:
+
+$$
+\mathcal {L} _ {K L} \left(q ^ {t} \| q ^ {s}\right) = \sum_ {i = 1} ^ {C} q _ {i} ^ {t} \cdot \log \left(\frac {q _ {i} ^ {t}}{q _ {i} ^ {s}}\right). \tag {1}
+$$
+
+Where  $q_{t}, q_{s} \in \mathbb{R}^{1 \times C}$  is the prediction vectors of teacher and student. The optimization goal of KL divergence is to achieve identical outputs between the student and teacher, thereby transferring as much knowledge as possible from the teacher to the student. However, the optimization process of KL divergence is not easy, as it is prone to suboptimal points, which can hinder further improvement in student performance. As illustrated in Fig. 1, compared to Student 2, Student 1 exhibits a smaller KL divergence with the teacher; however, Student 2 achieves the correct classification result
+
+![](images/605841a0222cfb9fbb3715ffd5b6db944595c4d2cfe285aa373c6df254bccc22.jpg)  
+Figure 2. Most Channels in the Distribution are Low-Probability Channels.
+
+consistent with the teacher, while Student 1 does not. This indicates that the optimization direction of KL divergence sometimes diverges from the task objective, leading students to suboptimal points.
+
+Intuitively, and as a preliminary observation, Eq. 1 shows that in the KL divergence each channel is weighted by the predicted probability of the teacher. Channels with small  $q_{i}^{t}$  therefore contribute marginally to the loss, causing the student to tend to largely ignore them (the formal derivation is provided in Appendix A.5). In practice, most channels in a single prediction have lower probabilities, as illustrated in Fig. 2. Neglecting these channels hampers knowledge transfer for two reasons:
+
+R1: Gradient attenuation. Because low-probability channels carry tiny weights, the corresponding gradients are strongly damped, which makes alignment with the teacher network difficult..
+
+R2: Impaired capture of inter-class information. Low-probability channels retain rich cross-category correlation information in the teacher logits; overlooking them deprives the student of these semantic correlations. For example, in Fig. 1, Student 1 may not learn the distinction between the cat and fox classes.
+
+Considering these challenges, we aim to find an auxiliary method that mitigates the issues caused by channel scale differences, learns inter-class relationship information, and avoids the suboptimal points that KL divergence might encounter while maintaining the optimization objective of KL divergence. To address this problem, we propose a ranking-based plug-and-play auxiliary loss. The benefits of imposing ranking constraints are as follows:
+
+B1: Gradients from the ranking term depend only weakly on the absolute channel scale.  
+B2: Ordered channel scores make cross-category relationships explicit.
+
+B3: Fixing relative channel order helps the model escape suboptimal solutions.
+
+Therefore, we propose to constrain the channel ranking similarity between student and teacher. We construct a plug-and-play ranking loss function based on Kendall's  $\tau$  Coefficient. This ranking loss can supplement the attention to low-probability channels in the logits and provide inter-class relationship information. We demonstrate that the gradients provided by KL divergence are related to channel scale, whereas the proposed ranking loss is not. We also show that the optimization objective of the proposed ranking loss is consistent with that of KL divergence, and visualize the loss landscape to illustrate that the ranking loss helps avoid suboptimal points in the early stage and does not alter the optimization objective in the end. Extensive experiments on CNN and ViT show that the proposed plug-and-play ranking loss can enhance the performance of various logit-based methods. In conclusion, our contributions are as follows:
+
+- We introduce a plug-and-play ranking loss for assisting knowledge distillation tasks, addressing the issues of KL divergence's neglect of low-probability channels and its tendency to fall into suboptimal points, while also learning inter-class relationship information to further enhance student performance.  
+- We demonstrate that the gradients provided by KL divergence exhibit sensitivity to channel scales, while the proposed ranking loss remains more robust to such variations. We also prove and visualize that the optimization objective of the proposed ranking loss is consistent with that of KL divergence and helps avoid the suboptimal points of KL divergence.  
+- Extensive experiments are conducted on a variety of CNN and ViT teacher-student architectures using the CIFAR-100, ImageNet, and MS-COCO datasets. Our findings confirm the widespread effectiveness of the proposed ranking loss in various distillation tasks, and its role as a plug-and-play auxiliary function provides substantial support for the training of distillation tasks.
+
+# 2. Related Works
+
+Knowledge Distillation. Knowledge distillation, initially proposed by Hinton (Hinton et al., 2015), serves as a method for model compression and acceleration, aiming to transfer knowledge from a heavy teacher model to a lightweight student model. By feeding the same samples, the teacher can produce soft labels, and training the student with these soft labels allows the transfer of knowledge from the teacher model to the student. Knowledge distillation tasks can be mainly divided into two categories: feature-based distillation (Chen et al., 2022; Zhang & Ma, 2021; Yang et al.,
+
+2022c; Guo et al., 2023; Park et al., 2019; Yang et al., 2022b) and logit-based distillation (Hinton et al., 2015; Li et al., 2023; Zhao et al., 2022; Jin et al., 2023; Sun et al., 2024; Chi et al., 2023; Wen et al., 2021; Yang et al., 2023b). Feature-based distillation additionally utilizes the model's intermediate features, providing more information to the student and enabling the student to learn at the feature level from the teacher, often resulting in better performance. Considering safety and privacy, the intermediate outputs of the model are often not obtainable; hence, logit-based methods that only use the model outputs for distillation offer better versatility and robustness. Our proposed method can act as a plug-and-play module added to logit-based methods, offering higher flexibility and further enhancing the performance of logit-based methods.
+
+Recent knowledge distillation methods have found that overly strict constraints can sometimes hinder the student's transfer of knowledge from the teacher's soft labels. For instance, (Cao et al., 2022) discovered that differences in feature sizes in feature-based methods could limit the student's learning; while (Sun et al., 2024) found that using the same temperature for both teacher and student in logit-based methods could affect further improvements in student performance. To reduce the learning difficulty for the student, (Sun et al., 2024; Cao et al., 2022) provided methods to ease the logit matching difficulty between student and teacher, yet we find that methods providing additional guidance to the student have not been fully explored.
+
+Ranking Loss in Knowledge Distillation. For knowledge distillation, ranking loss is a relaxed constraint that can provide rich inter-class information. It was first applied in the distillation of recommendation systems (Reddi et al., 2021; Tang & Wang, 2018; Choi et al., 2021; Qin et al., 2023; Yang et al., 2022a), (Li et al., 2022) explored the application of ranking loss in object detection tasks, and (Gao et al., 2020) discussed the role of ranking in the distillation of language model tasks. However, the exploration of ranking loss in logit-based image classification task distillation is not yet comprehensive. We find that the KL divergence used for distillation in classification tasks tends to overlook information from low-probability channels and may lead to suboptimal results. Our proposed method leverages ranking loss to balance the model's attention between high-probability and low-probability channels, while also using inter-class relationships to help the model avoid suboptimal outcomes.
+
+# 3. Preliminary
+
+Most logit-based distillation methods adhere to the original KD proposed by Hinton (Hinton et al., 2015), which transfers knowledge by matching the logit outputs of the student and teacher. This setting is more generalizable,
+
+allowing for distillation solely through outputs when the internal structures of the student and teacher are invisible. For a given dataset  $\mathcal{D}$ , assuming there are  $C$  categories and  $N$  samples, we possess a teacher model  $f_{t}$  and a student model  $f_{s}$ . For a given sample  $\mathcal{I} \in \mathcal{D}$ , we can obtain the outputs of the teacher and student model, denoted as  $z^{t} = f_{t}(\mathcal{I}), z^{s} = f_{s}(\mathcal{I})$  where  $z^{t}, z^{s} \in \mathbb{R}^{1 \times C}$ . Through a softmax function with temperature, the outputs are processed into the prediction vectors  $q_{t}, q_{s} \in \mathbb{R}^{1 \times C}$  finally:
+
+$$
+q _ {i} ^ {t} = \frac {\exp \left(z _ {i} ^ {t} / \mathcal {T}\right)}{\sum_ {j = 1} ^ {C} \exp \left(z _ {j} ^ {t} / \mathcal {T}\right)}, \quad q _ {i} ^ {t} = \frac {\exp \left(z _ {i} ^ {s} / \mathcal {T}\right)}{\sum_ {j = 1} ^ {C} \exp \left(z _ {j} ^ {s} / \mathcal {T}\right)} \tag {2}
+$$
+
+where  $\mathcal{T}$  is a temperature parameter,  $z_{i}$  represents the logit value of the  $i$ -th channel of the model output,  $q_{i}$  represents the predicted probability for the target being the  $i$ -th class. The KL divergence loss function used to constrain the student and teacher logits is of the following form:
+
+$$
+\mathcal {L} _ {K L} \left(q ^ {t} \| q ^ {s}\right) = \sum_ {i = 1} ^ {C} q _ {i} ^ {t} \cdot \log \left(\frac {q _ {i} ^ {t}}{q _ {i} ^ {s}}\right) \tag {3}
+$$
+
+It can be observed that in Eq. 3, the importance of the match between the student and teacher at the  $i$ -th channel is influenced by the coefficient  $q_{i}^{t}$ . This implies that channels with smaller logit values receive less attention, leading to the KL divergence's disregard for smaller channels.
+
+# 4. Ranking Loss Based on Kendall's  $\tau$  Coefficient
+
+In this section, we introduce the plug-and-play ranking loss function designed to constrain the ranking of channels in the logits output by the student model. With the aid of the ranking loss, the knowledge distillation task can balance the overemphasis on larger logit values and the neglect of smaller ones as measured by the KL divergence. Additionally, the ranking loss imposes a constraint on the leading channel value, which helps to correct the optimization direction and avoid suboptimal solutions. In Sec. 4.1, we will introduce the ranking loss function and employ Kendall's  $\tau$  coefficient to compute the ordinal consistency between the teacher and student logits. In Sec. 4.2, we discuss how ranking loss benefits optimizing logits distillation with KL divergence. Furthermore, we discuss the differentiable form of Kendall's  $\tau$  coefficient and, based on this, design three distinct forms of ranking loss functions in Appendix A.3.
+
+# 4.1. Differentiable Kendall's  $\tau$  Coefficient
+
+In order to make the loss function pay more attention to the information provided by low-probability channels as well as to help correct the suboptimal problem in Fig. 1, we introduce the ranking loss based on Kendall's  $\tau$  coefficient.
+
+By pairing each of the  $C$  channels, we can obtain  $\frac{C(C - 1)}{2}$  pairs, whose Kendall's  $\tau$  coefficient is expressed as follows:
+
+$$
+\tau = \frac {P _ {c} - P _ {d}}{\frac {1}{2} C (C - 1)} \tag {4}
+$$
+
+where  $P_{c}$  represents concordant pairs and  $P_{d}$  represents discordant pairs. Kendall's  $\tau$  coefficient provides an expression of ordinal similarity. For the teacher and student logits, a channel pair  $(i,j)$  is considered concordant if the signs of  $(z_i^t - z_j^t)$  for the teacher and  $(z_i^s - z_j^s)$  for the student are the same; otherwise, the pair is discordant. We substitute the logits of the teacher and student in the following manner:
+
+$$
+\tau = \frac {\sum_ {i} \sum_ {j <   i} \operatorname {s g n} \left(z _ {i} ^ {t} - z _ {j} ^ {t}\right) \cdot \operatorname {s g n} \left(z _ {i} ^ {s} - z _ {j} ^ {s}\right)}{\frac {1}{2} C (C - 1)} \tag {5}
+$$
+
+where  $\text{sgn}()$  represents sign function. While we have quantified the ordinal similarity between the teacher and student logits, the aforementioned formula is non-smooth. To utilize the ordinal similarity for gradient computation, we approximate the sign function with the tanh function to convert it into a differentiable form:
+
+$$
+\begin{array}{l} \tau_ {d} = \frac {\sum_ {i} \sum_ {j <   i} \tanh (k \cdot (z _ {i} ^ {t} - z _ {j} ^ {t})) \cdot \tanh (k \cdot (z _ {i} ^ {s} - z _ {j} ^ {s}))}{\frac {1}{2} C (C - 1)} \\ = \frac {2}{C (C - 1)} \cdot \sum_ {i} \sum_ {j <   i} \left(1 - \frac {2}{1 + e ^ {2 \cdot (z _ {i} ^ {t} - z _ {j} ^ {t}) \cdot k}}\right). \\ \left(1 - \frac {2}{1 + e ^ {2 \cdot \left(z _ {i} ^ {s} - z _ {j} ^ {s}\right) \cdot k}}\right) \tag {6} \\ \end{array}
+$$
+
+where  $k$  is a parameter that controls the steepness of the function; a larger  $k$  causes the tanh function to more closely approximate the sign function. By negating the similarity measure in Eq. 6, it can serve as a loss function (bounded within  $[-1,1]$ ) to enforce the consistency of the logits order between the teacher and student:
+
+$$
+\mathcal {L} _ {R K} = - \tau_ {d} \tag {7}
+$$
+
+The overall loss function is formulated as follows:
+
+$$
+\mathcal {L} = \alpha \mathcal {L} _ {K L} + \beta \mathcal {L} _ {C E} + \gamma \mathcal {L} _ {R K} \tag {8}
+$$
+
+where  $\mathcal{L}_{KL}$  denotes KL divergence loss,  $\mathcal{L}_{CE}$  denotes Cross-Entropy loss,  $\mathcal{L}_{RK}$  denotes our ranking loss, and  $\alpha, \beta, \gamma$  are hyper-parameters.
+
+# 4.2. How Ranking Loss Benefits Optimizing Logits Distillation with KL Divergence
+
+# 4.2.1. FROM THE PERSPECTIVE OF GRADIENT: RANKING LOSS CARES ABOUT LOW-PROBABILITY CHANNELS
+
+In this section, we analyze why ranking loss cares about low-probability channels. The gradient of KL divergence and
+
+![](images/6d996166169416a6aacda7279ed715322f343003ccb1c124e03326753b3fb70a.jpg)  
+Figure 3. The Gradient of Ranking Loss and KL Divergence for Different Values of Logits.
+
+![](images/77c91eb758cf0ccf240e299b052aea79831cb6b72c1c5e1bac9fc02d503d749f.jpg)
+
+ranking loss is written as follows. The specific calculation process of the gradient can be found in the appendix A.5:
+
+$$
+\frac {\partial \mathcal {L} _ {\mathrm {K D}}}{\partial z _ {i} ^ {s}} = - T \left(q _ {i} ^ {t} - q _ {i} ^ {s}\right). \tag {9}
+$$
+
+$$
+\begin{array}{l} \frac {\partial \mathcal {L} _ {R K}}{\partial z _ {i} ^ {s}} = - \frac {k}{C (C - 1)} \sum_ {j \neq i} \left[ 1 - \tanh ^ {2} \left(k (z _ {i} ^ {s} - z _ {j} ^ {s})\right) \right]. \\ \tanh  \left(k \left(z _ {i} ^ {t} - z _ {j} ^ {t}\right)\right) \tag {10} \\ \end{array}
+$$
+
+The gradient scale of the KL divergence is influenced by the scales of the teacher and student models. Consequently, for outputs from low-probability channels, when the scales of the teacher and student are similar, the KL divergence's gradient for these components becomes negligible, leading to the neglect of information from low-probability channels. Although temperature scaling is typically employed to address this issue, it uniformly amplifies the gradients of all logits, thereby continuing to overlook the outputs of low-probability channels. In contrast, the gradients produced by ranking loss will be less affected by the scale of the teacher's logits. The gradient of a logit channel primarily depends on the difference between its rank and the target rank, effectively harnessing the knowledge from low-probability channels. Therefore, In the early stage of training, ranking loss helps the model quickly learn the overall ranking of the teacher model, which helps the model converge to a better initial solution faster.
+
+In our study, we conduct a visualization of the gradients associated with both the ranking loss and the KL divergence loss in Fig .3. The logits of the student model are randomly generated but maintain the same scale as those of the teacher model across each channel. Our findings indicate that the ranking loss consistently provides gradients of similar scale across varying logit values, whereas the gradients from the KL loss are heavily dependent on the specific logit values. This suggests that the ranking loss offers a more uniform attention distribution across different logits compared to the KL loss. Notably, the ranking loss assigns low-probability gradients to logits with larger values, as they are the clas
+
+Table 1. CIFAR-100 Heterogeneous Architecture Results. The Top-1 Accuracy (%) is reported as the evaluation metric. The teacher and student have heterogeneous architectures. We incorporate the proposed ranking loss into the existing logit-based methods, with the performance gains indicated in parentheses. The best and second best results are emphasized in bold and underlined.  
+
+<table><tr><td rowspan="4">KD</td><td rowspan="2">Teacher</td><td>ResNet32×4</td><td>ResNet32×4</td><td>ResNet50</td><td>ResNet32×4</td><td>WRN-40-2</td></tr><tr><td>79.42</td><td>79.42</td><td>79.34</td><td>79.42</td><td>75.61</td></tr><tr><td rowspan="2">Student</td><td>WRN-16-2</td><td>WRN-40-2</td><td>MN-V2</td><td>SHN-V1</td><td>SHN-V1</td></tr><tr><td>73.26</td><td>75.61</td><td>64.60</td><td>70.50</td><td>70.50</td></tr><tr><td></td><td>FitNet (Adriana et al., 2015)</td><td>74.70</td><td>77.69</td><td>63.16</td><td>73.59</td><td>73.73</td></tr><tr><td></td><td>CRD (Tian et al., 2019)</td><td>75.65</td><td>78.15</td><td>69.11</td><td>75.11</td><td>76.05</td></tr><tr><td></td><td>ReviewKD (Chen et al., 2021)</td><td>76.11</td><td>78.96</td><td>69.89</td><td>77.45</td><td>77.14</td></tr><tr><td></td><td>DIST (Huang et al., 2022)</td><td>75.58</td><td>78.02</td><td>68.66</td><td>76.34</td><td>76.00</td></tr><tr><td></td><td>LSKD (Sun et al., 2024)</td><td>77.53</td><td>79.66</td><td>71.19</td><td>76.48</td><td>76.93</td></tr><tr><td></td><td>KD (Hinton et al., 2015)</td><td>74.9</td><td>77.7</td><td>67.35</td><td>74.07</td><td>74.83</td></tr><tr><td></td><td>KD+Ours</td><td>75.18(+0.28)</td><td>78.50(+0.80)</td><td>70.45(+3.10)</td><td>75.98(+1.91)</td><td>76.13(+1.30)</td></tr><tr><td></td><td>CTKD (Li et al., 2023)</td><td>74.57</td><td>77.66</td><td>68.67</td><td>74.48</td><td>75.61</td></tr><tr><td></td><td>CTKD+Ours</td><td>75.71(+1.14)</td><td>78.61(+0.95)</td><td>70.18(+1.51)</td><td>76.67(+2.19)</td><td>76.80(+1.19)</td></tr><tr><td></td><td>DKD (Zhao et al., 2022)</td><td>75.7</td><td>78.46</td><td>70.35</td><td>76.35</td><td>76.33</td></tr><tr><td></td><td>DKD+Ours</td><td>75.99(+0.29)</td><td>78.75(+0.29)</td><td>70.90(+0.55)</td><td>77.36(+1.01)</td><td>76.43(+0.10)</td></tr><tr><td></td><td>MLKD (Jin et al., 2023)</td><td>76.52</td><td>79.26</td><td>71.04</td><td>77.18</td><td>77.44</td></tr><tr><td></td><td>MLKD+Ours</td><td>76.83(+0.31)</td><td>79.86(+0.60)</td><td>71.66(+0.62)</td><td>77.63(+0.45)</td><td>77.87(+0.43)</td></tr></table>
+
+sification targets and reach the correct rank. Consequently, these logits receive less attention from the ranking loss.
+
+# 4.2.2. FROM THE PERSPECTIVE OF OPTIMAL SOLUTION DOMAIN: RANKING LOSS WON'T INTERFERE WITH KL DIVERGENCE AT THE END
+
+In this section, we analyze the optimal solution domain for Kullback-Leibler (KL) divergence and ranking loss to show how ranking loss affects the optimization process of KL divergence. For KL divergence, the optimal solution domain is achieved when the distribution of the student model aligns perfectly with that of the teacher model, which is equivalent to a linear mapping on the logits:
+
+$$
+q _ {i} ^ {t} = q _ {i} ^ {s} \quad \forall i \quad \Rightarrow \quad z _ {i} ^ {t} = z _ {i} ^ {s} + c \quad \forall i \tag {11}
+$$
+
+$q_{i}^{t}$  and  $q_{i}^{s}$  respectively represent the probability outputs of the teacher and the student for the i-th class, and  $c$  represents any constant number. For the ranking loss, the optimal solution domain is such that for any two indices, the order of the logits output by the teacher and the student is consistent:
+
+$$
+\begin{array}{l} s g n (z _ {i} ^ {t} - z _ {j} ^ {t}) \cdot s g n (z _ {i} ^ {s} - z _ {j} ^ {s}) = 1 \quad \forall i, j \\ \Longleftrightarrow \left(z _ {i} ^ {t} - z _ {j} ^ {t}\right) \cdot \left(z _ {i} ^ {s} - z _ {j} ^ {s}\right) > 0 \quad \forall i, j \tag {12} \\ \Longleftrightarrow z _ {i} ^ {t} = F \left(z _ {i} ^ {s}\right) \quad \forall i \quad w h e r e \quad F ^ {\prime} (x) > 0 \\ \end{array}
+$$
+
+It implies that the optimal solution domain for ranking loss is quite lenient and easily attainable, encompassing the optimal solution domain for KL divergence. Therefore, in the later stage of training, the ranking loss will not hinder the optimization of KL divergence, which means our ranking loss can perfectly be used as an auxiliary loss.
