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2025/Interactive Cross-modal Learning for Text-3D Scene Retrieval/38ab8253-143b-4733-8680-ad3c5b1a61ad_origin.pdf

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2025/Interactive Cross-modal Learning for Text-3D Scene Retrieval/full.md

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+# Interactive Cross-modal Learning for Text-3D Scene Retrieval
+
+Yanglin Feng $^{1}$ , Yongxiang Li $^{1}$ , Yuan Sun $^{2}$ , Yang Qin $^{1}$ , Dezhong Peng $^{1,3}$ , Peng Hu $^{1*}$
+
+<sup>1</sup>College of Computer Science, Sichuan University, Chengdu, China.
+
+$^{2}$ National Key Laboratory of Fundamental Algorithms and Models for Engineering Numerical Simulation, Sichuan University, Chengdu, China.
+
+$^{3}$ Tianfu Jincheng Laboratory, Chengdu, China.
+
+fcyzfyl@163.com, rhythmmli.scu@gmail.com, sunyuan_work@163.com, qinyang.gm@gmail.com, pengdz@scu.edu.cn, penghu.ml@gmail.com
+
+# Abstract
+
+Text-3D Scene Retrieval (T3SR) aims to retrieve relevant scenes using linguistic queries. Although traditional T3SR methods have made significant progress in capturing fine-grained associations, they implicitly assume that query descriptions are information-complete. In practical deployments, however, limited by the capabilities of users and models, it is difficult or even impossible to directly obtain a perfect textual query suited the entire scene and model, thereby leading to performance degradation. To address this issue, we propose a novel Interactive Text-3D Scene Retrieval Method (IDEal), which promotes the enhancement of the alignment between texts and 3D scenes through continuous interaction. To achieve this, we present an Interactive Retrieval Refinement framework (IRR), which employs a questioner to pose contextually relevant questions to an answerer in successive rounds that either promote detailed probing or encourage exploratory divergence within scenes. Upon the iterative responses received from the answerer, IRR adopts a retriever to perform both feature-level and semantic-level information fusion, facilitating scene-level interaction and understanding for more precise re-rankings. To bridge the domain gap between queries and interactive texts, we propose an Interaction Adaptation Tuning strategy (IAT). IAT mitigates the discriminability and diversity risks among augmented text features that approximate the interaction text domain, achieving contrastive domain adaptation for our retriever. Extensive experimental results on three datasets demonstrate the superiority of IDeal. Code is available at https://github.com/Yang11nFeng/IDEal.
+
+# 1 Introduction
+
+Recent years have witnessed natural language interfaces to embodied intelligence systems [1, 2, 3, 4, 5] become increasingly prevalent in our daily lives. This opens up further opportunities for natural language-based interaction with intelligent agents, such as a user verbally instructing agents to perform tasks in a specific scene. Before executing any task, an agent must first retrieve the scene relevant to the user's intent. This requirement has spurred recent works on Text-3D Scene Retrieval (T3SR) [6, 7], which enables the retrieval of 3D point-cloud scenes using linguistic queries. Such a language-scene alignment capability lays a critical foundation for enabling agents to generalize across scenes and environments.
+
+Although existing dedicated methods [6] achieve promising performance for T3SR by facilitating fine-grained query text and scene understanding, such success often relies on the assumption that
+
+![](images/9b7f7974724dcccb04353a97ded30ee08f46e6781d79d8cf1824981e381d98a3.jpg)  
+Figure 1: Overview of interactive text-3D scene retrieval. The above gray part illustrates the two specific challenges, while the below white part shows the illustrations of our interactive framework.
+
+the queries provided are information-complete. However, such an assumption is often violated in real-world scenarios due to the inherent limitations of text inputs and models, such as incomplete one-shot descriptions of user intent [8], ambiguous descriptions [6], domain shifts [9], and limited generalization of the models. As a result, the performance and robustness of the models remain persistently constrained, and relying solely on limited internal knowledge is insufficient to overcome this inherent bottleneck.
+
+To break through the bottleneck, recent studies [8, 10, 11] have explored integrating external knowledge from Large Language Models (LLMs) to enhance their understanding and alignment abilities. However, such approaches typically require prohibitively expensive fine-tuning or retraining of offline models [10, 12]. Some other attempts [8, 13, 14] have proposed interactive cross-modal retrieval frameworks that incorporate LLMs and Vision-Language Models (VLMs) to facilitate more fine-grained understanding and alignment, thereby iteratively evolving the retrieval performance. Although these methods have demonstrated remarkable effectiveness in image-text matching [8] and video-text retrieval [15, 16], they still face two tricky challenges in the T3SR setting, as shown in Figure 1. Firstly, since the scale and complexity of 3D scenes, these methods lack holistic perspectives beyond a localized focus during interaction, e.g., the LLMs tend to focus on the salient objects in the scene and ignore the fine-grained details at a scene-level perspective, limiting the depth and breadth of LLM interaction, as demonstrated in Table 1. Secondly, existing retrieval models exhibit limited generalization ability to biased text domains, limiting their effectiveness in handling realistic interaction texts that exhibit domain gaps.
+
+To address the aforementioned challenges, this paper proposes a novel Interactive Text-3D Scene Retrieval method (IDEal) to conduct continuous interaction between the T3SR models and external users (e.g., LLMs), achieving the active alignment between text queries and 3D scenes, as depicted in Figure 1. Our IDEal consists of two components: an Interactive Retrieval Refinement framework (IRR) and an Interaction Adaptation Tuning strategy (IAT), as illustrated in Figure 2. More specifically, IRR coordinates three specialized agents (i.e., questioner, answerer, and retriever) to perform multi-round interaction. First, the questioner adaptively determines whether to continue probing object details or to pursue divergence by exploring the broader scene, based on the assessment of the current round's description. Based on this, it continuously formulates context-relevant questions to the answerer. After receiving responses, the retriever iteratively integrates information at both the feature and semantic levels, facilitating comprehensive scene-level understanding for progressively precise re-rankings. To mitigate the domain shift between training queries and interactive texts, IAT proposes adapting the retriever toward the interaction text domain. Specifically, IAT leverages LLMs to generate more realistic augmented texts that closely resemble the interaction text domain. Subsequently, IAT robustly mitigates the discriminability and diversity theoretical risks in the features of the augmented texts for domain gap bridging, thereby ensuring an unbiased understanding of the interaction texts by the retriever. The contributions of this paper are as follows:
+
+- We propose a novel Interactive Text-3D Scene Retrieval Method (IDEal), which actively enhances alignment between text queries and 3D scenes through ongoing interactions.  
+- An Interactive Retrieval Refinement framework (IRR) is presented to enable a deep interaction for comprehensive scene exploration, leading to progressively improved retrieval.  
+- An Interaction Adaptation Tuning strategy (IAT) is proposed, which facilitates the transfer of the retriever to the interaction text domain, promoting improved interaction.  
+- We conduct extensive comparison experiments on text-3D scene datasets. Our IDEal remarkably outperforms the existing methods, demonstrating its superiority.
+
+# 2 Related Work
+
+Cross-Modal Retrieval. Cross-Modal Retrieval (CMR) [17, 18, 19, 20, 21, 22] aims to match corresponding results across modalities for a given query, bridging the gap caused by modal heterogeneity. In recent years, CMR has garnered significant attention in fields such as Image-Text Retrieval [14, 23, 24, 25], Video-Text Retrieval [26, 27], 2D-3D Retrieval [28, 29], Pointcloud-Text Matching [6]. The primary challenge of CMR lies in effectively aligning multimodal data. To address this issue, most existing works could be broadly categorized into two groups: 1) Coarse-grained retrieval [30, 31, 32] directly maps multimodal data into a shared space, aiming for a more straightforward and computationally efficient alignment. 2) Fine-grained retrieval [33, 34] seeks to establish local associations between fine-grained features across modalities (e.g., regions in images, words in texts). These local associations are then progressively integrated to form precise cross-modal correspondences. This paper focuses on a more challenging CMR task, i.e., Text-3D Scene Retrieval, involving obscure spatial cues and sophisticated 3D scenes (including issues such as viewpoints and occlusions [35, 36]). Although prior work [6] performs well with comprehensive descriptions, it struggles with online and blurry queries. To this end, we propose an Interactive T3SR solution that iteratively incorporates online feedback to achieve more precise and practical scene retrieval.