+
+# 4.2.3. FROM THE PERSPECTIVE OF CLASSIFICATION: RANKING LOSS HELPS STUDENT CLASSIFY CORRECTLY
+
+As illustrated in Fig. 1, using KL loss may lead to a smaller loss yet result in incorrect classification. The tendency to fall into local optima will hinder the optimization process in distillation. Although the use of cross-entropy loss can mitigate this issue, it does not incorporate information from the teacher model. Consequently, in experiments, cross-entropy loss is often assigned a very small weight, which also leads to a small gradient. For instance, KD (Hinton et al., 2015)  $\mathcal{L}_{\mathrm{CE}}: \mathcal{L}_{\mathrm{KL}} = 0.1: 0.9$ ; DKD (Zhao et al., 2022)  $\mathcal{L}_{\mathrm{CE}}: \mathcal{L}_{\mathrm{TCKL}}: \mathcal{L}_{\mathrm{NCKL}} = 1: 1: 8$ ; MLKD (Jin et al., 2023)  $\mathcal{L}_{\mathrm{CE}}: \mathcal{L}_{\mathrm{KL}} = 0.1: 9$ ; LSKD (Sun et al., 2024)  $\mathcal{L}_{\mathrm{CE}}: \mathcal{L}_{\mathrm{KL}} = 0.1: 9$ . As a result, cases with lower KL loss but incorrect classification still frequently occur. In contrast, by aligning the rank between student and teacher, ranking loss helps KL divergence avoid such suboptimal situations. Also, the ranking loss can incorporate information from the teacher model. In this way, ranking loss helps students classify correctly, avoiding the suboptimal case in logit distillation. Furthermore, a model with better generalization should have a more reasonable rank of logits. For instance, recognizing that a tiger is more similar to a cat than to a fish. By learning such inter-class relationships, the student can improve classification performance and enhance its representational capacity.
+
+# 5. Experiments
+
+Datasets. In order to validate the efficacy and robustness of our proposed method, we conduct extensive experiments
+
+Table 2. CIFAR-100 Homogenous Architecture Results. The Top-1 Accuracy (%) is reported as the evaluation metric. The teacher and student have homogeneous architectures. We incorporate the proposed ranking loss into the existing logit-based methods, with the performance gains indicated in parentheses. The best and second best results are emphasized in bold and underlined.  
+
+<table><tr><td rowspan="4">KD</td><td rowspan="2">Teacher</td><td>ResNet32×4</td><td>VGG13</td><td>WRN-40-2</td><td>ResNet110</td></tr><tr><td>79.42</td><td>74.64</td><td>75.61</td><td>74.31</td></tr><tr><td rowspan="2">Student</td><td>ResNet8×4</td><td>VGG8</td><td>WRN-40-1</td><td>ResNet20</td></tr><tr><td>72.50</td><td>70.36</td><td>71.98</td><td>69.06</td></tr><tr><td></td><td>FitNet (Adriana et al., 2015)</td><td>73.50</td><td>71.02</td><td>72.24</td><td>68.99</td></tr><tr><td></td><td>CRD (Tian et al., 2019)</td><td>75.51</td><td>73.94</td><td>74.14</td><td>71.46</td></tr><tr><td></td><td>ReviewKD (Chen et al., 2021)</td><td>75.63</td><td>74.84</td><td>75.09</td><td>71.34</td></tr><tr><td></td><td>DIST (Huang et al., 2022)</td><td>76.31</td><td>73.80</td><td>74.73</td><td>71.40</td></tr><tr><td></td><td>LSKD (Sun et al., 2024)</td><td>78.28</td><td>75.22</td><td>75.56</td><td>72.27</td></tr><tr><td></td><td>KD (Hinton et al., 2015)</td><td>73.33</td><td>72.98</td><td>73.54</td><td>70.67</td></tr><tr><td></td><td>KD+Ours</td><td>74.74(+1.41)</td><td>74.14(+1.16)</td><td>74.49(+0.95)</td><td>71.09(+0.42)</td></tr><tr><td></td><td>CTKD (Li et al., 2023)</td><td>73.39</td><td>73.52</td><td>73.93</td><td>70.99</td></tr><tr><td></td><td>CTKD+Ours</td><td>75.59(+2.2)</td><td>74.76(+1.24)</td><td>74.86(+0.93)</td><td>71.08(+0.09)</td></tr><tr><td></td><td>DKD (Zhao et al., 2022)</td><td>76.32</td><td>74.68</td><td>74.81</td><td>71.06</td></tr><tr><td></td><td>DKD+Ours</td><td>76.61(+0.29)</td><td>75.1(+0.42)</td><td>74.94 (+0.13)</td><td>71.84(+0.78)</td></tr><tr><td></td><td>MLKD (Jin et al., 2023)</td><td>77.08</td><td>75.18</td><td>75.35</td><td>71.89</td></tr><tr><td></td><td>MLKD+Ours</td><td>77.25(+0.17)</td><td>75.35(+0.17)</td><td>76.08(+0.73)</td><td>72.35(+0.46)</td></tr></table>
+
+on three datasets. 1) CIFAR-100 (Krizhevsky et al., 2009) is a significant dataset for image classification, comprising 100 categories, with 50,000 training images and 10,000 test images. 2) ImageNet (Russakovsky et al., 2015) is a large-scale dataset utilized for image classification, comprising 1,000 categories, with approximately 1.28 million training images and 50,000 test images. 3) MS-COCO (Lin et al., 2014) is a mainstream dataset for object detection comprising 80 categories, with 118,000 training images and 5,000 test images.
+
+Baselines. As a plug-and-play loss, we apply the proposed ranking loss to various logit-based methods, including KD (Hinton et al., 2015), CTKD (Li et al., 2023), DKD (Zhao et al., 2022), and MLKD (Jin et al., 2023), to verify whether it can bring performance gains to knowledge distillation methods. We also compare it with various feature-based methods, including FitNet (Adriana et al., 2015), CRD (Tian et al., 2019), and ReviewKD (Chen et al., 2021). Additionally, we compare it with other auxiliary losses and modified KL divergence methods, including DIST (Huang et al., 2022) and LSKD (Sun et al., 2024).
+
+Implementation Details. To ensure the robustness of the proposed plug-and-play loss without introducing excessive configurations, we maintain the same experimental settings as the baselines used (KD+Ours and KD share the same experimental setups for example). We set the batch size to 64 for CIFAR-100, 512 for ImageNet and 8 for COCO. We employ SGD (Sutskever et al., 2013) as the optimizer, with the number of epochs and learning rate settings consistent with the comparative baselines. The hyper-parameters  $\alpha$ ,  $\beta$
+
+in Eq. 6 are set to be the same as the compared baselines to maintain fairness, and  $\gamma$  are set equal to  $\alpha$ . We utilize 1 NVIDIA GeForce RTX 4090 to train models on CIFAR-100 and 4 NVIDIA GeForce RTX 4090 for training on ImageNet. The algorithm of our method can be found in Appendix A.4.
+
+Other Settings. In order to prevent the occasional gradient explosions at the start of optimization due to the too variant logits caused by random initialization of student, we add and only add normalization in ranking loss, which does not affect our fairness as a plug-and-play function. We conduct experiments about ablation of normalization in Appendix A.2, and the algorithm can be found in Appendix A.4.
+
+# 5.1. Main Result
+
+CIFAR-100 Results. In our study, we conduct a comparative analysis of Knowledge Distillation (KD) outcomes across various teacher-student (He et al., 2016; Simonyan & Zisserman, 2014; Zagoruyko & Komodakis, 2016; Zhang et al., 2018; Howard et al., 2017; Sandler et al., 2018) configurations. While Tab. 1 presents cases where the teacher and student models share the heterogeneous architecture, Tab. 2 illustrates instances of homogenous structures. Furthermore, as a plug-and-play method, we applied ranking loss in multiple logit-based distillation techniques. The incorporation of ranking loss resulted in an average improvement of  $1.83\%$  in vanilla KD. In the context of existing state-of-the-art (SOTA) logits-based methods, including DKD, CTKD, and MLKD, significant gains are made.
+
+ImageNet Results. The results on ImageNet of KD in terms of top-1 and top-5 accuracy are compared in Tab. 3.
+
+Table 3. The top-1 and top-5 accuracy  $(\%)$  on the ImageNet validation set. The teacher and student are ResNet50 and MN-V1  
+
+<table><tr><td></td><td>AT</td><td>OFD</td><td>CRD</td><td>KD</td><td>KD+Ours</td></tr><tr><td>Top-1</td><td>69.56</td><td>71.25</td><td>71.37</td><td>70.50</td><td>71.54(+1.04)</td></tr><tr><td>Top-5</td><td>89.33</td><td>90.34</td><td>90.41</td><td>89.80</td><td>90.84(+1.04)</td></tr></table>
+
+Table 4. Knowledge Distillation for Transformer. The Top-1 Accuracy (\%) on the validation set of CIFAR-100. The Teacher model is ResNet56.  
+
+<table><tr><td>Student</td><td>DeiT-Tiny 65.08</td><td>T2T-ViT-7 69.37</td><td>PiT-Tiny 73.58</td><td>PVT-Tiny 69.22</td></tr><tr><td>KD</td><td>71.11</td><td>71.72</td><td>74.03</td><td>72.46</td></tr><tr><td>KD+Ours</td><td>73.25(+2.14)</td><td>72.49(+0.77)</td><td>74.33(+0.30)</td><td>73.42(+0.96)</td></tr></table>
+
+Our proposed method can achieve consistent improvement on the large-scale dataset as well. Additional ImageNet experiments are provided in Appendix A.9
+
+# 5.2. Extensions
+
+Knowledge Distillation for Transformer. To validate the effectiveness of our plug-and-play ranking loss on ViT and to assess its performance when facing larger teacher-student structural differences, we conduct experiments using ViT students. The experimental results indicate that the incorporation of ranking loss yields significant improvements over the conventional knowledge distillation, as shown in Tab. 4. This denotes that our proposed method is also applicable to distillation challenges predicated on transformer architectures. The implementation details of the Transformer experiments are attached in the Appendix.
+
+Knowledge Distillation for Object Detection. To further verify the generality of our proposed method in downstream tasks, we also conduct experiments under the setting of object detection. The results show that our proposed ranking loss can improve the performance of knowledge distillation method in object detection and achieve better performance than the same period of the feature-based object detection method, which is shown in Tab. 5.
+
+Ablation of Weight and Steepness Parameter. We conduct extensive ablation studies to investigate the effectiveness of ranking loss under various settings of  $k$  and coefficients. Tab. 7 presents the distillation outcomes across different  $k$  configurations. It is observed that a larger  $k$  value yields superior performance, suggesting that the differential form of ranking loss more closely approximates the Kendall  $\tau$  coefficient, thereby imparting stronger ranking knowledge. Tab. 6 delineates the performance of ranking loss under varying coefficients. This setting effectively aligns the classification outcomes of the teacher and student models, thus significantly enhancing the distillation effect.
+
+Table 5. Knowledge Distillation for Object Detection. All experiments are conducted on COCO2017 with the teacher as ResNet50 and the student as MobileNet-V2.  
+
+<table><tr><td>Method</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>KD (Hinton et al., 2015)</td><td>30.13</td><td>50.28</td><td>31.35</td></tr><tr><td>FitNet (Adriana et al., 2015)</td><td>30.20</td><td>49.80</td><td>31.69</td></tr><tr><td>FGFI (Wang et al., 2019)</td><td>31.16</td><td>50.68</td><td>32.92</td></tr><tr><td>CTKD (Li et al., 2023)</td><td>31.39</td><td>52.34</td><td>31.35</td></tr><tr><td>KD+Ours</td><td>31.99(+1.86)</td><td>53.80(+3.52)</td><td>33.37(+2.02)</td></tr></table>
+
+Ablation of Temperature. There has been extensive and detailed research on the temperature of the KL loss (e.g., CTKD(Li et al., 2023), MLKD(Jin et al., 2023), LSKD(Sun et al., 2024)), achieving significant results. Unlike these studies, our approach aims to move away from focusing on the KL divergence and instead explore plug-and-play auxiliary losses to guide the KL divergence. Therefore, in our experiments, the temperature and other parameters of the KL divergence are kept the same as those in the respective baselines (with dynamic/multi-level temperatures used in CTKD+Ours and MLKD+Ours). The proposed ranking loss does not have a temperature parameter because the temperature does not affect the order of logits. Instead, the control over the distribution can be achieved by manipulating the steepness parameter  $k$  of the sign function, and ablation experiments for  $k$  are shown in Tab. 7. Meanwhile, although the temperature helps control the logit distribution, additional measures are needed to prevent the KL divergence from falling into suboptimal. We conducted an ablation study on the temperature, as shown in Tab. 8. The results demonstrate that ranking loss as a plug-and-play loss can bring further improvements under multiple temperature settings.
+
+Comparison between Ranking loss and KL Divergence. We separately evaluate distillation with KL divergence alone and with the proposed ranking loss alone; the results are presented in Tab. 9. We observe that using the ranking loss alone achieves performance comparable to using KL divergence alone, and that combining the two objectives yields further gains. These findings substantiate the efficacy of the proposed ranking loss.
+
+More Experiments. Additional experiments and discus  
+Table 6. Ablation of Weight. The Top-1 Accuracy (\%) on the validation set of CIFAR-100.  
+
+<table><tr><td rowspan="2">Teacher</td><td>ResNet32×4</td><td>WRN-40-2</td><td>ResNet32×4</td><td>ResNet50</td></tr><tr><td>79.42</td><td>75.61</td><td>79.42</td><td>79.34</td></tr><tr><td rowspan="2">Student</td><td>ResNet8×4</td><td>WRN-40-1</td><td>SHN-V2</td><td>MN-V2</td></tr><tr><td>72.50</td><td>71.98</td><td>71.28</td><td>64.60</td></tr><tr><td>KD(Baseline)</td><td>73.33</td><td>73.54</td><td>74.45</td><td>67.35</td></tr><tr><td>γ = 0.1</td><td>74.15</td><td>74.15</td><td>75.52</td><td>69.25</td></tr><tr><td>γ = 0.5</td><td>74.84</td><td>74.07</td><td>76.34</td><td>69.81</td></tr><tr><td>γ = 0.9</td><td>74.74</td><td>74.49</td><td>76.58</td><td>70.45</td></tr><tr><td>γ = 2</td><td>74.62</td><td>74.65</td><td>77.07</td><td>69.97</td></tr><tr><td>γ = 4</td><td>75.07</td><td>73.60</td><td>77.05</td><td>70.59</td></tr><tr><td>γ = 6</td><td>74.78</td><td>73.09</td><td>76.90</td><td>69.58</td></tr></table>
+
+![](images/1a4bc79df65bcf064177e6c12ee4924db67c7670d2f1b46400cfd7c9adc55175.jpg)  
+(a)
+
+![](images/c609a5c6dde8a7ffcb32c375246483af15254d0238fbfea6f721ac105f8f26a1.jpg)  
+(b)
+
+![](images/9b8b1c8f1638a40d26b04a2610d8ab4484bfe1efbb12abba84a9bea1f99577d2.jpg)  
+(c)  
+Figure 4. Ranking Loss using Top-K and Min-K channels. Top-1 Accuracy (\%) of Top-k/Min-k Ranking Loss. 0 and 100 on the x-axis represent KD and  $\mathrm{KD + Rank}$ , respectively.
+
+![](images/f5626e61056710a2a9862dab107e5eb24ff8ef7f90b8322ac5973a3556bdcc12.jpg)  
+(d)
+
+Table 7. Ablation of  $k$  . The Top-1 Accuracy (%) on the validation set of CIFAR-100.  
+
+<table><tr><td rowspan="2">Teacher</td><td>ResNet32×4</td><td>WRN-40-2</td><td>ResNet32×4</td><td>ResNet50</td></tr><tr><td>79.42</td><td>75.61</td><td>79.42</td><td>79.34</td></tr><tr><td rowspan="2">Student</td><td>ResNet8×4</td><td>WRN-40-1</td><td>SHN-V2</td><td>MN-V2</td></tr><tr><td>72.50</td><td>71.98</td><td>71.28</td><td>64.60</td></tr><tr><td>KD(Baseline)</td><td>73.33</td><td>73.54</td><td>74.45</td><td>67.35</td></tr><tr><td>k=0.1</td><td>73.13</td><td>73.75</td><td>75.47</td><td>68.9</td></tr><tr><td>k=0.5</td><td>74.36</td><td>74.17</td><td>75.73</td><td>68.87</td></tr><tr><td>k=1</td><td>74.74</td><td>74.49</td><td>76.58</td><td>70.45</td></tr><tr><td>k=2</td><td>74.79</td><td>74.87</td><td>77.06</td><td>70.53</td></tr><tr><td>k=4</td><td>75.56</td><td>74.48</td><td>77.21</td><td>70.81</td></tr><tr><td>k=6</td><td>75.74</td><td>74.11</td><td>76.11</td><td>70.32</td></tr></table>
+
+Table 8. Ablation of Temperature. All experiments are conducted on CIFAR-100 with the teacher as ResNet32×4 and the student as ResNet8×4.  
+
+<table><tr><td>Temperature</td><td>T = 4</td><td>T = 5</td><td>T = 6</td><td>T = 10</td></tr><tr><td>KD</td><td>73.33</td><td>73.39</td><td>73.43</td><td>73.55</td></tr><tr><td>KD+Ours</td><td>73.56(+0.23)</td><td>73.49(+0.10)</td><td>74.36(+0.93)</td><td>74.04(+0.49)</td></tr></table>
+
+sions, including Derivation of the KL Loss and Ranking Loss, Ablation of Normalization, and Different Forms of Ranking Loss, are provided in the Appendix. Please refer to the Sec. A for further details.
+
+# 5.3. Analysis
+
+Accuracy & Loss Curves with Ranking Loss. The accuracy and loss curves of KD and  $\mathrm{KD + Ours}$ , as shown in Fig. 5, demonstrate how ranking loss aids in optimization. The middle figure shows that the precise alignment of KL divergence also makes channel ranking more ordered, but adding ranking loss achieves a more consistent ranking more quickly. The right figure shows that in the early stages, ranking loss accelerates the reduction of KL loss and reduces its oscillation in suboptimal regions. In the later stages, ranking loss does not interfere with KL divergence and ultimately reaches a better position. The left figure shows that with ranking loss, student achieves leading accuracy at all stages. This indicates that the addition of the ranking loss helps
+
+Table 9. Experiments on Combining DIST.  
+
+<table><tr><td>Teacher
+Student</td><td>ResNet32×4
+SHN-V1</td><td>ResNet32×4
+WRN-40-2</td><td>WRN-40-2
+SHN-V1</td><td>VGG13
+VGG8</td><td>WRN-40-2
+WRN-40-1</td></tr><tr><td>KD (Only KL)</td><td>74.07</td><td>77.70</td><td>74.83</td><td>72.98</td><td>73.54</td></tr><tr><td>Only Ranking Loss</td><td>75.60</td><td>78.04</td><td>75.81</td><td>73.72</td><td>74.00</td></tr><tr><td>KL+Ranking Loss</td><td>75.98</td><td>78.50</td><td>76.13</td><td>74.14</td><td>74.49</td></tr></table>
+
+the model converge and achieve better generalization and performance.
+
+![](images/9a91756ddad704a85d5dbc34b4a9ec0a2c890967f41017546b168401ac42598f.jpg)  
+Figure 5. Accuracy & Loss Curves with Ranking Loss. Left: Top-1 Test Accuracy  $(\%)$  Curve. Middle: Loss Curve of Ranking Loss. Right: Loss Curve of KL Divergence.
+
+![](images/d245c456305a3385622a33ce1422237a6bcb0af57ee15fa11e42e83e903df9c3.jpg)
+
+![](images/896fc0fa70192e9c7d3677c208f3f42776f9e7f695d619ceb76eac15865d8db6.jpg)
+
+Visualization of Loss Landscape. To further investigate the role of ranking loss in the distillation process, we visualized the loss landscapes (Li et al., 2018) of student models with and without the application of ranking loss during Knowledge Distillation (KD), as depicted in Fig. 4. It is evident that the student models distilled with ranking loss exhibit a markedly flatter loss landscape and fewer local optima compared to those without it. We hypothesize that during the optimization process, ranking loss can filter out certain local optima that, despite presenting a better overall loss performance, yield poorer classification outcomes. Consequently, ranking loss can effectively enhance the generalization performance of student models.
+
+Top-k & Min-k Ranking. To further validate the beneficial knowledge present in low-probability channels, we conducted comparative experiments using the top  $10\%$ , top  $30\%$ , top  $50\%$ , and min  $10\%$ , min  $30\%$ , min  $50\%$  channels. The experiments were performed across four combinations of homogeneous and heterogeneous teacher-student pairs, and the results are presented in Fig. 4. We observed that
+
+![](images/1063357110aa39fc66b05247e8a921fe66e02e957cb3eb6e32c1dd8ce4ab5559.jpg)  
+Figure 6. Loss Landscape. Left: The left landscape shows the suboptimal solutions in the distillation task. Right: After adding our ranking loss, the suboptimal solution is significantly reduced, as shown on the right.
+
+![](images/fc0d853059063d6c6397e7f195d45f6a8cdf3adac73a30df72ce2940b52d43e3.jpg)  
+Figure 8. Sensitivity Analysis. Left: Sensitivity of  $\alpha -\gamma$  .Right: Sensitivity of  $\beta -\gamma$
+
+using the min-k channels achieves results similar to those obtained with top-k channels (as shown in Fig. 4(a),(b), and (c)). Additionally, in some cases, min-k channels provide even more beneficial information to aid student learning (as demonstrated in Fig. 4(d)), which indicates that the low-probability channels also contain rich knowledge.
+
+![](images/216767a3495b9689aac5eae3692cf815978d5df5af495e4d43602125133551ff.jpg)  
+Figure 7. Ranking Alignment Comparison. Left: KD. Right: KD+Ours.
+
+![](images/d8c4ac935dde8cde2e3bcaeb203f55aac2c389959d1f8da4b80b5798bd98e324.jpg)
+
+Further visualization of Ranking. We show in Fig. 3 that ranking loss balances the focus on high-probability and low-probability channels. To further illustrate the role of ranking loss, we present a visualization of the ranking results generated by KD and  $\mathrm{KD + Ours}$  in Fig. 7. It can be seen that after adding ranking loss, the channels are more ordered overall, which is more obvious in the low-probability channel part. This is because KL divergence does not pay attention to sorting and neglects low-probability channels. Although the large channels in KD receive more attention, the alignment value alone cannot achieve order consistency, thus losing some knowledge. The results indicate that our approach consistently enhances channel alignment, underscoring the robustness and general applicability of our method, particularly in its focused attention on low-probability channels.
+
+Further Ablation Study of the Hyperparameters. To further substantiate the generalization capability and robustness of our method, we conducted comprehensive ablation studies using different combinations of coefficients. As illustrated in Fig. 8, our method consistently maintains high performance across various settings, underscoring the strong generalization ability and universal applicability of our method.
+
+![](images/04f2fb0359b0bd23873e86dcd6fe7541b2e0bc80a3708d5764497efe1e7e0193.jpg)
+
+![](images/9dc4896f16479bb03589efd4bfbc722dfb4921ac44956d04152a8666167273fd.jpg)
+
+# 6. Conclusion
+
+In this paper, we investigate the optimization process of logit distillation and identify that the Kullback-Leibler divergence tends to overlook the knowledge embedded in the low-probability channels of the output. Moreover, KL divergence does not guarantee alignment between the classification results of the student and teacher models. To address this issue, we introduce a plug-and-play ranking loss based on Kendall's  $\tau$  Coefficient that encourages the student model to pay more attention to the knowledge contained within the low-probability channels, while also enforcing alignment with the teacher's predictive outcomes. We provide a theoretical analysis demonstrating that the gradients of the ranking loss are less affected by channel scale and that its optimization objective is consistent with that of KL divergence, making it an effective auxiliary loss for distillation. Extensive experiments validate that our approach significantly enhances the distillation performance across various datasets and teacher-student architectures.
+
+# Acknowledgements
+
+This work is supported by the National Key R&D Program of China (2022YFB4701400/4701402), SSTIC Grant (KJZD20230923115106012, KJZD20230923114916032, GJHZ20240218113604008), Beijing Key Lab of Networked Multimedia.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
+
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+
+# A. Appendix
+
+# A.1. Implementation Details for Transformer
+
+We use the AdamW optimizer and train for 300 epochs with an initial learning rate of 5e-4 and a weight decay of 0.05. The minimum learning rate is 5e-6, and the patch size is 16. We set  $\alpha = 1$ ,  $\beta = 1$ ,  $\gamma = 0.5$ , and batch size is 128. The value range of ranking loss is  $[-1, 1]$  We use a single RTX4090 for CIFAR-100 and 4 RTX4090 for ImageNet.
+
+# A.2. Ablation of Normalization
+
+Due to the random initialization of student, the output logits will be too variant at the start of optimization and occasionally lead to gradient explosions. Therefore, we add normalization to the ranking loss to stabilize the ranking loss optimization at the very beginning. We also supplemented a set of small-scale ablation experiments that showed that the improvement in ranking performance did not come from the normalization added to the ranking loss, as shown in Tab. 10 below:
+
+Table 10. Ablation of Normalization.  
+
+<table><tr><td rowspan="2">Teacher</td><td>ResNet32×4</td><td>WRN-40-2</td></tr><tr><td>79.42</td><td>75.61</td></tr><tr><td rowspan="2">Student</td><td>SHN-V1</td><td>WRN-40-1</td></tr><tr><td>70.30</td><td>71.98</td></tr><tr><td>KD</td><td>74.07</td><td>73.54</td></tr><tr><td>KD+Ours w/o Norm</td><td>76.38(+2.31)</td><td>74.07(+0.53)</td></tr><tr><td>KD+Ours w Norm</td><td>75.98(+1.91)</td><td>74.49(+0.95)</td></tr></table>
+
+# A.3. Different Forms of Ranking Loss
+
+In the initial application of the ranking loss(Zheng et al., 2023), it is necessary to compute the gradients for two input vectors separately. However, in the scenario of distillation, the soft labels from the teacher model do not require gradients. Therefore, we propose three variants of the diff-Kendall ranking loss suitable for distillation scenarios, aimed at further exploring the role of ranking loss in distillation. Since the sample pair  $(z_i^t - z_j^t)$  itself has a sign., we can derive an equivalent form from Eq. 5:
+
+$$
+\tau = \frac {\sum_ {i} \sum_ {j <   i} \operatorname {s g n} \left(\left(z _ {i} ^ {t} - z _ {j} ^ {t}\right) \cdot \left(z _ {i} ^ {s} - z _ {j} ^ {s}\right)\right)}{\frac {1}{2} C (C - 1)} \tag {13}
+$$
+
+For Eq. 5 and Eq. 13, we can similarly transform them into differential forms of ranking loss by replacing  $sgn$  with  $tanh$ , and gradients are not computed for the output part corresponding to the teacher, which is:
+
+$$
+L _ {\tau} ^ {f o r m 1} = \frac {\sum_ {i} \sum_ {j <   i} \tanh  \left(z _ {i} ^ {t} - z _ {j} ^ {t}\right) _ {\text {d e t a c h}} \cdot \tanh  \left(z _ {i} ^ {s} - z _ {j} ^ {s}\right)}{\frac {1}{2} C (C - 1)} \tag {14}
+$$
+
+$$
+L _ {\tau} ^ {f o r m 2} = \frac {\sum_ {i} \sum_ {j <   i} \tanh  \left[ \left(z _ {i} ^ {t} - z _ {j} ^ {t}\right) _ {\text {d e t a c h}} \cdot \left(z _ {i} ^ {s} - z _ {j} ^ {s}\right) \right]}{\frac {1}{2} C (C - 1)} \tag {15}
+$$
+
+Where  $(.)_{\text{detach}}$  means not participating in training. Further, since the gradient of the teacher's output is not required, we can directly use the sign function instead of the tanh function to obtain the ranking loss:
+
+$$
+L _ {\tau} ^ {f o r m 3} = \frac {\sum_ {i} \sum_ {j <   i} s g n \left(z _ {i} ^ {t} - z _ {j} ^ {t}\right) _ {\text {d e t a c h}} \cdot \tanh  \left(z _ {i} ^ {s} - z _ {j} ^ {s}\right)}{\frac {1}{2} C (C - 1)} \tag {16}
+$$
+
+Different Forms of Ranking Loss. In our investigation, we examined the performance of the three forms of distillation presented in Sec. A.3, as depicted in Tab. 11. Form1 exhibited the most superior performance, followed by Form2, with Form3 trailing yet still enhancing the efficacy of the original knowledge distillation. This suggests that within the ranking loss, it is imperative to optimize considering the magnitude of the teacher's logits differences as a coefficient for the loss, rather than merely optimizing as a sign function.
+
+Table 11. Different Forms of Ranking Loss. The experiments are conducted on the CIFAR-100, with 9 heterogeneous and 7 homogeneous architectures. The average Top-1 accuracy (%) is reported.  
+
+<table><tr><td>Loss Form</td><td>KD</td><td>KD+Form1</td><td>KD+Form2</td><td>KD+Form3</td></tr><tr><td>Similar Structure</td><td>72.01</td><td>73.47(+1.46)</td><td>73.42(+1.41)</td><td>73.38(+1.37)</td></tr><tr><td>Different Structure</td><td>72.74</td><td>73.50(+0.76)</td><td>73.36(+0.62)</td><td>73.05(+0.31)</td></tr></table>
+
+# A.4. Algorithm
+
+# Algorithm 1 Plug-and-Play Ranking Loss for Logit Distillation
+
+Input: Transfer set  $\mathcal{D}$  with samples of image-label pair  $\{x_{n},y_{n}\}_{n = 1}^{N}$ , base temperature  $T$ , teacher  $f_{t}$ , student  $f_{s}$ , knowledge distillation Loss  $\mathcal{L}_{\mathrm{KD}}$ , ranking Loss  $\mathcal{L}_{\mathrm{RK}}$ , the weight of ranking Loss  $\gamma$ .
+
+Output: Trained student model  $f_{s}$
+
+for  $(x_{n},y_{n})$  in  $\mathcal{D}$  do
+
+Get the logits of Teacher and student:  $z^t = f_t(x_n)$ ,  $z^s = f_s(x_n)$
+
+Calculate the probability with temperature:  $q^{t} = \text{softmax}\left(\frac{z^{t}}{T}\right)$ ,  $q^{s} = \text{softmax}\left(\frac{z^{s}}{T}\right)$ .
+
+Get the normalized logits of Teacher and student:  $\hat{z}^t = \frac{z^t - z^{\overline{t}}}{\text{std}(z^t)}$ ,  $\hat{z}^s = \frac{z^s - z^{\overline{s}}}{\text{std}(z^s)}$
+
+Update  $f_{s}$  towards minimizing:  $\mathcal{L}_{\mathrm{total}} = \mathcal{L}_{\mathrm{KD}}(q^t,q^s) + \gamma \cdot \mathcal{L}_{\mathrm{RK}}(\hat{z}^t,\hat{z}^s)$
+
+end
+
+# A.5. Derivation of the KL Loss and Ranking Loss
+
+# Derivation of the KL Divergence with Respect to Student Logits in Knowledge Distillation.
+
+Denote the teacher's logits as  $\mathbf{z}^t = [z_1^t, z_2^t, \ldots, z_C^t]$ .
+
+Denote the student's logits as  $\mathbf{z}^s = [z_1^s, z_2^s, \ldots, z_C^s]$ .
+
+Let  $T$  be the temperature scaling factor used in the softmax function.
+
+Teacher probabilities can be calculated as:
+
+$$
+q _ {i} ^ {t} = \frac {\exp \left(\frac {z _ {i} ^ {t}}{T}\right)}{\sum_ {j = 1} ^ {C} \exp \left(\frac {z _ {j} ^ {t}}{T}\right)}, \quad \text {f o r} i = 1, 2, \dots , C. \tag {17}
+$$
+
+Student probabilities can be calculated as:
+
+$$
+q _ {i} ^ {s} = \frac {\exp \left(\frac {z _ {i} ^ {s}}{T}\right)}{\sum_ {j = 1} ^ {C} \exp \left(\frac {z _ {j} ^ {s}}{T}\right)}, \quad \text {f o r} i = 1, 2, \dots , C. \tag {18}
+$$
+
+The loss function used in knowledge distillation is the scaled Kullback-Leibler (KL) divergence between the teacher and student probability distributions:
+
+$$
+L _ {\mathrm {K D}} = T ^ {2} \cdot \operatorname {K L} \left(q ^ {t} \| q ^ {s}\right) = T ^ {2} \sum_ {i = 1} ^ {C} q _ {i} ^ {t} \log \left(\frac {q _ {i} ^ {t}}{q _ {i} ^ {s}}\right). \tag {19}
+$$
+
+The derivative of the loss with respect to the student logits  $z_{i}^{s}$  is:
+
+$$
+\frac {\partial L _ {\mathrm {K D}}}{\partial z _ {i} ^ {s}} = - T ^ {2} \sum_ {k = 1} ^ {C} q _ {k} ^ {t} \frac {\partial \log q _ {k} ^ {s}}{\partial z _ {i} ^ {s}}. \tag {20}
+$$
+
+Taking the natural logarithm:
+
+$$
+\log q _ {k} ^ {s} = \frac {z _ {k} ^ {s}}{T} - \log \left(\sum_ {j = 1} ^ {C} \exp \left(\frac {z _ {j} ^ {s}}{T}\right)\right). \tag {21}
+$$
+
+Compute the partial derivative:
+
+$$
+\frac {\partial \log q _ {k} ^ {s}}{\partial z _ {i} ^ {s}} = \frac {\partial}{\partial z _ {i} ^ {s}} \left(\frac {z _ {k} ^ {s}}{T}\right) - \frac {\partial}{\partial z _ {i} ^ {s}} \left(\log \left(\sum_ {j = 1} ^ {C} \exp \left(\frac {z _ {j} ^ {s}}{T}\right)\right)\right). \tag {22}
+$$
+
+Compute each term separately:
+
+$$
+\frac {\partial}{\partial z _ {i} ^ {s}} \left(\frac {z _ {k} ^ {s}}{T}\right) = \frac {1}{T} \delta_ {i k}, \quad \delta_ {i k} = \left\{ \begin{array}{l l} 1, & \text {i f} i = k, \\ 0, & \text {i f} i \neq k. \end{array} \right. \tag {23}
+$$
+
+$$
+\left. \frac {\partial}{\partial z _ {i} ^ {s}} \left(\log \left(\sum_ {j = 1} ^ {C} \exp \left(\frac {z _ {j} ^ {s}}{T}\right)\right)\right) = \frac {1}{\sum_ {j = 1} ^ {C} \exp \left(\frac {z _ {j} ^ {s}}{T}\right)} \cdot \frac {\partial}{\partial z _ {i} ^ {s}} \left(\sum_ {j = 1} ^ {C} \exp \left(\frac {z _ {j} ^ {s}}{T}\right)\right). \right. \tag {24}
+$$
+
+The derivative inside the sum is:
+
+$$
+\frac {\partial}{\partial z _ {i} ^ {s}} \left(\sum_ {j = 1} ^ {C} \exp \left(\frac {z _ {j} ^ {s}}{T}\right)\right) = \exp \left(\frac {z _ {i} ^ {s}}{T}\right) \cdot \frac {1}{T} = \frac {\exp \left(\frac {z _ {i} ^ {s}}{T}\right)}{T}. \tag {25}
+$$
+
+Therefore, the second term becomes:
+
+$$
+\frac {1}{\sum_ {j = 1} ^ {C} \exp \left(\frac {z _ {j} ^ {s}}{T}\right)} \cdot \frac {\exp \left(\frac {z _ {i} ^ {s}}{T}\right)}{T} = \frac {1}{T} q _ {i} ^ {s}. \tag {26}
+$$
+
+Thus, the total derivative is:
+
+$$
+\frac {\partial \log q _ {k} ^ {s}}{\partial z _ {i} ^ {s}} = \frac {1}{T} \delta_ {i k} - \frac {1}{T} q _ {i} ^ {s} = \frac {1}{T} (\delta_ {i k} - q _ {i} ^ {s}). \tag {27}
+$$
+
+We now substitute  $\frac{\partial\log q_k^s}{\partial z_i^s}$  back into the expression for the derivative of the loss:
+
+$$
+\begin{array}{l} \frac {\partial L _ {\mathrm {K D}}}{\partial z _ {i} ^ {s}} = - T ^ {2} \sum_ {k = 1} ^ {C} q _ {k} ^ {t} \left(\frac {1}{T} \left(\delta_ {i k} - q _ {i} ^ {s}\right)\right) \tag {28} \\ = - T \sum_ {k = 1} ^ {C} q _ {k} ^ {t} \left(\delta_ {i k} - q _ {i} ^ {s}\right). \\ \end{array}
+$$
+
+Considering:
+
+$$
+\sum_ {k = 1} ^ {C} q _ {k} ^ {t} \delta_ {i k} = q _ {i} ^ {t}, \tag {29}
+$$
+
+$$
+\sum_ {k = 1} ^ {C} q _ {k} ^ {t} q _ {i} ^ {s} = q _ {i} ^ {s} \sum_ {k = 1} ^ {C} q _ {k} ^ {t} = q _ {i} ^ {s} \cdot 1 = q _ {i} ^ {s}, \quad \sum_ {k = 1} ^ {C} q _ {k} ^ {t} = 1. \tag {30}
+$$
+
+Therefore, the loss derivative simplifies to:
+
+$$
+\frac {\partial L _ {\mathrm {K D}}}{\partial z _ {i} ^ {s}} = - T \left(q _ {i} ^ {t} - q _ {i} ^ {s}\right). \tag {31}
+$$
+
+Derivation of the Ranking Loss with Respect to Student Logits in Knowledge Distillation.
+
+The Rank loss is calculated as:
+
+$$
+L _ {R K} = - \frac {\sum_ {i} \sum_ {j <   i} \tanh  \left(k \left(z _ {i} ^ {t} - z _ {j} ^ {t}\right)\right) \cdot \tanh  \left(k \left(z _ {i} ^ {s} - z _ {j} ^ {s}\right)\right)}{\frac {C (C - 1)}{2}} \tag {32}
+$$
+
+The derivation is calculated as:
+
+$$
+\begin{array}{l} \frac {\partial L _ {R K}}{\partial z _ {i} ^ {s}} = \frac {1}{\frac {C (C - 1)}{2}} \sum_ {j \neq i} \frac {\partial}{\partial z _ {i} ^ {s}} \frac {1}{2} \cdot \left[ \tanh  \left(k \left(z _ {i} ^ {s} - z _ {j} ^ {s}\right)\right) \tanh  \left(k \left(z _ {i} ^ {t} - z _ {j} ^ {t}\right)\right) \right] (33) \\ = \frac {1}{C (C - 1)} \sum_ {j \neq i} \frac {\partial}{\partial z _ {i} ^ {s}} \left[ \tanh  \left(k \left(z _ {i} ^ {s} - z _ {j} ^ {s}\right)\right) \tanh  \left(k \left(z _ {i} ^ {t} - z _ {j} ^ {t}\right)\right) \right] (34) \\ \end{array}
+$$
+
+Denote  $\phi_{ij}$  as:
+
+$$
+\phi_ {i j} = \tanh  \left(k \left(z _ {i} ^ {s} - z _ {j} ^ {s}\right)\right) \tanh  \left(k \left(z _ {i} ^ {t} - z _ {j} ^ {t}\right)\right) \tag {35}
+$$
+
+Its derivative w.r.t.  $z_i^s$  is:
+
+$$
+\frac {\partial \phi_ {i j}}{\partial z _ {i} ^ {s}} = \left[ 1 - \tanh ^ {2} \left(k (z _ {i} ^ {s} - z _ {j} ^ {s})\right) \right] \cdot k \cdot \tanh \left(k (z _ {i} ^ {t} - z _ {j} ^ {t})\right) \tag {36}
+$$
+
+Finally, the gradient of  $L_{RK}$  with respect to  $z_i^s$  is:
+
+$$
+\frac {\partial L _ {R K}}{\partial z _ {i} ^ {s}} = - \frac {k}{C (C - 1)} \sum_ {j \neq i} \left[ 1 - \tanh  ^ {2} \left(k \left(z _ {i} ^ {s} - z _ {j} ^ {s}\right)\right) \right] \tanh  \left(k \left(z _ {i} ^ {t} - z _ {j} ^ {t}\right)\right) \tag {37}
+$$
+
+# A.6. Derivation of the Differential Ranking Loss
+
+In this section, we explain how we get the final form of Eq. 6. Noticed that:
+
+$$
+\tanh  (x) = \frac {e ^ {x} - e ^ {- x}}{e ^ {x} + e ^ {- x}} = \frac {e ^ {2 x} - 1}{e ^ {2 x} + 1} = 1 - \frac {2}{e ^ {2 x} + 1} \tag {38}
+$$
+
+Then, we can get the final form of Eq. 6 by expanding the tanh function.
+
+$$
+\begin{array}{l} \tau_ {d} = \frac {\sum_ {i} \sum_ {j <   i} \tanh  \left(k \cdot \left(z _ {i} ^ {t} - z _ {j} ^ {t}\right)\right) \cdot \tanh  \left(k \cdot \left(z _ {i} ^ {s} - z _ {j} ^ {s}\right)\right)}{\frac {1}{2} C (C - 1)} (39) \\ = \frac {2}{C (C - 1)} \cdot \sum_ {i} \sum_ {j <   i} \left(1 - \frac {2}{1 + e ^ {2 \cdot \left(z _ {i} ^ {t} - z _ {j} ^ {t}\right) \cdot k}}\right) \cdot \left(1 - \frac {2}{1 + e ^ {2 \cdot \left(z _ {i} ^ {s} - z _ {j} ^ {s}\right) \cdot k}}\right) (40) \\ \end{array}
+$$
+
+# A.7. Does Temperature Solve the Problem of Ignoring Low-Probability Channels
+
+Some methods (Wei et al., 2025; Cheng et al., 2025) address the issue of neglecting low-probability channels by employing temperature scaling. We examined the probability distribution of LSKD (Sun et al., 2024), an approach using adaptive temperature that has shown promising results. As shown in Fig .9, while LSKD somewhat alleviates the problem of low-probability channels, the transformed probabilities still predominantly occupy these low-probability channels. Thus, using temperature scaling does not effectively resolve the issue of neglecting low-probability channels.
+
+![](images/784fe22ef35e9312af3b85c655bd1cf5d86505eb8d1952e550ca5e5790e7d334.jpg)  
+Figure 9. probability produced by KD and LSKD
+
+![](images/8f04980be1a652f8cc2a3919e850f5f4d9a8fece39b5ce2fe9a6f59c1b2c8ece.jpg)
+
+![](images/60d56a86375501faa214f10623e8e925bbe0b373b24ef440cb76a21e434505ad.jpg)
+
+# A.8. Combining with Other Correlation Coefficients.
+
+DIST (Huang et al., 2022) explores the use of the Pearson correlation coefficient for knowledge distillation. However, unlike our proposed method, the Pearson correlation coefficient measures the linear relationship between two variables. Our method focuses on the ranking relationship of logits, which is not strictly a linear relationship. Therefore, Kendall's  $\tau$  correlation coefficient may be more suitable for capturing this non-linear ranking information. To further investigate the combined effect of using both linear and non-linear auxiliary functions, we incorporate ranking loss into DIST, which improves its performance. The results are shown in Tab. 12.