+
+Interactive Learning. Unlike traditional learning paradigms [6, 37], interactive learning emphasizes the continuous improvement of a model's behavior through ongoing interactions with the environment or users. Specifically, several pioneering works [38, 39] leverage simple forms of user feedback (e.g., preferred sample selection and relevance scoring) to iteratively achieve improved training quality or better satisfy user-specific requirements during testing. With the development of Large Language Models (LLMs), other studies [8, 40, 41] have begun exploring question-answering interactions through free-form text dialogue, closely replicating natural human communication. For example, several methods leverage iterative interactions to continuously refine the retrieval query for better reranking. Recently, more advanced methods such as PlugIR [13], MERLIN [16], ICL [42], and LLaVA-ReID [43] have integrated LLMs for context-aware question generation, mining more visual details. However, these methods cannot be effectively generalized to T3SR due to the differences in tasks and data domains shown in Figure 1. In this paper, we develop an interactive framework tailored for T3SR to help the offline models adapt to complex scene perception.
+
+# 3 Method
+
+# 3.1 Problem Formulation
+
+Given a text query set  $\mathcal{T} = \{t_i\}_{i=1}^{n_t}$  and a 3D scene gallery  $\mathcal{C} = \{c_j\}_{j=1}^{n_c}$ , where  $t_i$  and  $c_j$  represent  $i$ -th text and  $j$ -th scene,  $n_t = |\mathcal{T}|$  and  $n_c = |\mathcal{C}|$  means the sample number, and  $|\cdot|$  denotes the volume of set. The purpose of T3SR is to use the text query to match the ideal 3D scenes from the gallery, where there exists correspondence  $y_{i,j} \in \{0,1\}$ , indicating whether the points are matched (i.e.,  $y_{ij} = 1$ ), or unmatched (i.e.,  $y_{ij} = 0$ ). Existing methods [6, 37] typically train an offline model to achieve encoding of multimodal data, followed by meticulous single-turn retrieval. They assume that user-provided text queries are information complete, overlooking the practical fact that queries are often partial, ambiguous, or even exhibit domain shift.
+
+To address these issues, an interactive Text-3D scene retrieval method, i.e., IDEal, is proposed to bridge the interaction between the retrieval models and external agents, progressively overcoming the aforementioned query limitations and achieving improved alignment between texts and 3D scenes. More specifically, IDEal asks question  $q_{i}^{l}$  ( $l \in \{1, \dots, r\}$ ) about  $i$ -th sample based on the users' previous response  $a_{i}^{l-1}$  ( $a_{i}^{0} = t_{i}$ ), where  $r$  is the upper limit of rounds. Subsequently, the external agents recall details and answer  $a_{i}^{l}$  of the target 3D scene  $c_{j}$  ( $y_{ij} = 1$ ), forming a dialogue context  $\mathcal{D}_{i} = \{a_{i}^{0}, (q_{i}^{1}, a_{i}^{1}), \dots, (q_{i}^{r}, a_{i}^{r})\}$  composed of question-answer pairs ( $q_{i}, a_{i}$ ). Both the  $j$ -round response text and 3D scenes are projected into a shared feature space by a trained retrieval model, which can be:  $\pmb{u}_{i}^{j} = f_{r}(a_{i}^{j}; \pmb{\theta})$  and  $\pmb{v}_{i} = f_{r}(c_{i}; \pmb{\theta})$ , where  $\pmb{\theta}$  denotes the learnable parameters.
+
+![](images/d8c936e403e3c7922a02ceb81d615befe1a1f2119538172abac16c6dd5c4f197.jpg)  
+Figure 2: (Left) Pipeline of the proposed IDeal. A questioner employs a router to adaptively pose either probe or divergent questions, which require responses from an answerer. After receiving iterative responses, a retriever projects the multimodal data into a shared feature space, performing feature-/semantic-level fusion (Right) to enable progressively precise scene retrieval.