+
+Table 12. Experiments on Combining DIST.  
+
+<table><tr><td>Teacher
+Student</td><td>ResNet32×4
+WRN-16-2</td><td>ResNet32×4
+WRN-40-2</td><td>WRN-40-2
+SHN-V1</td><td>VGG13
+VGG8</td><td>WRN-40-2
+WRN-40-1</td></tr><tr><td>DIST (Huang et al., 2022)</td><td>75.58</td><td>78.02</td><td>76.00</td><td>73.80</td><td>74.73</td></tr><tr><td>DIST+Ours</td><td>75.85(+0.27)</td><td>78.54(+0.52)</td><td>76.23(+0.23)</td><td>74.06(+0.26)</td><td>74.86(+0.13)</td></tr></table>
+
+# A.9. More ImageNet Results
+
+We further augment CTKD (Li et al., 2023) and NKD (Yang et al., 2023a) with the proposed ranking loss and evaluate them on the ImageNet dataset; the results are reported in Tab. 13.
+
+Table 13. The top-1 and top-5 accuracy (%) on the ImageNet validation set. The teacher and student are ResNet34 and ResNet18  
+
+<table><tr><td>Method</td><td>Top-1 Acc.</td><td>Top-5 Acc.</td></tr><tr><td>AT (Zagoruyko &amp; Komodakis, 2017)</td><td>70.69</td><td>90.01</td></tr><tr><td>OFD (Heo et al., 2019)</td><td>70.81</td><td>89.98</td></tr><tr><td>CRD (Tian et al., 2019)</td><td>71.17</td><td>90.13</td></tr><tr><td>CTKD (Li et al., 2023)</td><td>71.32</td><td>90.27</td></tr><tr><td>CTKD+Ours</td><td>71.68(+0.36)</td><td>90.30(+0.03)</td></tr><tr><td>NKD (Yang et al., 2023a)</td><td>71.96</td><td>90.27</td></tr><tr><td>NKD+Ours</td><td>72.06(+0.10)</td><td>90.49(+0.22)</td></tr></table>

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+# Enhancing Spectral GNNs: From Topology and Perturbation Perspectives
+
+Taoyang Qin<sup>1</sup> Ke-Jia CHEN<sup>1,2</sup> Zheng Liu<sup>1</sup>
+
+# Abstract
+
+Spectral Graph Neural Networks process graph signals using the spectral properties of the normalized graph Laplacian matrix. However, the frequent occurrence of repeated eigenvalues limits the expressiveness of spectral GNNs. To address this, we propose a higher-dimensional sheaf Laplacian matrix, which not only encodes the graph's topological information but also increases the upper bound on the number of distinct eigenvalues. The sheaf Laplacian matrix is derived from carefully designed perturbations of the block form of the normalized graph Laplacian, yielding a perturbed sheaf Laplacian (PSL) matrix with more distinct eigenvalues. We provide a theoretical analysis of the expressiveness of spectral GNNs equipped with the PSL and establish perturbation bounds for the eigenvalues. Extensive experiments on benchmark datasets for node classification demonstrate that incorporating the perturbed sheaf Laplacian enhances the performance of spectral GNNs.
+
+# 1. Introduction
+
+Both spatial and spectral Graph Neural Networks (GNNs) have shown outstanding performance in various graph tasks (Kipf & Welling, 2017; Gilmer et al., 2017; Huang et al., 2023; Jin et al., 2024). Unlike spatial GNNs, spectral GNNs conduct frequency-domain convolution on normalized Laplacian matrices of graphs. However, in most graph datasets, the multiplicity of eigenvalues of the normalized graph Laplacian matrices is not small. The repeated eigenvalues limit the expressive power of spectral GNNs, resulting in a deteriorated performance (Wang & Zhang, 2022; Lu et al., 2024).
+
+$^{1}$ School of Computer Science, Nanjing University of Posts and Telecommunications, Nanjing, China  $^{2}$ Jiangsu Key Laboratory of Big Data Security & Intelligent Processing, Nanjing, China. Correspondence to: Ke-Jia CHEN <chenkj@njupt.edu.cn>, Zheng Liu <zliu@njupt.edu.cn>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+![](images/820f71eb6480327264e6f9b857561e626166ddfdc24e7d8ed7948703244a2678.jpg)  
+Figure 1. The perturbed sheaf Laplacian  $\overline{\mathcal{L}_{\mathcal{F}}}$  can be viewed as a special perturbation of  $\mathcal{L}_{block}$  ( $\mathcal{L}$  is the normalized graph Laplacian,  $\otimes$  denotes the Kronecker product,  $I$  is the  $d \times d$  identity matrix, and all restriction maps are orthogonal). Specifically, for each  $(v,z) \in E$ , if  $Q_{vz}$  slightly deviates from the identity  $I$  while satisfying the sheaf Laplacian constraints, the resulting matrix is both a valid normalized sheaf Laplacian and a perturbed version of  $\mathcal{L}_{block}$ .
+
+Wang and Zhang (Wang & Zhang, 2022) highlight the issue of repeated eigenvalues and propose an orthogonal Jacobi basis approach in which each prediction dimension has a flexible filter design. However, the individual filter design does not fully resolve the performance degeneracy caused by different eigenvectors sharing the same eigenvalue. Lu et al. (Lu et al., 2024) tackle the issue by adding a diagonal matrix of uniformly sampled eigenvalues to the normalized Laplacian matrix. The polynomial filters integrated with the corrected eigenvalues yield improved node classification results. Although this method helps mitigate repeated eigenvalues, it also compromises the original topological information encoded in the normalized Laplacian matrix by reducing the operator's spectral gap, which reflects graph's connectivity and information flow (see Appendix F.2).
+
+Besides the above issues, both works focus only on linear settings, leaving the impact of nonlinear activation function unexplored.
+
+We focus on the challenge of repeated eigenvalues in a nonlinear context. We propose to utilize a higher dimensional sheaf Laplacian matrix, which guarantees a higher upper bound on the number of distinct eigenvalues (Hansen & Ghrist, 2018; Hansen, 2020). Specifically, we normalize the sheaf Laplacian matrix by applying perturbation on the block form of the normalized graph Laplacian matrix. The designed perturbation of the identity matrix serves as the restriction map, which increases the diversity of eigenvalue
+
+while retaining the topological information of the normalized Laplacian matrix (as shown in Figure 1). To summarize, our contributions are as follows:
+
+- We propose a perturbed sheaf Laplacian (PSL) matrix which has more distinct eigenvalues by carefully designing the perturbation on the basis of perturbation theory.  
+- We achieve optimal model performance by applying the PSL matrix on popular spectral GNNs architectures and optimizing the perturbation process.  
+- We theoretically analyze the expressiveness of the PSL-based GNNs, and characterize the impact of the perturbations by establishing the perturbation bounds for the eigenvalues.
+
+Experiments on benchmark datasets demonstrate the superiority of PSL-based spectral GNNs. The increased number of distinct eigenvalues of the learned perturbed sheaf Laplacian matrices validates the effectiveness of the proposed perturbation method.
+
+# 2. Preliminaries
+
+This section provides a brief introduction to spectral GNNs and highlights the key aspects of the theory of cellular sheaves. All important notations are given in Appendix A.
+
+# 2.1. Spectral Graph Neural Networks
+
+Spectral GNNs build on spectral graph theory (Chung, 1997), which applies the graph Fourier transform to define frequency-domain convolutions. For a graph  $G = (V,E)$  with  $n$  nodes, the normalized Laplacian is  $\mathcal{L} = I - D^{-\frac{1}{2}}AD^{-\frac{1}{2}}$ . Its eigendecomposition is  $\mathcal{L} = U\Lambda U^T$ , where  $\Lambda$  is diagonal and  $U$  holds the eigenvectors. The graph Fourier transform of a signal  $X \in \mathbb{R}^{n \cdot f}$  is  $\hat{X} = U^T X$ , with each row of  $\hat{X}$  representing a distinct frequency component. The output  $Z$  of a spectral convolution can be expressed as:
+
+$$
+Z = \mathbf {g} * _ {G} X = U \left(\left(U ^ {T} \mathbf {g}\right) \odot \left(U ^ {T} X\right)\right) = U \hat {G} U ^ {T} X \tag {1}
+$$
+
+where  $\mathbf{g}$  is the spatial kernel, and  $\hat{G} = \mathrm{diag}(\hat{\mathbf{g}}_1,\dots ,\hat{\mathbf{g}}_n)$  denotes the spectral kernel coefficients. For spectral GNNs, a polynomial function is commonly used to approximate different kernels, which can be expressed as follows:
+
+$$
+\begin{array}{l} Z = \varphi (U g (\Lambda) U ^ {T} X W) \\ = \varphi (U \left(\sum_ {k = 0} ^ {K} \alpha_ {k} \Lambda^ {k}\right) U ^ {T} X W) \tag {2} \\ = \varphi (\sum_ {k = 0} ^ {K} \alpha_ {k} \mathcal {L} ^ {k} X W), \\ \end{array}
+$$
+
+where  $g$  is the filter function,  $W$  is a learnable weight matrix,  $\alpha_{k}$  is a trainable coefficient for the  $k$ -th order polynomial approximation, and  $\varphi$  is a function like multi-layer perceptrons (MLPs).
+
+Note that, if  $\varphi$  is a linear function, spectral GNNs are termed linear spectral GNNs; if  $\varphi$  is a nonlinear function, they are termed nonlinear spectral GNNs.
+
+# 2.2. Theory of Cellular Sheaves
+
+The theory of cellular sheaves, rooted in algebraic topology, has been incorporated into GNNs to additionally augment the graph's topological structure and thereby improve the performance of GNNs (Bodnar et al., 2022; Duta et al., 2023a).
+
+Definition 2.1. A cellular sheaf  $(G,\mathcal{F})$  over an undirected graph  $G = (V,E)$  consists of:
+
+- A vector space  $\mathcal{F}(v)$  associated with each node  $v \in V$ .  
+- A vector space  $\mathcal{F}(e)$  associated with each edge  $e\in E$  
+- A linear map  $\mathcal{F}_{v\leq e}$  for each incident node-edge pair  $v\leq e$  , mapping vectors from  $\mathcal{F}(v)$  to  $\mathcal{F}(e)$
+
+Here, the vector spaces for nodes and edges are termed stalks, and the linear maps between them are termed restriction maps. We set all the stalks to be  $\mathbb{R}^d$  and confine all restriction maps to orthogonal mappings. The restriction map  $\mathcal{F}_{v\leq e}:\mathcal{F}(v)\to \mathcal{F}(e)$  maps vectors from node stalk to edge stalk and  $\mathcal{F}_{v\leq e}^{\top}:\mathcal{F}(e)\to \mathcal{F}(v)$  maps vectors from edge stalk to node stalk. It follows that  $\mathcal{F}_{u\leq (v,u)}^{\top}\mathcal{F}_{v\leq (v,u)}$  maps vectors from  $\mathcal{F}(v)$  to  $\mathcal{F}(u)$ , where  $(v,u)\in E$ .
+
+Definition 2.2. The sheaf Laplacian matrix of a sheaf over an undirected graph  $G = (V,E)$  is a block matrix  $\mathcal{L}_{\mathcal{F}}$ . The diagonal blocks are  $\mathcal{L}_{\mathcal{F},vv} = \sum_{v\leq e}\mathcal{F}_{v\leq e}^{\top}\mathcal{F}_{v\leq e}$ , while the non-diagonal blocks are  $\mathcal{L}_{\mathcal{F},vu} = -\mathcal{F}_{v\leq e}^{\top}\mathcal{F}_{u\leq e}$ , where  $(v,u)\in E$ .
+
+The sheaf Laplacian matrix is positive semidefinite, which means that the eigenvalues are non-negative (Hansen, 2020; Bodnar et al., 2022). Additionally, when all the restriction maps are identity maps, the sheaf Laplacian matrix degenerates into the block form of the graph Laplacian matrix.
+
+Definition 2.3. The normalized form of the sheaf Laplacian matrix is given by  $\overline{\mathcal{L}_{\mathcal{F}}} = D^{-\frac{1}{2}}\mathcal{L}_{\mathcal{F}}D^{-\frac{1}{2}}$ , where  $D_v = \mathcal{L}_{\mathcal{F},vv}$  and the diagonal matrix of the sheaf Laplacian matrix is  $D = \mathrm{diag}(D_1, \ldots, D_n)$ .
+
+In the following, we will refer to the normalized sheaf Laplacian matrix simply as the sheaf Laplacian.
+
+![](images/c2c04f04726583c0214c6edce2e0e6baabd61892b1743e947653038ad4038643.jpg)  
+(a) Limited predictive ability.
+
+![](images/07b3488a581aef0f0fa1899bdeadf99e019e570e33cabf6d14bc9be95232eecc.jpg)  
+(b) Loss of frequency components.  
+Figure 2.  $X$  is the graph data, and  $\hat{X}_{\lambda}$  its Fourier-transformed matrix.  $g$  is the filter function. (a) Because the graph has only two distinct nonzero eigenvalues, each element in  $Z$  is a linear combination of just two types of frequency components, thereby limiting the expressive power of spectral GNNs. (b) Poorly designed filter functions can lead to the loss of frequency components, thereby limiting the expressive power of spectral GNNs.
+
+# 3. Motivation
+
+In this section, we summarize and extend the discussion of repeated eigenvalue issues as presented by Wang and Zhang (Wang & Zhang, 2022) and Lu et al. (Lu et al., 2024), which contribute to the motivation for our work.
+
+# 3.1. Limited Predictive Ability
+
+The phenomenon of repeated eigenvalues limits the predictive ability of spectral GNNs, thus affecting their expressive power (as shown in Figure 2 (a)). Firstly, we use a lemma to describe this issue.
+
+Lemma 3.1. When there are only  $k$  distinct eigenvalues of the normalized Laplacian matrix, linear spectral GNNs can produce at most  $k$  different filter coefficients, and thus can only generate one-dimensional predictions with a maximum of  $k$  arbitrary elements.
+
+The proof of this lemma can be found in Appendix A.1 of Lu et al. (Lu et al., 2024). This lemma reveals a key problem: the predictive ability of linear Spectral GNNs is limited by the number of distinct eigenvalues of the normalized graph Laplacian matrix.
+
+In this paper, we extend the lemma to the case of nonlinear spectral GNNs which use an activation function (Rosenblatt, 1958; Nair & Hinton, 2010; Goodfellow et al., 2016).
+
+Theorem 3.2. When there are only  $k$  distinct eigenvalues of the normalized Laplacian matrix, and the nonlinear real-valued function  $\sigma$  satisfies  $\sigma(x) \neq 0$  for  $x \neq 0$ , nonlinear spectral GNNs can produce at most  $k$  different filter coefficients, and thus can only generate one-dimensional predictions with a maximum of  $k$  arbitrary elements.
+
+Theorem 3.2 shows that, even with real-valued activation functions such as tanh and sigmoid, the predictive ability of nonlinear spectral GNNs remains equivalent to that of linear spectral GNNs, both of which are similarly limited. This limitation arises from the issue of repeated eigenval-
+
+ues of the normalized graph Laplacian matrix. The proof is in Appendix B.
+
+# 3.2. Loss of Frequency Components
+
+Another issue caused by repeated eigenvalues is that if the filter function  $g(\lambda)$  is poorly designed, setting a specific eigenvalue to zero can lead to the disappearance of corresponding frequency components (As shown in Figure 2 (b)).
+
+Interestingly, for many commonly used graph datasets, the eigenvalue 1 of the corresponding normalized graph Laplacian matrix has the highest multiplicity (see Appendix F.5 for details). This observation may also provide a new perspective on why adding self-loops to the adjacency matrix leads to better GCN performance on certain datasets (Kipf & Welling, 2017; Wu et al., 2019). Given that the filter function of GCN is  $1 - \lambda$ , it filters out all frequency components associated with the eigenvalue 1. By introducing self-loops, the distribution of eigenvalues is altered, thereby causing GCN to lose fewer frequency components.
+
+# 3.3. Restricted Frequency Processing Capability
+
+The phenomenon of repeated eigenvalues also restricts the frequency processing capability of spectral GNNs. Consider  $\hat{X} = U^T X W$ , where  $g(\Lambda)$  represents the filtering function, with  $\Lambda$  being the eigenvalue matrix of the normalized Laplacian. Evidently,  $g(\Lambda) \hat{X}$  is equivalent to  $g(\lambda_i) \odot \hat{X}_i$ , where  $g(\lambda_i)$  scales each  $\hat{X}_i$  element-by-element. Thus, for identical eigenvalues  $\lambda_i = \lambda_j$ , the associated frequency components  $\hat{X}_i$  and  $\hat{X}_j$  are scaled by the same factor, which limits the expressiveness of the spectral GNNs.
+
+# 3.4. Why Perturbed Sheaf Laplacian
+
+The solution to the above issues is to construct a Laplacian matrix with more distinct eigenvalues. The sheaf Lapla
+
+![](images/8b25e717e5b1e0e3de9b3c78aeae5fa38c9cc2b3b3ebbb25710d56031479d069.jpg)  
+Figure 3. Model workflow: The convert operation initially reduces feature dimensionality and reshapes the input to  $X \in \mathbb{R}^{nd \times k}$ . The  $l$ -th aggregation step,  $\mathrm{avg}^{(l)}$ , is performed as  $\mathrm{Combine}(\overline{\mathcal{L}_{\mathcal{F}}}^{(l)}, X^{(l)}, W^{(l)})$ . After final aggregation, and the convert operation means reshaping the output back to  $n$  rows of  $d \times k$  columns, followed by a linear transformation to yield  $Z$  with columns equal to the number of node classes.
+
+cian is from the theory of cellular sheaves. When applied to graphs, the sheaf Laplacian naturally provides a larger set of eigenvalues while maintaining a positive semidefinite nature and the eigenvalue range in [0, 2] (Hansen, 2020).
+
+We carefully design the restriction map as a slight perturbation of the identity matrix in order to construct a sheaf Laplacian with more distinct eigenvalues. We call it the perturbed sheaf Laplacian (PSL). The perturbed sheaf Laplacian matrix achieves two advantages: (1) By aligning with perturbation theory, the PSL matrix increases the eigenvalue diversity; (2) the PSL matrix is structurally close to the normalized graph Laplacian matrix, retaining its topological information.
+
+# 4. Method
+
+This section introduces the proposed PSL-GNN model (with its general workflow illustrated in Figure 3). The complexity of our method is illustrated in Appendix F.1.
+
+# 4.1. Preprocessing of Graph Data
+
+To make the first dimension of the input matrix  $X_0 \in \mathbb{R}^{n \times f_1}$  compatible with the second dimension of the sheaf Laplacian matrix  $\overline{\mathcal{L}_{\mathcal{F}}} \in \mathbb{R}^{nd \times nd}$ , we first apply a linear projection:
+
+$$
+X _ {0} W _ {1} \in \mathbb {R} ^ {n \times d f _ {3}}, \tag {3}
+$$
+
+where  $W_{1}\in \mathbb{R}^{f_{1}\times f_{2}}$  with  $f_{2} = d\times f_{3}$ . Next, we reshape the resulting matrix:
+
+$$
+X = \mathcal {R} \left(X _ {0} W _ {1}, \mathbb {R} ^ {n d \times f _ {3}}\right), \tag {4}
+$$
+
+where  $\mathcal{R}(X_0W_1,\mathbb{R}^{nd\times f_3})$  represents reshaping  $X_0W_1$  into a matrix with nd rows and  $f_{3}$  columns.
+
+# 4.2. Construction of Perturbed Sheaf Laplacian
+
+When there exists a restriction map that is not an identity map,  $\overline{\mathcal{L}_{\mathcal{F}}}$  can be viewed as the result of a special perturbation applied to the block form of the normalized graph Laplacian matrix  $\mathcal{L}_{\mathrm{block}}$ . For each block element  $\mathcal{L}_{\mathrm{block},ij}$ , the corresponding perturbation matrix  $Q_{ij} = \mathcal{F}_{i\triangleleft e}^{\top}\mathcal{F}_{j\triangleleft e}$  is applied, resulting in  $\overline{\mathcal{L}_{\mathcal{F}}}$ . According to Section 3.4, when  $\mathcal{F}_{i\triangleleft e}^{\top}\mathcal{F}_{j\triangleleft e}$  slightly deviates from the identity matrix, the corresponding sheaf Laplacian matrix is regarded as a perturbed sheaf Laplacian matrix.
+
+Based on this,  $Q_{ij}$  can be viewed as a perturbation applied to  $L_{\mathrm{block},ij}$ . To ensure that the overall perturbation effect increases the number of distinct eigenvalues of  $\overline{\mathcal{L}_{\mathcal{F}}}$ , we propose the following theorem to guide the perturbations.
+
+Theorem 4.1. Let  $P = \overline{\mathcal{L}_{\mathcal{F}}} - \mathcal{L}_{block}$  be the perturbation matrix applied to  $\mathcal{L}_{block}$ , and let  $\phi = \min_{i,j; i \neq j} |\lambda_i - \lambda_j|$  where  $\lambda_i$  are the eigenvalues of  $\mathcal{L}_{block}$ . If  $\| P \|_2 < \frac{\phi}{2}$ , then the multiplicity of eigenvalues of  $\mathcal{L}_{\mathcal{F}}$  will decrease.
+
+The proof is in Appendix B. According to Theorem 4.1, to reduce the multiplicity of eigenvalues, the orthogonal matrices of different blocks should be restricted to slightly deviate from the identity matrices. Considering that the reflection matrix can deviate from the identity matrix to varying extents by adjusting the reflection vector, we constrain every restriction map to the reflection matrix. Specifically, we assign every restriction map to  $\mathcal{F}_{i\leq (i,j)} = I - 2\frac{\mathbf{u}_{ij}\mathbf{u}_{ij}^{\top}}{\|\mathbf{u}_{ij}\|^{2}}$  where  $\mathbf{u}_{ij}$  is a vector of dimensions  $d$  ( $d$  represents the dimension of the stalks). Here,  $\mathbf{u}_{ij}$  is parameterized and learnable, allowing  $Q_{ij}$  to deviate from the identity matrix. To ensure that the applied perturbations remain minimal, we constrain each component of  $\mathbf{u}_{ij}$  within  $[\frac{\eta}{10},\eta ]$  for the reflection matrix, where  $\eta$  is a sufficiently small value.
+
+# 4.3. PSL-GNN: Perturbed Sheaf Laplacian-Based Graph Neural Network
+
+The proposed perturbed sheaf Laplacian matrix can be integrated with any spectral GNNs. In this section, we apply it within the GCN (Kipf & Welling, 2017) and GPRGNN (Chien et al., 2021) respectively as examples.
+
+# 4.3.1.PSL-GCN
+
+The architecture of GCN is often represented as:
+
+$$
+X _ {0} ^ {(l + 1)} = \sigma ((I - \Delta) X _ {0} ^ {(l)} W), \tag {5}
+$$
+
+where  $\Delta$  represents the normalized graph Laplacian matrix,  $l$  denotes the layer index, and  $X^{(l)}$  represents the feature matrix at layer  $l$ . Through integration, we obtain PSL-GCN, which can be expressed as follows:
+
+$$
+X ^ {(l + 1)} = \sigma \left(\left(I - \Delta_ {\mathcal {F}}\right) X ^ {(l)} W _ {2}\right), \tag {6}
+$$
+
+where  $X^{(l + 1)}\in \mathbb{R}^{nd\times f_4}$  and  $X^{(l)}\in \mathbb{R}^{nd\times f_3}$ .  $\Delta_{\mathcal{F}}\in$ $\mathbb{R}^{nd\times nd}$  is the PSL matrix, and  $W_{2}\in \mathbb{R}^{f_{3}\times f_{4}}$  is the parameter matrix used for linear transformations within the network.
+
+We need to reshape the output of the last layer and perform a series of transformations such as  $W_{last} \in \mathbb{R}^{df_{last} \times c}$  and  $\mathcal{R}$  to obtain the final prediction matrix  $Z$ :
+
+$$
+Z = \mathcal {R} \left(X ^ {(l a s t)}, \mathbb {R} ^ {n \times d f _ {l a s t}}\right) W _ {l a s t}. \tag {7}
+$$
+
+# 4.3.2. PSL-GPR
+
+The structure of GPRGNN is represented as follows:
+
+$$
+Z = \sum_ {k = 0} ^ {K} \alpha_ {k} (I - \mathcal {L}) ^ {k} X _ {0}, \tag {8}
+$$
+
+where  $X_0 \in \mathbb{R}^{n \times f_1}$  represents the feature matrix, and  $\mathcal{L}$  represents the normalized graph Laplacian matrix. According to Equation (4), we have the converted feature matrix  $X \in \mathbb{R}^{nd \times f_3}$ . Following the architecture of GPRGNN, our model, namely PSL-GPR, is expressed as:
+
+$$
+Y = \sum_ {k = 0} ^ {K} \alpha_ {k} \left(I - \overline {{\mathcal {L} _ {\mathcal {F}}}}\right) ^ {k} X. \tag {9}
+$$
+
+In this equation,  $Y \in \mathbb{R}^{nd \times f_3}$  represents the output feature matrix,  $I \in \mathbb{R}^{nd \times nd}$  is the identity matrix,  $\overline{\mathcal{L}_{\mathcal{F}}} \in \mathbb{R}^{nd \times nd}$  is the PSL matrix,  $K$  is the order of the polynomial, and  $\alpha_k$  are the trainable coefficients.
+
+We reshape  $Y$  into  $\mathbb{R}^{n \times df_3}$  and apply a linear transformation with  $W_3 \in \mathbb{R}^{df_3 \times c}$  to obtain the final prediction matrix  $Z \in \mathbb{R}^{n \times c}$ , where  $c$  represents the number of classes. This process can be expressed as:
+
+$$
+Z = \mathcal {R} (Y, \mathbb {R} ^ {n \times d f _ {3}}) W _ {3}. \tag {10}
+$$
+
+# 5. Theoretical Analysis
+
+In this section, we conduct a theoretical analysis of the expressive power of PSL-GNN and establish the perturbation bounds of the eigenvalues. All proofs are in Appendix C and Appendix D.
+
+# 5.1. Expressiveness of PSL-GNN
+
+We first discuss the expressive power of the block matrix form  $\mathcal{L}_{\mathrm{block}}$  of the normalized graph Laplacian matrix  $\mathcal{L}$ .
+
+Proposition 5.1. The number of distinct eigenvalues of  $\mathcal{L}_{block} \in \mathbb{R}^{nd \times nd}$  is the same as that of  $\mathcal{L} \in \mathbb{R}^{n \times n}$ , where  $\mathcal{L}$  is the normalized graph Laplacian matrix and  $\mathcal{L}_{block} = \mathcal{L} \otimes I$ , with  $I \in \mathbb{R}^{d \times d}$  being the identity matrix. Also, their corresponding GNNs, which share the same architecture, are mutually equivalent.
+
+Proposition 5.1 indicates that when the applied orthogonal matrices are the identity matrices, PSL-GNN is equivalent to the corresponding GNN.
+
+Corollary 5.2. PSL-GNN has stronger one-dimensional prediction ability.
+
+For a given normalized graph Laplacian matrix, its PSL possesses a greater number of distinct eigenvalues, which underpins its enhanced predictive ability.
+
+Corollary 5.3. PSL-GNN loses fewer frequency components.
+
+With fewer repeated eigenvalues than the normalized Laplacian, PSL-GNN remains robust even if high-multiplicity frequency components are discarded.
+
+Corollary 5.4. PSL-GNN has a stronger capability in processing frequency components.
+
+The PSL possesses a greater number of distinct eigenvalues, allowing a more diverse treatment of frequency components.
+
+# 5.2. Perturbation Bounds of Eigenvalues
+
+We establish the perturbation bounds for the eigenvalues based on perturbation theory (Weyl, 1912; Dailey et al., 2014).
+
+Theorem 5.5. Let  $\mathcal{L}_{\text{block}}$  and  $\mathcal{L}_{\mathcal{F}} = \mathcal{L}_{\text{block}} + P$  be  $N \times N$  symmetric positive semidefinite matrices, with eigenvalues  $\lambda_1, \ldots, \lambda_N$  and  $\tilde{\lambda}_1, \ldots, \tilde{\lambda}_N$ , respectively, where  $\mathcal{L}_{\text{block}}$  is diagonally dominant. Suppose the perturbation matrix  $P$  satisfies  $|P_{ij}| \leq \epsilon |(L_{\text{block}})_{ij}|$  for all  $i \neq j$  for all  $i$ , where  $0 < \epsilon < \frac{1}{2}$ . Hence, for each  $i$ ,  $|\tilde{\lambda}_i - \lambda_i| \leq \frac{2\epsilon}{1 - 2\epsilon} \tilde{\lambda}_{\max}$ .
+
+As the size of the perturbation  $\eta$  decreases, the corresponding  $\epsilon$  also decreases, resulting in a smaller range of eigenvalue perturbations.
+
+# 6. Experiment
+
+In this section, we give the results of the performance comparison between PSL-GNN and GNN baselines on the node classification task. Besides, we validate the effectiveness of the proposed perturbed sheaf Laplacian matrix through ablation studies. We also perform a statistical analysis of the independent eigenvalues of the final learned perturbed sheaf Laplacian matrices. The experimental setup is in Appendix E. Additional experimental studies can be found in Appendix F.
+
+# 6.1. Evaluation on Real-World Datasets
+
+We conduct experiments on 7 benchmark graph datasets (all information about the datasets is summarized in Appendix E.2.), and the results are presented in Table 1. We evaluate the accuracy of the PSL-GNNs by comparing them with their corresponding GNN models. We also calculate the vertical average gain (VAG) of the perturbed sheaf Laplacian matrix for all models on each dataset. The formula for VAG is given as follows:
+
+$$
+V A G = \frac {1}{| \mathcal {M} |} \sum_ {m _ {i} \in \mathcal {M}} \left(\operatorname {A C C} \left(m _ {i} (\mathcal {F})\right) - \operatorname {A C C} \left(m _ {i} (G)\right)\right), \tag {11}
+$$
+
+where  $\mathcal{M}$  is the set of GNN models, ACC is the short form of accuracy,  $G$  is the original graph, and  $\mathcal{F}$  is a sheaf over the graph. We also calculate the horizontal average gain (HAG) of the perturbed sheaf Laplacian matrix for all datasets on each model. The formula for HAG is given as follows:
+
+$$
+H A G = \frac {1}{| \mathcal {D} |} \sum_ {d _ {i} \in \mathcal {D}} \left(\operatorname {A C C} \left(d _ {i} (\mathcal {F})\right) - \operatorname {A C C} \left(d _ {i} (G)\right)\right), \tag {12}
+$$
+
+where  $\mathcal{D}$  is the set of used datasets.
+
+By analyzing the results in Table 1, we have the following observations: (1) All models are enhanced by the perturbed sheaf Laplacian matrices. This improvement can be attributed to the fact that the perturbed sheaf Laplacian matrices have more distinct eigenvalues. Consequently, the PSL-GNNs have a stronger capability to handle various frequency components, thereby improving their performance in the node classification task; (2) For Texas and Cornell, which are characterized by a relatively small number of nodes, the performance improvement is particularly significant. The observed phenomenon can be attributed to the inherently lower structural stability of smaller graphs, which makes them more sensitive to perturbations. This heightened sensitivity facilitates the splitting of eigenvalues into distinct values, thereby significantly increasing the number of distinct eigenvalues; (3) The GCN shows the highest average gain compared to other baselines. This can be explained from the perspective of suppressing the loss
+
+of frequency components: Unlike the variable filter functions in other baselines, the GCN filter function is fixed, i.e.,  $g(\lambda) = 1 - \lambda$ , where  $\lambda$  represents the eigenvalue of the normalized graph Laplacian matrix. The multiplicity of  $\lambda = 1$  is relatively high (see Figure 5 for details), which leads to a significant loss of frequency components corresponding to  $\lambda = 1$ , thereby severely affecting the expressive power of spectral GNNs. In contrast, the perturbed sheaf Laplacian can reduce the multiplicity of  $\lambda = 1$ , resulting in a higher average gain.
+
+# 6.2. Ablation Study
+
+To further study the effectiveness of the perturbed sheaf Laplacian matrix, we compare it with both the general sheaf Laplacian matrix (GSL) (Bodnar et al., 2022) and the normalized graph Laplacian matrix.
+
+The construction of the GSL involves using  $\frac{d(d - 1)}{2}$  parameters to generate each restriction map  $\mathcal{F}_{v\leq e}$ . Besides, each parameter is no longer constrained by  $\eta$ . Based on the above setup, the restriction maps are allowed to fully explore the orthogonal group. For the construction of normalized graph Laplacian matrix, we set  $\eta = 0$ , which means no perturbation is applied. Consequently, the perturbed sheaf Laplacian matrix degenerates into the block matrix form of the normalized graph Laplacian matrix. We integrate these three types of Laplacian into vanilla GNN, creating PSL-GNN, GSL-GNN, GNN (vanilla). Subsequently, we compare their performance on node classification. All models share the same parameters, except  $\eta$ , which is tuned for PSL-GPR and PSL-GCN ( $\{1e - 1, 1e - 2, 1e - 3, 1e - 4\}$ ) and set to 0 for GPR-GNN and GCN. This ensures fair and reliable comparisons.
+
+We conduct multiple training runs and averaged various metrics to obtain the reliable results in Table 2. The results demonstrate that: (1) GSL-GNN outperforms vanilla GNN, probably because the general sheaf Laplacian generalizes the graph Laplacian, offering greater learnability and flexibility for downstream tasks; (2) The performance of PSL-GNN is comparable to that of GSL-GNN. It indicates that the structure learned by the general sheaf Laplacian might align closely with that learned by the perturbed sheaf Laplacian.
+
+# 6.3. The Number of Distinct Eigenvalues
+
+We measure the number of distinct eigenvalues for both the normalized graph Laplacian matrix and the final learned perturbed sheaf Laplacian matrix when  $\eta \geq 1\mathrm{e} - 3$ , respectively. In Table 3,  $N_{G}$  denotes the number of distinct eigenvalues of the normalized graph Laplacian matrix, while the other entries represent the number of distinct eigenvalues of the learned perturbed sheaf Laplacian matrix. As the
+
+Table 1. Experimental results on node classification task: mean accuracy  $(\%)\pm 95\%$  confidence interval.  
+
+<table><tr><td>Datasets</td><td>Cora</td><td>Pubmed</td><td>Citeseer</td><td>Photo</td><td>Cornell</td><td>Texas</td><td>Actor</td><td>HAG</td></tr><tr><td>GCN</td><td>86.62±1.06</td><td>86.51±0.47</td><td>75.32±1.91</td><td>87.52±0.58</td><td>75.32±1.98</td><td>78.12±1.87</td><td>34.23±1.54</td><td rowspan="2">1.93↑</td></tr><tr><td>PSL-GCN</td><td>88.09±1.75</td><td>88.21±2.43</td><td>76.75±2.32</td><td>90.12±0.90</td><td>77.89±2.45</td><td>80.45±2.21</td><td>35.67±1.83</td></tr><tr><td>APPNP</td><td>88.11±1.35</td><td>87.39±0.88</td><td>76.34±1.71</td><td>90.23±0.50</td><td>78.56±2.01</td><td>80.12±1.83</td><td>34.89��1.73</td><td rowspan="2">1.05↑</td></tr><tr><td>PSL-APPNP</td><td>89.03±1.79</td><td>88.23±2.39</td><td>76.92±2.36</td><td>90.58±0.94</td><td>79.89±2.37</td><td>82.34±2.19</td><td>36.01±2.11</td></tr><tr><td>Graph-Heat</td><td>86.85±1.29</td><td>87.12±1.69</td><td>74.75±2.44</td><td>89.32±0.74</td><td>75.78±1.88</td><td>79.45±1.72</td><td>34.01±1.62</td><td rowspan="2">1.22↑</td></tr><tr><td>PSL-Heat</td><td>87.38±1.75</td><td>88.74±2.42</td><td>76.72±2.52</td><td>89.43±0.93</td><td>77.34±2.29</td><td>80.67±2.14</td><td>35.56±1.94</td></tr><tr><td>BernNet</td><td>88.29±0.63</td><td>88.50±1.19</td><td>76.78±0.49</td><td>94.02±1.26</td><td>86.23±2.14</td><td>85.67±1.99</td><td>36.12±1.49</td><td rowspan="2">1.13↑</td></tr><tr><td>PSL-BernNet</td><td>89.67±1.79</td><td>89.54±2.40</td><td>77.46±2.38</td><td>94.19±0.94</td><td>88.34±2.48</td><td>86.89±2.32</td><td>37.45±1.91</td></tr><tr><td>GPRGNN</td><td>87.59±1.31</td><td>88.01±1.14</td><td>76.83±1.67</td><td>92.98±1.09</td><td>86.32±2.11</td><td>85.17±1.87</td><td>36.42±1.65</td><td rowspan="2">1.21↑</td></tr><tr><td>PSL-GPR</td><td>88.45±1.48</td><td>89.11±1.89</td><td>77.42±2.02</td><td>94.23±1.47</td><td>88.67±2.33</td><td>86.45±2.04</td><td>37.45±1.92</td></tr><tr><td>Jacobi</td><td>88.83±1.65</td><td>88.03±0.75</td><td>76.89±1.84</td><td>93.72±1.53</td><td>87.11±2.07</td><td>84.45±1.96</td><td>36.56±1.78</td><td rowspan="2">1.08↑</td></tr><tr><td>PSL-Jacobi</td><td>89.01±1.80</td><td>89.02±2.42</td><td>77.71±2.50</td><td>94.43±0.96</td><td>88.98±2.42</td><td>86.23±2.15</td><td>37.78±2.05</td></tr><tr><td>VAG</td><td>0.89↑</td><td>1.22↑</td><td>1.01↑</td><td>0.86↑</td><td>1.96↑</td><td>1.68↑</td><td>1.28↑</td><td></td></tr></table>
+
+table shows, the number of distinct eigenvalues of the perturbed sheaf Laplacian matrix significantly exceeds that of the normalized graph Laplacian matrix.
+
+# 7. Related work
+
+In this section, we outline the research developments related to the issues of interest, along with the background and tools employed.
+
+# 7.1. Expressiveness of GNNs
+
+The research on expressive power of GNNs originates from the Weisfeiler-Lehman (WL) test for graph isomorphism (Weisfeiler & Leman, 1968). Building on this foundation, GIN (Xu et al., 2019b) establishes a direct connection between the expressive power of GNNs and the 1-WL test. Since then, various works have attempted to analyze GNNs with the WL test and graph isomorphism testing (Morris et al., 2019; Maron et al., 2019a; Chen et al., 2019; Zhang et al., 2021). Besides, some studies measure the expressive power of GNNs by using alternative approaches, such as expressing universal invariant functions (Maron et al., 2019b) and counting graph's substructures (Chen et al., 2020). Balcilar et al.(Balcilar et al., 2021) first analyze the expressive power of GNNs from a spectral perspective. Wang and Zhang (Wang & Zhang, 2022) show that the number of distinct eigenvalues in the normalized graph Laplacian matrix impacts the expressive power of linear spectral GNNs. Besides, they also build a bridge between the expressive power of spectral GNNs and 1-WL test. Lu et al. (Lu et al., 2024) further analyze that the presence of repeated eigenvalues imposes limitations on the predictive ability of linear spectral GNNs. In this paper, we summarize and extend the work of Wang and Zhang, as well as Lu et al., to address the limitations posed by repeated eigen-
+
+values of normalized graph Laplacian matrix.
+
+# 7.2. Sheaf Neural Networks on Graphs
+
+Sheaf Neural Networks (SNNs), introduced by Hansen and Gebhart (Hansen & Gebhart, 2020), extend graph neural networks by using the sheaf Laplacian matrix for dealing with non-constant, heterogeneous, and signed node relationships. Building on this foundation, Bodnar et al. (Bodnar et al., 2022) apply sheaf Laplacian matrix to address heterophily and oversmoothing problems in graph neural networks. This is achieved by concatenating node features as parameters, which are then used to construct restriction maps. Barbero et al. (Barbero et al., 2022) generate a sheaf Laplacian matrix through a pre-trained approach, which is suitable for manifold data and effectively incorporates the underlying geometric characteristics of the data. Duta et al. (Duta et al., 2023b) construct the hyper sheaf Laplacian to capture both linear and non-linear diffusion by selecting the most discrepant feature pairs within hyperedges. Battiloro et al. (Battiloro et al., 2024) use VDM (Singer & Wu, 2012) to construct the sheaf Laplacian and design a tangent bundle filter. This method requires a large amount of specialized data and involves high complexity. Our method simplifies the process by perturbing the block form of the normalized graph Laplacian, which has been proven both efficient and effective.
+
+# 7.3. Eigenvalue Perturbation Theory
+
+Early research mainly focuses on eigenvalue perturbation of general symmetric matrices. Tools such as Weyl's theorem (Weyl, 1912) and Gershgorin's disk theorem (Gerschgorin, 1931) are applied to derive the fundamental effects of perturbations on eigenvalues. Due to the stability of semidefinite matrices and diagonally dominant matrices,
+
+Table 2. Comparison of models based on three Laplacian matrices (GNN vs. GSL-GNN vs. PSL-GNN): mean accuracy  $(\%)$  ±95  $(\%)$  on seven datasets.  