+
+![](images/4c9c3530aea8ab27900360f0f8b2c37cee6e47949ae537a36b112c0700c3d0f2.jpg)
+
+# 3.2 Interactive Retrieval Refinement Framework
+
+In this section, we introduce the Interactive Retrieval Refinement framework (IRR), which coordinates three agents (i.e., questioner, answerer, and retriever) to enable iterative interaction. In the following sections, we will elaborate on them by introducing their interaction process.
+
+# 3.2.1 Adaptive Questioning
+
+To achieve a deep interaction for T3SR, we present an adaptive questioner, which enables pertinent questioning to the answerer for both detailed and comprehensive exploration of complex 3D scenes. Firstly, a question router is employed, which determines the focus of questioning based on the feature distribution in the shared space. More specifically, the router assesses whether the previous-round description is informative by computing a Cross-modal Affinity Entropy  $(\mathcal{E})$ , formulated as:
+
+$$
+\mathcal {E} \left(\boldsymbol {u} _ {i} ^ {r - 1}\right) = - \sum_ {j \in N _ {k} \left(\boldsymbol {u} _ {i} ^ {r - 1}\right)} p \left(\boldsymbol {u} _ {i} ^ {r - 1}, \boldsymbol {v} _ {j}\right) \log p \left(\boldsymbol {u} _ {i} ^ {r - 1}, \boldsymbol {v} _ {j}\right), \tag {1}
+$$
+
+where  $N_{k}(\pmb{u}_{i}^{r - 1})$  means the  $k$ -nearest-neighbor index of  $\pmb{u}_i^{r - 1}$ , and affinity probability  $p(\pmb{u}_i^{r - 1},\pmb{v}_j)$  is formulated as:
+
+$$
+p \left(\boldsymbol {u} _ {i} ^ {r - 1}, \boldsymbol {v} _ {j}\right) = \frac {\exp \left(\mathcal {S} \left(\boldsymbol {u} _ {i} ^ {r - 1} , \boldsymbol {v} _ {j}\right) / \tau\right)}{\sum_ {l \in N _ {k} \left(\boldsymbol {u} _ {i} ^ {r - 1}\right)} \exp \left(\mathcal {S} \left(\boldsymbol {u} _ {i} ^ {r - 1} , \boldsymbol {v} _ {l}\right) / \tau\right)}, \tag {2}
+$$
+
+where  $S$  represents the computation of similarity between features, and  $\tau$  is a temperature parameter.
+
+However, the distribution of scene features extracted by the trained retrieval models is fixed and inherently non-uniform, with regions of over-density and under-density introducing a structural density bias [44]. To mitigate the bias, we introduce a Density Compensated Factor for each scene feature, which is formulated as:  $\rho (\pmb {v}_i) = \frac{1}{(1 / k)\sum_{j\in N_k(\pmb{v}_j)}\mathcal{D}(\pmb{v}_i,\pmb{v}_j) + \epsilon}$ , where  $\mathcal{D}$  represents the distance calculation and  $\epsilon$  is a minimal constant for numerical stability. Based on this, we try to approximately correct the original similarity score  $S(\pmb{u}_i^{r - 1},\pmb{v}_j)$  by incorporating the Density Compensated Factor  $\rho (\pmb {v}_i)$ , which could be written as:  $\tilde{S} (\pmb{u}_i^{(r - 1)},\pmb {v}_j) = S(\pmb{u}_i^{(r - 1)},\pmb {v}_j) / \sqrt{\rho(\pmb{v}_j)}$ . Subsequently, this corrected similarity is brought back into Equations (1) and (2) to obtain a Density Compensated Affinity Entropy, denoted as  $\tilde{\mathcal{E}}$ . This process approximates a Bayesian Correction leveraging prior density estimation<sup>2</sup>, mitigating the impact of the inherent bias in scene feature distribution.