+
+<table><tr><td>Datasets</td><td>Cora</td><td>Pubmed</td><td>Citeseer</td><td>Photo</td><td>Texas</td><td>Cornell</td><td>Actor</td></tr><tr><td>GCN</td><td>86.8±1.1</td><td>86.4±0.8</td><td>75.4±2.2</td><td>87.6±1.5</td><td>78.4±1.9</td><td>75.2±1.7</td><td>34.1±1.5</td></tr><tr><td>GSL-GCN</td><td>87.6±0.9</td><td>86.9±2.0</td><td>76.4±2.1</td><td>89.4±1.1</td><td>80.1±1.3</td><td>77.7±1.3</td><td>35.3±1.9</td></tr><tr><td>PSL-GCN</td><td>88.1±0.8</td><td>87.9±1.8</td><td>77.4±1.8</td><td>90.1±1.1</td><td>80.2±1.7</td><td>77.8±1.4</td><td>35.5±2.2</td></tr><tr><td>APPNP</td><td>88.0±1.5</td><td>87.4±0.9</td><td>76.2±1.5</td><td>90.1±0.6</td><td>78.5±2.0</td><td>80.1±1.6</td><td>34.9±1.5</td></tr><tr><td>GSL-APPNP</td><td>89.0±1.4</td><td>88.1±2.1</td><td>77.0±2.6</td><td>90.6±0.9</td><td>79.9±2.1</td><td>82.4±2.5</td><td>36.0±2.3</td></tr><tr><td>PSL-APPNP</td><td>89.0±1.1</td><td>88.0±2.6</td><td>77.0±2.3</td><td>90.3±1.0</td><td>79.7±2.1</td><td>82.3±1.9</td><td>36.0±1.6</td></tr><tr><td>Graph-Heat</td><td>86.6±1.4</td><td>87.1±1.6</td><td>74.8±2.5</td><td>89.2±1.1</td><td>75.8±2.0</td><td>79.3±1.6</td><td>34.0±1.6</td></tr><tr><td>GSL-Heat</td><td>87.4±1.8</td><td>88.7±2.5</td><td>76.6±2.2</td><td>89.4±1.3</td><td>77.4±1.9</td><td>80.7±2.1</td><td>35.6±2.1</td></tr><tr><td>PSL-Heat</td><td>87.2±1.4</td><td>88.3±1.8</td><td>76.7±2.4</td><td>89.1±1.5</td><td>76.6±2.9</td><td>80.4±1.1</td><td>35.3±1.9</td></tr><tr><td>BernNet</td><td>88.2±0.7</td><td>88.4±1.3</td><td>76.6±0.7</td><td>94.0±1.2</td><td>86.3±1.9</td><td>85.8±1.4</td><td>36.0±1.6</td></tr><tr><td>GSL-BernNet</td><td>89.7±1.6</td><td>89.5±2.4</td><td>77.6±2.0</td><td>94.1±1.0</td><td>88.3±2.4</td><td>86.8±1.7</td><td>37.3±2.2</td></tr><tr><td>PSL-BernNet</td><td>89.9±1.3</td><td>89.7±1.9</td><td>77.3±1.8</td><td>94.2±1.0</td><td>88.4±2.8</td><td>86.9±1.2</td><td>37.5±2.3</td></tr><tr><td>GPRGNN</td><td>87.1±0.9</td><td>87.9±1.1</td><td>76.7±1.5</td><td>92.8±1.2</td><td>85.2±1.7</td><td>86.2±2.1</td><td>36.3±1.5</td></tr><tr><td>GSL-GPR</td><td>88.1±1.1</td><td>88.6±2.2</td><td>76.8±1.9</td><td>93.6±1.0</td><td>86.1±1.8</td><td>87.8±2.0</td><td>36.7±2.4</td></tr><tr><td>PSL-GPR</td><td>88.4±1.2</td><td>89.3±1.6</td><td>77.4±1.8</td><td>93.2±1.6</td><td>86.7±1.5</td><td>88.2±2.4</td><td>37.0±2.5</td></tr><tr><td>Jacobi</td><td>88.7±1.4</td><td>88.0±0.7</td><td>76.7±1.8</td><td>93.6±1.3</td><td>87.4±1.8</td><td>84.2±2.1</td><td>36.5±1.7</td></tr><tr><td>GSL-Jacobi</td><td>89.0±1.8</td><td>89.1±2.2</td><td>77.7±2.2</td><td>94.4±1.0</td><td>89.0±2.3</td><td>86.3±2.0</td><td>37.7±2.0</td></tr><tr><td>PSL-Jacobi</td><td>89.0±1.4</td><td>89.3±1.9</td><td>77.8±2.3</td><td>94.2±1.3</td><td>88.9±2.4</td><td>86.6±1.1</td><td>37.8±1.5</td></tr></table>
+
+Table 3. Distinct eigenvalues statistics of normalized Laplacian on real-world datasets across multiple training runs using PSL-GNN models (95% confidence interval).  
+
+<table><tr><td>Datasets</td><td>Cora</td><td>Pubmed</td><td>Citeseer</td><td>Photo</td><td>Texas</td><td>Cornell</td><td>Actor</td></tr><tr><td>||V||</td><td>2708</td><td>19717</td><td>3327</td><td>7650</td><td>183</td><td>183</td><td>7600</td></tr><tr><td>NG</td><td>2262</td><td>7647</td><td>1969</td><td>7511</td><td>73</td><td>60</td><td>4837</td></tr><tr><td>NPSL-GCN</td><td>2389±33</td><td>9031±45</td><td>2339±43</td><td>7604±19</td><td>147±11</td><td>142±16</td><td>6181±42</td></tr><tr><td>NPSL-APPNP</td><td>2408±31</td><td>8977±60</td><td>2300±41</td><td>7602±15</td><td>145±6</td><td>147±10</td><td>6234±37</td></tr><tr><td>NPSL-Heat</td><td>2376±25</td><td>8981±54</td><td>2301±37</td><td>7604±13</td><td>145±8</td><td>148±9</td><td>6225±40</td></tr><tr><td>NPSL-BernNet</td><td>2434±36</td><td>9018±56</td><td>2308±35</td><td>7609±10</td><td>144±13</td><td>149±6</td><td>6240±46</td></tr><tr><td>NPSL-GPR</td><td>2436±35</td><td>9024±57</td><td>2311±34</td><td>7603±19</td><td>145±12</td><td>148±7</td><td>6243±45</td></tr><tr><td>NPSL-Jacobi</td><td>2442±36</td><td>9030±58</td><td>2316±35</td><td>7567±30</td><td>147±13</td><td>150±8</td><td>6249±46</td></tr></table>
+
+some works focus on establishing perturbation bounds of these two matrices. Dailey et al. (Dailey et al., 2014) conduct in-depth research on the perturbation bounds of symmetric positive semidefinite matrices and propose a strong relative perturbation bounds. For diagonally dominant matrices, a strong relative perturbation bound is proposed by Bai et al. (Ye, 2009). In this paper, we primarily focus on symmetric positive semidefinite and also diagonally dominant matrices, aiming to study the changes in their eigenvalues after perturbation.
+
+# 8. CONCLUSION
+
+In this paper, we propose a novel sheaf Laplacian by perturbing the normalized graph Laplacian, which enhances the spectral GNNs. Our approach addresses the limitations posed by repeated eigenvalues in the normalized
+
+graph Laplacian, thereby improving the expressive power of GNNs. Extensive experiments on benchmark datasets demonstrate the effectiveness of our method, showing notable improvements in node classification performance. Although our approach has been validated through both theoretical analysis and experiments, further exploration is required to fully reduce the multiplicity of eigenvalues and refine the matrix construction methods. Additionally, we have not yet investigated the impact of the perturbed sheaf Laplacian (PSL) on graph-level tasks, which remains an interesting direction for future research.
+
+# Acknowledgements
+
+We sincerely thank the anonymous reviewers for their constructive comments and thoughtful suggestions.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
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+
+# A. IMPORTANT NOTATIONS.
+
+Table 4. IMPORTANT NOTATIONS  
+
+<table><tr><td>Symbol</td><td>Definition</td></tr><tr><td>G,F</td><td>a graph, vector spaces over the graph</td></tr><tr><td>V,E</td><td>set of nodes, set of edges</td></tr><tr><td>A,D</td><td>adjacency matrix, degree matrix</td></tr><tr><td>L,LF</td><td>normalized graph laplacian, sheaf laplacian</td></tr><tr><td>Lblock,LF</td><td>block form of normalized graph laplacian, perturbed sheaf laplacian</td></tr><tr><td>n,N,d</td><td>the dimension of L , the dimension of LF , the dimension of stalks</td></tr><tr><td>λ,λ</td><td>the eigenvalue of Lblock, the eigenvalue of LF</td></tr><tr><td>X,X̂</td><td>feature matrix (input matrix), the input matrix after the Fourier transform</td></tr><tr><td>Y,Z</td><td>the output matrix after the Laplace transform, the final predication matrix</td></tr><tr><td>P,Q</td><td>the applied perturbation matrix, the transport matrix (the transpose product of the restriction maps)</td></tr><tr><td>g,R</td><td>the filter function, the reshape operation</td></tr><tr><td>u</td><td>reflection vector</td></tr><tr><td>φ,φ</td><td>the minimum eigenvalue gap in Lblock, a function like multi-layer perceptrons</td></tr><tr><td>ε,η</td><td>the maximum perturbation ratio for every block, the upper limit of perturbations for all parameters in u</td></tr></table>
+
+# B. Proofs.
+
+# B.1. Proof of Theorem 3.2
+
+Theorem B.1. When there are only  $k$  distinct eigenvalues of the normalized Laplacian matrix, and the nonlinear real-valued function  $\sigma$  satisfies  $\sigma(x) \neq 0$  for  $x \neq 0$ , nonlinear spectral GNNs can produce at most  $k$  different filter coefficients, and thus can only generate one-dimensional predictions with a maximum of  $k$  arbitrary elements.
+
+Proof. Let  $Z = [z_{1}, \ldots, z_{n}]^{T}$  be the output, where  $z_{i} \in \mathbb{R}$  and  $z_{i} \neq 0$  for  $i = 1, 2, \ldots, n$ . Here,  $k$  is the number of distinct eigenvalues of the normalized graph Laplacian matrix. Assume that the filter function is  $g(\lambda)$ , where  $\lambda$  is the eigenvalue of the normalized graph Laplacian matrix. When  $\lambda_{i} = \lambda_{j}$ ,  $g(\lambda_{i}) = g(\lambda_{j})$ . Therefore, when the number of distinct eigenvalues of the Laplacian matrix is  $k$ , there are only  $k$  different filter coefficients.
+
+For the Fourier transform  $Z = U\mathrm{diag}(g(\lambda))U^{\top}XW$ , we assume that all elements of  $X$  are non-negative. Let the vector  $\mathbf{m} = U^{\top}XW$  and  $\mathbf{q} = \mathrm{diag}(g(\lambda))\cdot \mathbf{m}$ . Then  $Z = U\mathbf{q}$ , where  $Z\in \mathbb{R}^{n\times 1}$ . If  $\lambda_{i} = \lambda_{j}$ , then  $q_{i} = q_{j}\frac{m_{i}}{m_{j}}$ . Since there are  $k$  independent eigenvalues, there are only  $k$  arbitrary elements in the vector  $\mathbf{q}$ . Therefore, any element in the vector  $\mathbf{q}$  can be represented as a linear combination of these  $k$  arbitrary elements, that is:
+
+$$
+q _ {i} = \alpha_ {1} q _ {\ell_ {1}} + \alpha_ {2} q _ {\ell_ {2}} + \dots + \alpha_ {k} q _ {\ell_ {k}}, \quad \forall i \in [ 1, n ] \tag {13}
+$$
+
+where  $q_{\ell_1}, \ldots, q_{\ell_k}$  are the  $k$  arbitrary elements in  $\mathbf{q}$ . Then we can use a matrix  $U'$  with  $n$  rows and  $k$  columns, and a vector  $\mathbf{q}'$  with  $k$  elements to represent  $Z$ , i.e.  $U' \mathbf{q}' = Z$ . To illustrate this process, we group all eigenvectors according to their corresponding eigenvalues. For each distinct eigenvalue  $\lambda_{\ell_s}$  (where  $s = 1, 2, \ldots, k$ ), define the set:  $S_s = \{j \mid \lambda_j = \lambda_{\ell_s}\}$ .
+
+$Z$  could be expressed as follow:
+
+$$
+\left[ \begin{array}{c c c} \sum_ {j \in S _ {1}} U _ {1, j} \frac {m _ {j}}{m _ {\ell_ {1}}} & \dots & \sum_ {j \in S _ {k}} U _ {1, j} \frac {m _ {j}}{m _ {\ell_ {k}}} \\ \sum_ {j \in S _ {1}} U _ {2, j} \frac {m _ {j}}{m _ {\ell_ {1}}} & \dots & \sum_ {j \in S _ {k}} U _ {2, j} \frac {m _ {j}}{m _ {\ell_ {k}}} \\ \vdots & \vdots & \vdots \\ \sum_ {j \in S _ {1}} U _ {n, j} \frac {m _ {j}}{m _ {\ell_ {1}}} & \dots & \sum_ {j \in S _ {k}} U _ {n, j} \frac {m _ {j}}{m _ {\ell_ {k}}} \end{array} \right] \left[ \begin{array}{c} q _ {\ell_ {1}} \\ q _ {\ell_ {2}} \\ \vdots \\ q _ {\ell_ {k}} \end{array} \right] = \left[ \begin{array}{c} z _ {1} \\ z _ {2} \\ \vdots \\ z _ {n} \end{array} \right] \tag {14}
+$$
+
+Since  $U'$  has rank at most  $k$ , it spans a subspace of dimension at most  $k$ . Thus,  $Z = U' \mathbf{q}'$  is confined within this subspace. We can choose up to  $k$  linearly independent elements  $z_{\ell_1}, z_{\ell_2}, \ldots, z_{\ell_k}$  as a basis. Therefore, any  $z_i$  can be expressed as a linear combination:
+
+$$
+z _ {i} = \beta_ {1} z _ {\ell_ {1}} + \beta_ {2} z _ {\ell_ {2}} + \dots + \beta_ {k} z _ {\ell_ {k}}, \quad \text {f o r} \beta_ {j} \in \mathbb {R} \tag {15}
+$$
+
+Let the activation function  $\sigma(x)$  be a real-valued function, where  $\sigma(x) \neq 0$  for  $x \neq 0$ . Since the elements in  $Z$  are nonzero, let  $Z' = \sigma(Z)$  and  $\operatorname{rank}(U') = k$ , then  $z_i' = r_i \cdot z_i$ , where  $r_i \in \mathbb{R}$ . According to Equation (15), we have:
+
+$$
+\begin{array}{l} z _ {i} ^ {\prime} = \beta_ {1} \cdot \frac {r _ {i}}{r _ {\ell_ {1}}} \cdot z _ {\ell_ {1}} ^ {\prime} + \dots + \beta_ {k} \cdot \frac {r _ {i}}{r _ {\ell_ {k}}} \cdot z _ {\ell_ {k}} ^ {\prime} \tag {16} \\ = \beta_ {1} ^ {\prime} \cdot z _ {\ell_ {1}} ^ {\prime} + \dots + \beta_ {k} ^ {\prime} \cdot z _ {\ell_ {k}} ^ {\prime} \\ \end{array}
+$$
+
+Therefore with the nonlinear activation function, spectral GNNs can still only generate at most  $k$  arbitrary elements.
+
+# B.2. Proof of Theorem 4.1
+
+Theorem B.2. Let  $P = \overline{\mathcal{L}_{\mathcal{F}}} - \mathcal{L}_{block}$  be the perturbation matrix applied to  $\mathcal{L}_{block}$ , and let  $\phi = \min_{i,j; i \neq j} |\lambda_i - \lambda_j|$ , where  $\lambda_i$  are the eigenvalues of  $\mathcal{L}_{block}$ . If  $\| P \|_2 < \frac{\phi}{2}$ , then the eigenvalue multiplicity of  $\mathcal{L}_{\mathcal{F}}$  will decrease.
+
+Proof. When the perturbation matrix  $P$  is applied to  $\mathcal{L}_{\text{block}}$ , the new eigenvalues  $\mu_i$  and  $\mu_j$  of  $\overline{\mathcal{L}_{\mathcal{F}}}$  corresponding to  $\lambda_i$  and  $\lambda_j$  will lie within a neighborhood of these eigenvalues. According to Weyl's inequality, we have:  $|\mu_i - \lambda_i| \leq \|P\|_2$  and  $|\mu_j - \lambda_j| \leq \|P\|_2$ . This implies that  $\mu_i$  lies in the interval:  $[\lambda_i - \|P\|_2, \lambda_i + \|P\|_2]$ , and  $\mu_j$  lies in the interval:  $[\lambda_j - \|P\|_2, \lambda_j + \|P\|_2]$ .
+
+Now, since  $\phi = \min_{i,j; i \neq j} |\lambda_i - \lambda_j|$  and we assume  $\|P\|_2 < \frac{\phi}{2}$ , the length of each interval is  $2\|P\|_2$ , which is strictly less than  $\phi$ . The distance between the centers of these intervals is  $\phi$ , which is greater than the sum of the half-lengths of the intervals,  $2\|P\|_2$ . Thus,  $[\lambda_i - \|P\|_2, \lambda_i + \|P\|_2]$  and  $[\lambda_j - \|P\|_2, \lambda_j + \|P\|_2]$  do not overlap. This guarantees that the perturbed eigenvalues  $\mu_i$  and  $\mu_j$  remain distinct.
+
+Since  $\lambda_{i}$  and  $\lambda_{j}$  represent the pair of eigenvalues with the smallest distance  $\phi$ , if their perturbation intervals do not overlap, then for any other pair of eigenvalues  $\lambda_{k}$  and  $\lambda_{l}$  (where  $k \neq l$ ), their perturbation intervals will also not overlap due to  $|\lambda_{k} - \lambda_{l}| \geq \phi$ .
+
+Therefore, no two perturbation intervals overlap, ensuring that the eigenvalue multiplicity of  $\overline{\mathcal{L}_{\mathcal{F}}}$  will decrease as eigenvalues associated with higher multiplicities split into distinct values within their respective intervals.
+
+# C. Expressiveness of PSL-GNN
+
+# C.1. Proof of Proposition 5.1
+
+Proposition C.1. The number of distinct eigenvalues of  $\mathcal{L}_{block} \in \mathbb{R}^{nd \times nd}$  is the same as that of  $\mathcal{L} \in \mathbb{R}^{n \times n}$ , where  $\mathcal{L}$  is the normalized graph Laplacian matrix and  $\mathcal{L}_{block} = \mathcal{L} \otimes I$ , with  $I \in \mathbb{R}^{d \times d}$  being the identity matrix. Also, their corresponding GNNs, which share the same architecture, are mutually equivalent.
+
+Proof. We prove the first part of the proposition as follows. Let  $\mathcal{L}$  have eigenvalues  $\{\lambda_1, \lambda_2, \dots, \lambda_n\}$ , and let  $I \in \mathbb{R}^{d \times d}$  be the identity matrix with eigenvalues  $\{1, 1, \dots, 1\}$ . According to the eigenvalue property of the Kronecker product, the eigenvalues of  $\mathcal{L}_{\mathrm{block}} = \mathcal{L} \otimes I$  are  $\lambda_i \cdot 1 = \lambda_i$  for each  $i = 1, 2, \dots, n$ . Since  $I$  only introduces eigenvalues of 1, the
+
+Kronecker product does not add new eigenvalues. Each eigenvalue  $\lambda_{i}$  of  $\mathcal{L}$  is repeated  $d$  times, meaning the number of distinct eigenvalues of  $\mathcal{L}_{\mathrm{block}}$  is the same as that of  $\mathcal{L}$ .
+
+Consider an input matrix  $\mathbf{X} \in \mathbb{R}^{n \times f}$ , where  $f = d \times k$ . Reshape  $\mathbf{X}$  into  $\mathbf{X}' \in \mathbb{R}^{nd \times k}$ . Let  $\mathbf{Y} = \mathcal{L}\mathbf{X}$  and  $\mathbf{Y}' = \mathcal{L}_{\mathrm{block}}\mathbf{X}'$ . We partition  $\mathbf{X}$  into  $n$  row blocks:  $\mathbf{X} = \left[(\mathbf{X}^{(1)})^\top, \ldots, (\mathbf{X}^{(n)})^\top\right]^\top$ , where each  $\mathbf{X}^{(i)} \in \mathbb{R}^{1 \times f}$ . Reshape each  $\mathbf{X}^{(i)}$  into a block matrix  $\mathbf{X}'^{(i)} \in \mathbb{R}^{d \times k}$ . The reshaped matrix  $\mathbf{X}' \in \mathbb{R}^{nd \times k}$  is formed by stacking these blocks.
+
+Now compute  $\mathbf{Y}' = \mathcal{L}_{\mathrm{block}}\mathbf{X}'$ . The Kronecker product  $\mathcal{L}_{\mathrm{block}} = \mathcal{L} \otimes I_d$  implies that each element  $\mathcal{L}_{ij}$  of  $\mathcal{L}$  is multiplied by the  $d \times d$  identity matrix  $I_d$ . Thus,  $\mathbf{Y}'$  can be expressed as:  $\mathbf{Y}' = \left[\sum_{j=1}^{n} \mathcal{L}_{1j}\mathbf{X}'^{(j)} \dots \sum_{j=1}^{n} \mathcal{L}_{nj}\mathbf{X}'^{(j)}\right]^\top$ .
+
+Next, we reshape  $\mathbf{Y}'$  back to  $\mathbf{Y}_{\mathrm{reshaped}}'\in \mathbb{R}^{n\times f}$ . For each  $i,s,t$ , the element  $\mathbf{Y}_{i,(s - 1)d + t}$  in  $\mathbf{Y}$  corresponds to  $\mathbf{Y}_{s,t}^{\prime (i)}$  in  $\mathbf{Y}'$ . Thus, we have:  $\mathbf{Y}_{i,(s - 1)d + t} = \mathbf{Y}_{\mathrm{reshaped}}'(i,(s - 1)d + t)$ .
+
+Since this holds for all  $i, s, t$ , we conclude that  $\mathbf{Y} = \mathbf{Y}_{\mathrm{reshaped}}'$ .
+
+# C.2. Proof of Corollary 5.2
+
+Corollary C.2. PSL-GNN has stronger one-dimensional prediction ability.
+
+Proof. Consider a block form matrix  $\mathcal{L}_{block}$  and its corresponding perturbed sheaf Laplacian matrix  $\overline{\mathcal{L}_{\mathcal{F}}}$ . Let the input matrix be  $X \in \mathbb{R}^{n \times 1}$ . Their outputs are given as  $Z_{1} = U_{1}g(\lambda)U_{1}^{\top}XW_{1}$  and  $Z_{2} = U_{2}g(\lambda)U_{2}^{\top}XW_{2}$ , respectively. Define vectors  $\mathbf{m}_1 = XW_1$  and  $\mathbf{m}_2 = XW_2$ , and let  $\mathbf{q}_1 = g(\lambda) \cdot \mathbf{m}_1$  and  $\mathbf{q}_2 = g(\lambda) \cdot \mathbf{m}_2$ .
+
+Suppose the number of distinct eigenvalues of  $\overline{\mathcal{L}_{\mathcal{F}}}$  and  $\mathcal{L}$  are  $k_{1}$  and  $k_{2}$ , respectively. According to the Proposition 5.1, the number of distinct eigenvalues of  $\mathcal{L}_{\mathrm{block}}$  is equal to that of  $\mathcal{L}$ , so the number of distinct eigenvalues of  $\mathcal{L}_{\mathrm{block}}$  is  $k_{2}$ . When  $\lambda_{i} = \lambda_{j}$ ,  $g(\lambda_{i}) = g(\lambda_{j})$ .
+
+Therefore, the outputs can be expressed as  $Z_{1} = U_{1}^{\prime}\mathbf{q}_{1}^{\prime}$  and  $Z_{2} = U_{2}^{\prime}\mathbf{q}_{2}^{\prime}$ , where  $U_{1}^{\prime}$  is an  $n\times k_{1}$  matrix,  $U_{2}^{\prime}$  is an  $n\times k_{2}$  matrix, and  $\mathbf{q}_1^\prime$  and  $\mathbf{q}_2^\prime$  are vectors with  $k_{1}$  and  $k_{2}$  elements, respectively.
+
+Since the ranks of  $U_1'$  and  $U_2'$  are at most  $k_1$  and  $k_2$ ,  $Z_1$  and  $Z_2$  each have at most  $k_1$  and  $k_2$  arbitrary elements, respectively. Given that  $k_1 > k_2$ , the number of arbitrary elements in  $Z_1$  is greater than that in  $Z_2$ .
+
+# C.3. Proof of Corollary 5.3
+
+Corollary C.3. PSL-GNN loses fewer frequency components.
+
+Proof. Consider a block form matrix  $\mathcal{L}_{block}$  and its corresponding perturbed sheaf Laplacian matrix  $\overline{\mathcal{L}_{\mathcal{F}}}$ . Let the maximum multiplicity of the eigenvalues of  $\mathcal{L}_{block}$  be  $m$ , and the maximum multiplicity of the eigenvalues of  $\overline{\mathcal{L}_{\mathcal{F}}}$  be  $\hat{m}$ . Since  $\hat{m} < m$ , PSL-GNN is able to lose fewer frequency components even in the worst-case design of the filtering function.
+
+# C.4. Proof of Corollary 5.4
+
+Corollary C.4. PSL-GNN has a stronger capability in processing frequency components.
+
+Proof. Consider a block form matrix  $\mathcal{L}_{block}$  and its corresponding perturbed sheaf Laplacian matrix  $\overline{\mathcal{L}_F}$ . Let the number of distinct eigenvalues of  $\mathcal{L}_{block}$  and  $\overline{\mathcal{L}_F}$  be  $k_{1}$  and  $k_{2}$ , respectively. Since  $k_{1} < k_{2}$ , the PSL-GNN filter has more coefficients, implying that it can handle frequency components in a more diverse manner.
+
+Remark C.5. We explain Corollary 5.4 more clearly: the eigenvalues of  $\mathcal{L}_{\mathrm{block}}$  are divided into  $k_{1}$  groups, while those of  $\mathcal{L}_{\mathcal{F}}$  are divided into  $k_{2}$  groups, with  $k_{2} > k_{1}$ . Accordingly, the filter coefficients are divided into  $k_{1}$  and  $k_{2}$  groups, with identical coefficients within each group. This means that the frequency components associated with each eigenvalue in a group are scaled by the same factor. Since  $k_{2} > k_{1}$ , PSL-GNN can independently scale more frequency components.
+
+# D. Perturbation Bounds of Eigenvalues
+
+Theorem D.1. Let  $\mathcal{L}_{block}$  and  $\mathcal{L}_{\mathcal{F}} = \mathcal{L}_{block} + P$  be  $N\times N$  symmetric positive semidefinite matrices, with eigenvalues  $\lambda_1,\ldots ,\lambda_N$  and  $\tilde{\lambda}_1,\dots ,\tilde{\lambda}_N$  , respectively, where  $\mathcal{L}_{block}$  is diagonally dominant. Suppose the perturbation matrix  $P$  satisfies  $|P_{ij}|\leq \epsilon |(L_{block})_{ij}|$  for all  $i\neq j$  for all  $i$  , where  $0 <   \epsilon <  \frac{1}{2}$  . Hence, for each  $i$ $|\tilde{\lambda}_i - \lambda_i|\leq \frac{2\epsilon}{1 - 2\epsilon}\tilde{\lambda}_{\mathrm{max}}.$
+
+Proof. First, we bound the spectral norm of  $P$ . For each row  $i$ , we have:
+
+$$
+\sum_ {j = 1} ^ {n} \left| P _ {i j} \right| = \left| P _ {i i} \right| + \sum_ {j \neq i} \left| P _ {i j} \right| \leq \epsilon v _ {i} + \epsilon \sum_ {j \neq i} \left| \left(\mathcal {L} _ {\mathrm {b l o c k}}\right) _ {i j} \right| = \epsilon \left(v _ {i} + \sum_ {j \neq i} \left| \left(\mathcal {L} _ {\mathrm {b l o c k}}\right) _ {i j} \right|\right) = \epsilon \left(\mathcal {L} _ {\mathrm {b l o c k}}\right) _ {i i}. \tag {17}
+$$
+
+Therefore:
+
+$$
+\| P \| _ {\infty} \leq \epsilon \max  _ {1 \leq i \leq n} \left(\mathcal {L} _ {\text {b l o c k}}\right) _ {i i}. \tag {18}
+$$
+
+Since  $\mathcal{L}_{\mathrm{block}}$  is diagonally dominant and positive semidefinite:
+
+$$
+\left\| \mathcal {L} _ {\text {b l o c k}} \right\| _ {\infty} = \max  _ {1 \leq i \leq n} \sum_ {j = 1} ^ {n} | (\mathcal {L} _ {\text {b l o c k}}) _ {i j} | \leq 2 \max  _ {1 \leq i \leq n} (\mathcal {L} _ {\text {b l o c k}}) _ {i i} \leq 2 \| \mathcal {L} _ {\text {b l o c k}} \| _ {2}, \tag {19}
+$$
+
+where  $\| \mathcal{L}_{\mathrm{block}}\| _2$  denotes the spectral norm of  $\mathcal{L}_{\mathrm{block}}$ . Thus:
+
+$$
+\| P \| _ {2} \leq \| P \| _ {\infty} \leq \epsilon \| \mathcal {L} _ {\mathrm {b l o c k}} \| _ {\infty} \leq 2 \epsilon \| \mathcal {L} _ {\mathrm {b l o c k}} \| _ {2}. \tag {20}
+$$
+
+Next, we apply Weyl's theorem, which states that for symmetric matrices  $\mathcal{L}_{\mathrm{block}}$  and  $P$ :  $|\tilde{\lambda}_i - \lambda_i| \leq \| P\|_2$ . Substituting the norm bound derived above:
+
+$$
+\left| \tilde {\lambda} _ {i} - \lambda_ {i} \right| \leq \| P \| _ {2} \leq 2 \epsilon \| \mathcal {L} _ {\text {b l o c k}} \| _ {2}. \tag {21}
+$$
+
+We now relate  $\| \mathcal{L}_{\mathrm{block}}\| _2$  to  $\tilde{\lambda}_i$ . Since  $\mathcal{L}_{\mathcal{F}} = \mathcal{L}_{\mathrm{block}} + P$ :
+
+$$
+\left\| \mathcal {L} _ {\text {b l o c k}} \right\| _ {2} \leq \left\| \mathcal {L} _ {\mathcal {F}} \right\| _ {2} + \| P \| _ {2} \leq \left\| \mathcal {L} _ {\mathcal {F}} \right\| _ {2} + 2 \epsilon \left\| \mathcal {L} _ {\text {b l o c k}} \right\| _ {2}. \tag {22}
+$$
+
+Rearranging gives:
+
+$$
+\left\| \mathcal {L} _ {\text {b l o c k}} \right\| _ {2} \leq \frac {\left\| \mathcal {L} _ {\mathcal {F}} \right\| _ {2}}{1 - 2 \epsilon}. \tag {23}
+$$
+
+Given that  $\| \mathcal{L}_{\mathcal{F}}\| _2 = \tilde{\lambda}_{\mathrm{max}}\geq \tilde{\lambda}_i$  , it follows that:
+
+$$
+\left| \tilde {\lambda} _ {i} - \lambda_ {i} \right| \leq 2 \epsilon \| \mathcal {L} _ {\text {b l o c k}} \| _ {2} \leq \frac {2 \epsilon}{1 - 2 \epsilon} \| \mathcal {L} _ {\mathcal {F}} \| _ {2} = \frac {2 \epsilon}{1 - 2 \epsilon} \tilde {\lambda} _ {\max } \tag {24}
+$$
+
+Thus, the bound is proven.
+
+# E. Experimental Setup
+
+# E.1. Experimental Details
+
+We construct every reflection matrix  $\mathcal{F}_{i\leq (i,j)}$  based on the method in (Obukhov, 2021). For downstream task, we update the perturbed sheaf Laplacian matrix every 10 epochs. Given its high dimensionality, we employ sparse matrix storage to reduce computational overhead. Additionally, we apply an early stopping mechanism with a maximum of 1500 epochs and a patience threshold of 100. For each PSL-GNN and its corresponding GNN, we train them for the same number of times and take the average of their respective accuracies as the final result for each model. For model training, we use the Adam optimizer to optimize all models on an NVIDIA GeForce RTX 4090 GPU.
+
+# E.2. Datasets
+
+We use seven benchmark datasets, categorized as follows: (1) Citation Networks: Cora (McCallum et al., 2000), Citeseer (Giles et al., 1998), and Pubmed (Sen et al., 2008) are citation networks with nodes as publications, edges as citation links, and labels indicating topics or fields; (2) Co-purchase Networks: Photo (Shchur et al., 2018) is a co-purchase network with nodes as products, edges as co-purchases, and labels categorizing products; (3) Webpage Networks: Texas and Cornell
+
+(Shchur et al., 2018) are webpage networks with nodes as webpages, edges as hyperlinks, and labels indicating categories; (4) Actor Co-occurrence Network: Actor (Shchur et al., 2018) is a co-occurrence network with nodes as actors, edges as co-occurrences, and labels classifying actors.
+
+The information of each dataset is shown in Table 5. Additionally, due to the diagonal dominance of the normalized graph Laplacian  $\mathcal{L}$ , its eigenvalue  $\lambda = 1$  has higher multiplicity. Thus we also compute the proportion  $P_{\lambda=1}$  of this multiplicity for analysis.
+
+Table 5. The statistical information of the datasets.  
+
+<table><tr><td>Datasets</td><td>Cora</td><td>Pubmed</td><td>CiteSeer</td><td>Photo</td><td>Texas</td><td>Cornell</td><td>Actor</td></tr><tr><td>Nodes</td><td>2708</td><td>19717</td><td>3327</td><td>7650</td><td>183</td><td>183</td><td>7600</td></tr><tr><td>Edges</td><td>5278</td><td>44324</td><td>4552</td><td>245861</td><td>287</td><td>278</td><td>26705</td></tr><tr><td>Features</td><td>1433</td><td>500</td><td>3703</td><td>745</td><td>1703</td><td>1703</td><td>932</td></tr><tr><td>Classes</td><td>7</td><td>3</td><td>6</td><td>8</td><td>5</td><td>5</td><td>5</td></tr><tr><td>\(P_{\lambda=1}\)</td><td>0.125</td><td>0.639</td><td>0.171</td><td>0.048</td><td>0.362</td><td>0.349</td><td>0.155</td></tr></table>
+
+# E.3. GNN Baselines
+
+To evaluate the performance of our proposed method, we choose six different spectral GNNs as follows.
+
+GCN (Kipf & Welling, 2016) is a spectral graph neural network that uses a mean aggregator to combine information from neighboring nodes, bridging the gap between spectral and spatial GNN methods. APPNP (Klicpera et al., 2018) employs personalized PageRank to propagate node features, addressing the oversmoothing problem in deep networks. Graph-Heat (Xu et al., 2019a) incorporates the heat kernel for graph signal filtering, providing a natural mechanism to smooth or diffuse signals over the graph. BernNet (He et al., 2021) improves spectral graph convolutions using Bernstein polynomials, offering better flexibility to handle varying graph structures. GPRGNN (Chien et al., 2021) generalizes PageRank algorithms to enhance performance on graphs with heterophilic structures. Lastly, JacobiConv (Wang & Zhang, 2022) leverages orthogonal Jacobi polynomials to enhance expressiveness and optimization in spectral graph neural networks.
+
+For all baseline models, we use either their official implementations or implementations provided by PyTorch Geometric to ensure fair comparisons.
+
+# E.4. Hyper-Parameters
+
+To ensure a fair comparison, we keep all hyper-parameters the same for each PSL-GNN and its corresponding baseline GNN, except for  $\eta$  and  $d$ . Specifically, for all models, we set the learning rate to 0.05, weight decay to  $5e - 4$ , and the number of hidden units to 64. We set the order of the polynomial filters—BernNet, APPNP, Graph-Heat, and Jacobi—to 10. For all PSL-GNN models, we search  $\eta$  within  $\{1e - 1, 1e - 2, 1e - 3, 1e - 4\}$  and  $d$  within  $\{2, 3, 4\}$  to achieve the best model performance.
+
+# F. Additional Experimental Studies
+
+# F.1. Complexity Analysis
+
+We first analyze the space and time costs of constructing the perturbed sheaf Laplacian, then the per-epoch cost of PSL-GCN (with and without PSL reconstruction), and finally report empirical runtimes.
+
+# F.1.1. SPACE COMPLEXITY & PARAMETERS.
+
+Each restriction map uses  $d$  free parameters, so forming the edge-incidence matrix  $\mathcal{B}_{\mathcal{F}}$  (with two entries per edge) requires space  $= O(2E \cdot d^2 + n \cdot d)$ ,
+
+# F.1.2. PSL CONSTRUCTION TIME.
+
+We build the signed incidence matrix  $\mathcal{B}_{\mathcal{F}}$  (size  $nd\times \mathrm{nnz}_{\mathcal{B}}$ ), then compute  $\overline{\mathcal{L}_{\mathcal{F}}} = \mathcal{B}_{\mathcal{F}}\mathcal{B}_{\mathcal{F}}^{T}$ . Since  $\mathcal{B}_{\mathcal{F}}$  has  $\mathrm{nnz}_{\mathcal{B}}$  nonzeros, this sparse-sparse product costs  $O\big(\mathrm{nnz}_{\mathcal{B}}^2\big)$ .
+
+# F.1.3. PER-EPOCH COST OF PSL-GCN.
+
+Every 10 epochs we rebuild the PSL; thus per-epoch we either include or omit the  $O(\mathrm{m} \mathrm{z}_{B}^{2})$  term:
+
+- Including PSL construction:  $O\left( {{\mathrm{{nnz}}}_{\mathcal{B}}^{2}}\right)  + O\left( {n{f}_{1}{df}_{3}}\right)  + O\left( {{2nd}{f}_{3}\left( {r + {f}_{4}}\right) }\right)  + O\left( {{nd}{f}_{\text{last }}c}\right)$  .  
+- Amortized (excluding PSL construction):  $O\left( {{n}^{2}{d}_{1}{d}_{3} + {n}^{2}{d}_{2}{d}_{4}}\right)  + O\left( {{n}^{2}{d}_{3}\left( {r + {f}_{4}}\right) }\right)  + O\left( {{n}^{2}{d}_{4}\left( {r + {f}_{3}}\right) }\right)$  .
+
+# F.1.4. EMPIRICAL RUNTIMES.
+
+Table 6 reports average epoch time (ms) / total runtime per fold (s) for PSL-GCN vs. Neural Sheaf Diffusion (Diag-NSD, O(d)-NSD, Gen-NSD) (Bodnar et al., 2022):
+
+Table 6. Average epoch time (ms) / total runtime per fold (s).  
+
+<table><tr><td>Dataset</td><td>Cora</td><td>Pubmed</td><td>Citeseer</td><td>Photo</td><td>Texas</td><td>Cornell</td><td>Actor</td></tr><tr><td>Diag-NSD</td><td>26.4/5.9</td><td>124.3/34.7</td><td>25.2/5.8</td><td>90.6/17.4</td><td>8.3/1.6</td><td>8.1/1.5</td><td>98.4/21.5</td></tr><tr><td>O(d)-NSD</td><td>57.0/12.3</td><td>204.5/66.3</td><td>64.1/13.6</td><td>143.5/26.3</td><td>26.8/6.7</td><td>27.4/7.2</td><td>128.2/31.9</td></tr><tr><td>Gen-NSD</td><td>85.3/17.6</td><td>231.1/74.3</td><td>92.4/20.6</td><td>166.7/34.6</td><td>34.3/14.4</td><td>31.4/14.2</td><td>177.6/45.2</td></tr><tr><td>PSL-GCN</td><td>16.0/4.6</td><td>83.7/17.4</td><td>18.3/5.2</td><td>65.3/15.8</td><td>6.3/1.1</td><td>6.4/1.1</td><td>72.7/17.3</td></tr></table>
+
+# F.2. Comparison of Spectral Gap
+
+The Laplacian spectrum measures graph connectivity. Cheeger's inequality  $2h_{G} > \lambda_{1} > \frac{h_{G}^{2}}{2}$  shows that a larger spectral gap  $\lambda_{1}$  implies better connectivity (Jamadandi et al., 2024). The Cheeger constant  $h_{G}$  quantifies the tightest connectivity bottleneck in a graph. In (Lu et al., 2024), the new operator's (H) eigenvalues are defined as  $u_{i} = \beta \lambda_{i} + (1 - \beta)v_{i}$ , where  $v_{i} = 2i / n - 2$ . When  $v_{1} < \lambda_{1}$ , it follows that  $u_{1} < \lambda_{1}$  (because  $u_{1} - \lambda_{1} = (1 - \beta)(v_{1} - \lambda_{1})$  with  $\beta < 1$ ), which reduces the spectral gap and impairs information flow, especially in large graphs, where  $v_{1} = 2 / n - 2$  is very, very small. However, our approach perturbs the normalized Laplacian in a way that maintains subtle eigenvalue differences regardless of graph size, thereby preserving the topological information.
+
+Below, we verify our argument above by presenting the spectral gaps' orders of magnitude for the normalized graph Laplacian and PSL across various datasets. As shown in the Table 7, for each dataset,  $v_{1} < \lambda_{1}$ ,  $u_{1} < \lambda_{1}$ , and  $\lambda_{1} \leq \hat{\lambda}_{1}$  which indicates that Lu et al.'s approach (Lu et al., 2024) indeed reduces the spectral gap, thus compromising the original topological information encoded in the normalized Laplacian matrix.
+
+Table 7. Comparison of spectral gap  
+
+<table><tr><td></td><td>Cora</td><td>Pubmed</td><td>Citeseer</td><td>Photo</td><td>Texas</td><td>Cornell</td><td>Actor</td></tr><tr><td>n</td><td>2708</td><td>19717</td><td>3327</td><td>7650</td><td>183</td><td>183</td><td>7600</td></tr><tr><td>v1</td><td>7e-4</td><td>1e-4</td><td>6e-4</td><td>2e-4</td><td>1e-2</td><td>1e-2</td><td>2e-4</td></tr><tr><td>λ1</td><td>4e-3</td><td>1e-2</td><td>1e-3</td><td>1e-3</td><td>5e-2</td><td>7e-2</td><td>3e-2</td></tr><tr><td>ˆλ1</td><td>5e-3</td><td>1e-2</td><td>2e-3</td><td>1e-3</td><td>5e-2</td><td>7e-2</td><td>3e-2</td></tr></table>
+
+# F.3. Node Classification in A Fully-Supervised Split
+
+We train all models in a fully-supervised split (60% / 20% / 20%). For each baseline (including the EC-GNN variants), we adopt the best hyperparameter settings reported in the original paper. For all PSL-GNN variants, we use the same settings as their corresponding baselines, with  $\eta = 1 \times 10^{-3}$  and  $d = 2$ . The results are summarized in Table 8 and Table 9.
+
+Table 8. Results on real-world datasets in the dense splitting  $\left( {{60}\% /{20}\% /{20}\% }\right)$  .  
+
+<table><tr><td>Datasets</td><td>Cora</td><td>Citeseer</td><td>PubMed</td><td>Computers</td><td>Photo</td></tr><tr><td>GCN</td><td>87.16±1.03</td><td>79.84±0.73</td><td>86.79±0.32</td><td>83.31±1.01</td><td>88.24±0.78</td></tr><tr><td>EC-GCN</td><td>88.73±0.89</td><td>80.63±1.01</td><td>87.68±0.83</td><td>83.94±1.17</td><td>90.45±1.20</td></tr><tr><td>PSL-GCN</td><td>89.09±1.14</td><td>81.04±0.63</td><td>88.75±1.24</td><td>84.12±1.65</td><td>91.48±0.81</td></tr><tr><td>APPNP</td><td>88.23±1.05</td><td>80.61±0.94</td><td>88.17±0.38</td><td>85.35±0.38</td><td>90.48±0.57</td></tr><tr><td>EC-APPNP</td><td>89.54±1.43</td><td>81.48±1.23</td><td>88.42±1.36</td><td>86.10±1.07</td><td>90.75±1.06</td></tr><tr><td>PSL-APPNP</td><td>90.91±1.14</td><td>81.92±1.36</td><td>88.88±1.42</td><td>86.43±0.68</td><td>91.82±0.96</td></tr><tr><td>Graph-Heat</td><td>87.34±1.41</td><td>78.82±1.26</td><td>87.52±0.89</td><td>83.23±1.37</td><td>89.97±0.82</td></tr><tr><td>EC-Heat</td><td>88.13±1.02</td><td>79.44±1.12</td><td>88.93±0.91</td><td>83.86±1.24</td><td>90.57±0.86</td></tr><tr><td>PSL-Heat</td><td>88.51±1.38</td><td>80.02±1.31</td><td>89.21±1.57</td><td>84.64±1.58</td><td>90.87±0.91</td></tr><tr><td>BernNet</td><td>88.48±1.14</td><td>80.08±1.07</td><td>88.73±1.42</td><td>87.68±0.53</td><td>94.31±0.86</td></tr><tr><td>EC-BernNet</td><td>88.64±0.55</td><td>80.30±0.98</td><td>89.07±1.37</td><td>88.34±1.03</td><td>94.50±1.01</td></tr><tr><td>PSL-BernNet</td><td>90.01±0.93</td><td>80.86±1.15</td><td>90.13±1.44</td><td>88.74±0.56</td><td>94.79±1.03</td></tr><tr><td>GPRGNN</td><td>88.54±0.82</td><td>80.09±1.03</td><td>88.52±0.46</td><td>87.01±0.74</td><td>93.87±0.34</td></tr><tr><td>EC-GPR</td><td>89.41±0.69</td><td>80.66±1.01</td><td>89.64±0.53</td><td>89.91±0.68</td><td>94.76±1.02</td></tr><tr><td>PSL-GPR</td><td>90.13±0.92</td><td>81.11±0.76</td><td>89.82±1.89</td><td>89.93±1.70</td><td>94.87±0.96</td></tr><tr><td>Jacobi</td><td>88.96±0.68</td><td>80.73±0.88</td><td>89.67±0.82</td><td>90.42±0.31</td><td>95.52±0.33</td></tr><tr><td>EC-Jacobi</td><td>89.06±0.67</td><td>81.28±0.96</td><td>89.87±0.42</td><td>90.33±0.28</td><td>95.54±0.36</td></tr><tr><td>PSL-Jacobi</td><td>90.73±1.34</td><td>81.54±1.12</td><td>90.42±1.13</td><td>90.83±0.61</td><td>95.69±0.52</td></tr></table>
+
+Table 9. Results on real-world datasets in the dense splitting  $\left( {{60}\% /{20}\% /{20}\% }\right)$  .  
+
+<table><tr><td>Datasets</td><td>Chameleon</td><td>Actor</td><td>Squirrel</td><td>Texas</td><td>Cornell</td></tr><tr><td>GCN</td><td>59.65±2.52</td><td>34.63±1.62</td><td>46.64±1.28</td><td>78.65±2.98</td><td>76.42±5.21</td></tr><tr><td>EC-GCN</td><td>61.81±1.94</td><td>36.12±1.44</td><td>50.31±0.84</td><td>80.72±2.39</td><td>77.38±4.37</td></tr><tr><td>PSL-GCN</td><td>62.04±2.71</td><td>36.74±1.43</td><td>52.73±1.54</td><td>81.45±2.43</td><td>78.13±4.32</td></tr><tr><td>APPNP</td><td>51.78±1.91</td><td>39.59±0.83</td><td>34.74±0.60</td><td>90.89±1.53</td><td>91.84±2.12</td></tr><tr><td>EC-APPNP</td><td>52.93±2.12</td><td>40.32±0.94</td><td>35.33±0.84</td><td>91.32±1.34</td><td>91.97±1.64</td></tr><tr><td>PSL-APPNP</td><td>54.49±1.74</td><td>40.67±0.91</td><td>35.63±0.58</td><td>91.53±1.49</td><td>92.07±1.94</td></tr><tr><td>Graph-Heat</td><td>63.46±0.92</td><td>35.74±1.31</td><td>44.35±2.14</td><td>80.06±0.94</td><td>76.87±0.85</td></tr><tr><td>EC-Heat</td><td>64.86±1.22</td><td>36.51±0.94</td><td>45.01±1.87</td><td>81.25±1.36</td><td>77.91±0.63</td></tr><tr><td>PSL-Heat</td><td>65.60±1.23</td><td>36.93±1.45</td><td>45.32±1.89</td><td>81.49±1.10</td><td>80.94±1.05</td></tr><tr><td>BernNet</td><td>68.25±1.59</td><td>41.82±0.99</td><td>51.37±0.75</td><td>94.14±1.67</td><td>92.16±1.77</td></tr><tr><td>EC-BernNet</td><td>74.20±1.33</td><td>41.87±1.02</td><td>62.79±0.78</td><td>94.37±1.44</td><td>93.77±1.26</td></tr><tr><td>PSL-BernNet</td><td>75.09±1.46</td><td>41.93±1.30</td><td>63.82±1.33</td><td>94.54±1.80</td><td>94.01±0.93</td></tr><tr><td>GPRGNN</td><td>67.14±1.10</td><td>39.92±0.65</td><td>50.08±1.95</td><td>92.97±1.41</td><td>91.32±2.02</td></tr><tr><td>EC-GPR</td><td>74.24±1.06</td><td>40.42±0.77</td><td>62.48±2.03</td><td>92.27±1.92</td><td>90.79±2.22</td></tr><tr><td>PSL-GPR</td><td>74.78±1.34</td><td>41.17±0.96</td><td>63.65±0.87</td><td>94.74±1.65</td><td>92.44±1.85</td></tr><tr><td>Jacobi</td><td>74.23±1.45</td><td>41.16±0.70</td><td>57.38±1.24</td><td>93.45±2.03</td><td>92.94±2.38</td></tr><tr><td>EC-Jacobi</td><td>75.64±1.51</td><td>41.01±0.74</td><td>59.87±0.91</td><td>93.48±1.49</td><td>93.29±2.33</td></tr><tr><td>PSL-Jacobi</td><td>75.87±1.44</td><td>41.94±0.75</td><td>61.47±0.98</td><td>94.20±1.75</td><td>93.91±1.96</td></tr></table>
+
+From the above results, PSL-GNNs consistently outperform the baselines, which validates the effectiveness of the perturbed sheaf Laplacian (PSL).
+
+# F.4. Eigenvalue Perturbation Study
+
+Every restriction map  $\mathcal{F}_{v\leq e}$  is generated using  $d$  parameters, and is bounded by  $\eta$ , where  $d$  represents the dimension of the stalks. To evaluate the impact of different perturbation scales  $\eta$  on the eigenvalues, we compare the maximum relative eigenvalue perturbation between the resulting sheaf Laplacian matrix and the normalized graph Laplacian matrix. The
+
+maximum relative perturbation is defined as:
+
+$$
+\Delta \lambda = \max  \left| \frac {\hat {\lambda} _ {i} - \lambda_ {i}}{\lambda_ {i}} \right|, \quad \text {f o r} i = 1, 2, \dots , N, \tag {25}
+$$
+
+where  $\mathbf{N}$  is the dimension of the matrix,  $\hat{\lambda}_i$  denotes the eigenvalues of the sheaf Laplacian matrix, and  $\lambda_{i}$  represents the
+
+![](images/3b8c00a6bdfe3678366fada980ec0770e48b7d8514c2a7a4cd21d0f746f4ccc5.jpg)  
+$\Delta \lambda :d = 2$
+
+![](images/086cbf108a039f57c25073ba29afdf414abbd90285d2244af1617586f659f047.jpg)  
+$\Delta \lambda :d = 3$
+
+![](images/5ed2bcc169501240a41452060a5ada118b417d65b431c76b9f08ffdc40e9c6af.jpg)  
+$\Delta \lambda :d = 4$
+
+![](images/00d424db4e02979d02ffd8a1d8a7a1a76d5e2c188e9d301da7ec1c1476b8318e.jpg)  
+proportion :  $d = 2$
+
+![](images/074aaaa73a29f9f7e6606e184705e4463c74dc9d609c996719aaef86fb1f3859.jpg)  
+proportion:  $d = 3$
+
+![](images/a7f3d0e17b10f8657b7587d3f8b36d5b1ee292b72530f41a22311fb2c7568f68.jpg)  
+proportion:  $d = 4$  
+Figure 4. The relative eigenvalue perturbations and the proportion of distinct eigenvalues of the perturbed sheaf Laplacian across different datasets under varying  $\eta$  and  $d$ .
+
+eigenvalues of the normalized graph Laplacian matrix. We also examine how the proportion of distinct eigenvalues, defined as  $k / |v|$  (where  $k$  is the number of distinct eigenvalues), varies with different  $\eta$ . The results for each dataset are shown in the Figure 4.
+
+As shown in Figure 4, when  $\eta$  decreases,  $\Delta \lambda$  also decreases, which is consistent with the theoretical analysis for the perturbation of eigenvalues in Theorem 5.5. At  $\eta = 1\mathrm{e} - 3$ , the number of distinct eigenvalues peaks but decreases for  $\eta > 1\mathrm{e} - 3$ , as larger perturbations may cause eigenvalues to overlap, violating Theorem 4.1. Furthermore, the impact of  $d$  is negligible.
+
+# F.5. Visualization of Eigenvalues Distribution
+
+To better illustrate whether the multiplicities of eigenvalues in the perturbed sheaf Laplacian  $\overline{\mathcal{L}_{\mathcal{F}}}$  are reduced across different datasets, we visualize its eigenvalue distribution alongside that of the normalized graph Laplacian  $\mathcal{L}$ . Figure 5 shows the eigenvalues distributions under a perturbation scale of  $\eta = 1e - 3$ , where the vertical axis (density) represents the proportion of each distinct eigenvalue relative to the total, and the horizontal axis ( $\lambda$ ) represents the eigenvalues. Based on the analysis of these figures, we observe the following:
+
+1. For almost every graph dataset, the perturbed sheaf Laplacian matrix consistently exhibits fewer repeated eigenvalues compared to the corresponding normalized graph Laplacian matrix.  
+2. On small graph datasets, the reduction in eigenvalue multiplicity is more pronounced, which aligns with the greater average performance gains observed in Table 1.  
+3. For every graph dataset, the eigenvalue 1 always has the highest multiplicity. This explains why GCN performs worse than other baselines, as its filtering function removes frequency components associated with eigenvalue 1. In contrast,  $\mathcal{L}_{\mathcal{F}}$  reduces the multiplicity of eigenvalue 1, which explains the highest average improvement for GCN, as it preserves the frequency components.
+
+![](images/60383e782d2e9a76a05a20cf4ba581410c40be8a697c8284c935ed261c7b67d6.jpg)
+
+![](images/eade96d5548e8d8f73937236407d1b1f5aa3030d49b0b4bafd05ce258f1812b1.jpg)
+
+![](images/0cdd7f955a9a030541690892407e61b3e42ed7245123d935da437b3c94c34e7b.jpg)
+
+![](images/19ff4f3d5bca154b1ab4ec6e2d41e538a1acb46b9b64df6c86cdff7359ad831c.jpg)
+
+![](images/401d418880c0c76dc18021cb6b4bdf00e31b2ad482dcdece2db155ca1b3354dd.jpg)
+
+![](images/81cf916cc7fe3c29e116fbcb57e5eaa4d5b855955c9b6b9a97e48b09e420b49b.jpg)
+
+![](images/53ed1f4b22a278538b600d89d4b17fc4837dd9b9fc8d0d5df0aa77ef90ee7117.jpg)
+
+![](images/a05eac8c2daf6cd1908bbc808928e14636a6962277e818ed6d1a2b71707cee29.jpg)
+
+![](images/5aed5d928d9e5c9b8a115c3f22fac0d2e1ff86866524ecfa2383029383b884bc.jpg)  
+$\overline{\mathcal{L}_{\mathcal{F}}}$  on Cora
+
+![](images/8e4076ada7a832c134d0d24824687b0ab1a316439fe9e3fbf3de3463d8abf2b7.jpg)
+
+![](images/a631dd8bbf9290cb49329fe1c05cb7c135488f098ced53d7d8bbad3b7c2ec588.jpg)  
+$\overline{\mathcal{L}_{\mathcal{F}}}$  on Photo
+
+![](images/530df7f81b6593c58969bfbdfbdea261a825126dba73874336f89b402f3dc2c6.jpg)  
+$\overline{\mathcal{L}_{\mathcal{F}}}$  on Texas
+
+![](images/894d80de00f7403dba45801cb66cef3a33d70b37806c8499f6be4eb7f55a4c54.jpg)  
+L
+
+![](images/ef3c477a3b9efdd7dd7e195bfb2044a7186413ba915019e78f077ad9dfc2c2ff.jpg)  
+$\overline{\mathcal{L}_{\mathcal{F}}}$  on Actor  
+Figure 5. Eigenvalues distribution of normalized graph Laplacian matrices and perturbed sheaf Laplacian matrices.
+
+# F.6. Hyper-Parameters Study
+
+To demonstrate how hyper-parameters influence the performance of the model, we study the effect of the two main hyperparameters of PSL-GPF and PSL-GCN, i.e. perturbation size  $\eta$  and stalk dimension  $d$ . For the remaining parameters, we set the same configuration across Cora, Texas, and Actor.
+
+The results are shown in the Figure 6. From the results, we have the following observations:
+
+1. These models perform best when  $\eta = 1e - 3$  and  $\eta = 1e - 2$ . This could be explained by Figure 5: when  $\eta = 1e - 3$ , the proportion of distinct eigenvalues in the fixed perturbed sheaf Laplacian matrix is maximized. Parameters of the restriction mapping are constrained within the range  $[1e - 4, 1e - 3]$ , and the parameters might gradually approach the upper bound of  $1e - 3$ . When  $\eta = 1e - 2$ , the performance is similar, as the parameters are constrained within the range  $[1e - 3, 1e - 2]$ , they might gradually approach the lower bound of  $1e - 3$ .  
+2. When  $\eta = 1e - 4$ , the model's performance is similar to that when no perturbation is applied (i.e.,  $\eta = 0$ ). As shown in Figure 5, the perturbation effect is nearly negligible in this case, and the perturbed sheaf Laplacian matrix becomes almost identical to the normalized graph Laplacian matrix.
+
+# F.7. Comparison between PSL and GSL
+
+In the fully-supervised split, we compared the performance of PSL and GSL. The results are shown below:
+
+![](images/351a278f090dcc1e10254ad55cdc7c30e579663f5e17a2426161cc9d25c74c79.jpg)
+
+![](images/a3ef5c601be5e58fda301c326a730b719fdf0a696f76cc39e624459d04181491.jpg)
+
+![](images/c4656a706aed3e07b37da80f93dcd1f0a2c95ca07f2bae0e78256e113e8504a5.jpg)
+
+![](images/6af8e2859714db913d09fe178a6ddb2f3f1fcc30d7e7f0e58fce33d7ea4bcca4.jpg)  
+PSL-GPR on Cora
+
+![](images/45b49aa1a12fa9f13a1dc9b38d0fa1f591ac0be7a96bd54e961a833d05492fa0.jpg)  
+PSL-GPR on Texas
+
+![](images/38a54031fdaf2949e51522239a7a7e96db5809700f82afd0da5f9108ebcfb63a.jpg)  
+PSL-GPR on Actor
+
+![](images/11d5233928fe7c73ceef6a3d9278b0fc4a04dab20a8a4cdac52afc3ed2ff0bcc.jpg)  
+PSL-GCN on Cora
+
+![](images/9e29c9bac597486504c8458ab70aba4b2852f0feb0f32f132223f18b5c4010d7.jpg)  
+PSL-GCN on Texas
+
+![](images/4ed8dc6fbe396d124a9e878ce115e4c39f1ad85d128dec960a64655a56be5058.jpg)  
+PSL-GCN on Actor
+
+![](images/f99baaed1edb80ec27f1417013087c35a038272b16448259dc6c6f78c5bb969c.jpg)  
+PSL-APPNP on Cora
+
+![](images/71f88356d7b731d6bd00fc4666fe70023574cd46a12847f5cdd437df2e59732f.jpg)  
+PSL-APPNP on Texas
+
+![](images/cbffcc476b7df7f63ecb2161861848a565ff8c1ec8e188c95f9235f95c25edb3.jpg)  
+PSL-APPNP on Actor
+
+![](images/56b03c6131eee4535d2837555848f7fbe0824302ca77da5b22811d0874ebf000.jpg)  
+PSL-Heat on Cora
+
+![](images/8441de66bc061eae231cf97a196213cac56c7bebb730f29acdd86f8a3ab7ebcc.jpg)  
+PSL-Heat on Texas
+
+![](images/1677b5739290292374cd796dda96fb67d9b83a9d67f462452140eee8afe15c85.jpg)  
+PSL-Heat on Actor
+
+![](images/80dd44a336c399b96643bcfe3243db2e7aa02e3f416fa341e3096a4025828caa.jpg)  
+PSL-BernNet on Cora  
+PSL-Jacobi on Cora
+
+![](images/b30644066a39005b3bdbe73d2e3174c02185f83e73ccde56982d7497ccda645c.jpg)  
+PSL-BernNet on Texas  
+PSL-Jacobi on Texas  
+Figure 6. Hyper-Parameters Study.
+
+![](images/ef59f8c4268cfd24902d696d9fac9c4701c2e62d5f29446e91608f0876ee2e70.jpg)  
+PSL-BernNet on Actor  
+PSL-Jacobi on Actor
+
+Table 10. Comparison of models based on three Laplacian matrices (GNN vs. GSL-GNN vs. PSL-GNN): Mean accuracy  $(\%)$  ±95  $(\%)$  on seven datasets (in dense splitting:  $60\% /20\% /20\%$ ).  
+
+<table><tr><td>Datasets</td><td>Cora</td><td>Pubmed</td><td>Citeseer</td><td>Photo</td><td>Texas</td><td>Cornell</td><td>Actor</td></tr><tr><td>GCN</td><td>87.16±1.03</td><td>86.79±0.32</td><td>79.84±0.73</td><td>88.24±0.78</td><td>78.65±2.98</td><td>76.42±5.21</td><td>34.63±1.62</td></tr><tr><td>GSL-GCN</td><td>88.73 ±0.98</td><td>87.04 ±0.97</td><td>80.89 ±0.51</td><td>89.67 ±0.88</td><td>80.75 ±2.64</td><td>77.93 ±4.37</td><td>35.94 ±1.29</td></tr><tr><td>PSL-GCN</td><td>89.09±1.14</td><td>88.75±1.24</td><td>81.04±0.63</td><td>91.48±0.81</td><td>81.45±2.43</td><td>78.13±4.32</td><td>36.74±1.43</td></tr><tr><td>APPNP</td><td>88.23±1.05</td><td>88.17±0.38</td><td>80.61±0.94</td><td>90.48±0.57</td><td>90.89±1.53</td><td>91.84±2.12</td><td>39.59±0.83</td></tr><tr><td>GSL-APPNP</td><td>89.88±0.95</td><td>88.46±1.38</td><td>81.52±1.16</td><td>90.89±0.93</td><td>91.19±1.47</td><td>91.93±2.23</td><td>39.81±0.84</td></tr><tr><td>PSL-APPNP</td><td>90.91±1.14</td><td>88.88±1.42</td><td>81.92±1.36</td><td>91.82±0.96</td><td>91.53±1.49</td><td>92.07±1.94</td><td>40.67±0.91</td></tr><tr><td>Graph-Heat</td><td>87.34±1.41</td><td>87.52±0.89</td><td>78.82±1.26</td><td>89.97±0.82</td><td>80.06±0.94</td><td>76.84±0.85</td><td>35.74±1.31</td></tr><tr><td>GSL-Heat</td><td>87.71±1.22</td><td>89.14±0.89</td><td>79.23±1.02</td><td>90.34±1.31</td><td>80.79±1.9</td><td>80.87±0.70</td><td>36.36±1.87</td></tr><tr><td>PSL-Heat</td><td>88.51±1.38</td><td>89.21±1.57</td><td>80.02±1.31</td><td>90.87±0.91</td><td>81.49±1.10</td><td>80.94±1.05</td><td>36.93±1.45</td></tr><tr><td>BernNet</td><td>88.48±1.14</td><td>88.73±1.42</td><td>80.08±1.07</td><td>94.31±0.86</td><td>94.14±1.67</td><td>92.16±1.77</td><td>41.82±0.99</td></tr><tr><td>GSL-BernNet</td><td>89.96±1.11</td><td>89.84±1.50</td><td>80.77±0.81</td><td>94.67±1.10</td><td>94.23±1.76</td><td>92.68±1.07</td><td>41.88±1.44</td></tr><tr><td>PSL-BernNet</td><td>90.01±0.93</td><td>90.13±1.44</td><td>80.86±1.15</td><td>94.79±1.03</td><td>94.54±1.80</td><td>94.01±0.93</td><td>41.93±1.30</td></tr><tr><td>GPRGNN</td><td>88.54±0.82</td><td>88.52±0.46</td><td>80.09±1.03</td><td>93.87±0.34</td><td>92.97±1.41</td><td>91.32±2.02</td><td>39.92±0.65</td></tr><tr><td>GSL-GPR</td><td>89.82±1.23</td><td>89.04±1.01</td><td>80.56±0.93</td><td>94.12±1.25</td><td>93.47±1.38</td><td>91.87±1.72</td><td>40.64±0.82</td></tr><tr><td>PSL-GPR</td><td>90.13±0.92</td><td>89.82±1.89</td><td>81.11±0.76</td><td>94.87±0.96</td><td>94.74±1.65</td><td>92.44±1.85</td><td>41.17±0.96</td></tr><tr><td>Jacobi</td><td>88.96±0.68</td><td>89.67±0.82</td><td>80.73±0.88</td><td>95.52±0.33</td><td>93.45±2.03</td><td>92.94±2.38</td><td>41.16±0.70</td></tr><tr><td>GSL-Jacobi</td><td>89.43±0.94</td><td>89.92±1.33</td><td>81.24±1.16</td><td>95.55±0.36</td><td>93.84±1.97</td><td>93.45±1.71</td><td>41.21±0.45</td></tr><tr><td>PSL-Jacobi</td><td>90.73±1.34</td><td>90.42±1.13</td><td>81.54±1.12</td><td>95.69±0.52</td><td>94.20±1.75</td><td>93.91±1.96</td><td>41.94±0.75</td></tr></table>
+
+It can be observed that PSL outperforms GSL, further validating the effectiveness of the perturbed sheaf Laplacian.

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