+
+Leveraging the fairer metric  $\tilde{\mathcal{E}}$ , our questioner categorize descriptions with  $\tilde{\mathcal{E}} > \beta$  as uninformative, prompting an LLM to generate questions for detail probe (i.e.,  $\mathcal{Q}_1$ ) for attribute and spatial relationship detail refinements within the described area. Conversely, when  $\tilde{\mathcal{E}} \leq \beta$ , the descriptions are considered informative, triggering the adopting of questions for diverge exploration (i.e.,  $\mathcal{Q}_2$ ) that inquire about object arrangements not previously discussed in the dialogue.
+
+# 3.2.2 Iterative Retrieval
+
+After completing the questioning, an LLM is employed to simulate the external user acting as an answerer to answer the questions, following existing interactive approaches [8, 13]. It receives multi-round questions and provides responses based on its memory. In this paper, we adopt text modality to simulate the memory, which better approximates how humans recall information in mind.
+
+Upon receiving the response descriptions from the answerer, we construct a retriever that can be seamlessly integrated with existing cross-modal models [37, 45], enabling iterative scene retrieval. It can obtain the scene retrieval predictions for any given text within the shared feature space. Specifically, given a text feature  $\boldsymbol{u}_i$ , the prediction can be formulated as:
+
+$$
+\hat {\boldsymbol {p}} (\boldsymbol {u} _ {i}) = \left[ \hat {p} (\boldsymbol {u} _ {i}, \boldsymbol {v} _ {1}), \hat {p} (\boldsymbol {u} _ {i}, \boldsymbol {v} _ {2}), \dots , \hat {p} (\boldsymbol {u} _ {i}, \boldsymbol {v} _ {n _ {c}}) \right] ^ {\top}, \tag {3}
+$$
+
+where  $\hat{p}(\boldsymbol{u}_i, \boldsymbol{v}_j) = \exp(\mathcal{S}(\boldsymbol{u}_i, \boldsymbol{v}_j)) / \sum_{l=1}^{n_c} \exp(\mathcal{S}(\boldsymbol{u}_i, \boldsymbol{v}_l))$  denotes the probability that the  $i$ -th text retrieves the  $j$ -th scene. Accordingly, our retriever first utilizes the initial query to compute an initial retrieval prediction  $\hat{p}_1(\boldsymbol{u}_i)$ . Subsequently, the feature-level and semantic-level fusion of interactive responses is conducted to achieve more precise scene retrieval.
+
+For the fusion of interactive response features, on one hand, considering responses to  $\mathcal{Q}_1$  are refinements of the previous-round descriptions, we apply a weighted linear fusion to incorporate supplementary information. This strategy enables the preservation of core semantic cues from previous rounds while emphasizing newly introduced details:  $\pmb{u}_i^j = \alpha \pmb{u}_i^j + (1 - \alpha) \pmb{u}_i^{j-1}$ , where  $q_i^j \in \mathcal{Q}_1$ ,  $\alpha$  is a trade-off weight. On the other hand, benefiting from  $\mathcal{Q}_2$ , the other response features and the aforementioned fused features capture variations across different regions of the scene. To fuse them into a comprehensive feature, inspired by [46], we model their distribution around the target scene by encapsulating them within a minimum enclosing hypersphere:
+
+$$
+\left(\boldsymbol {o} _ {i} ^ {*}, R _ {i} ^ {*}\right) = \arg \min  _ {\boldsymbol {o} _ {i}, R _ {i}} \left\{R _ {i}: \boldsymbol {u} _ {i} ^ {j} \boldsymbol {o} _ {i} ^ {\top} \leq R _ {i}, \forall j \right\}, \tag {4}
+$$
+
+where  $\pmb{o}_i^*$  and  $R_i^*$  are the center and radius of the hypersphere, respectively. Based on this, features near the hypersphere boundary are grouped into a boundary set  $\mathcal{U}_i^1$ , while the remainder constitute the central set  $\mathcal{U}_i^2$ . To balance fusion robustness and feature discrimination, potentially noisy boundary features in  $\mathcal{U}_i^1$  are aggregated at the hypersphere center and averaged with cleaner features in  $\mathcal{U}_i^2$  to yield the final fused response feature:  $\bar{\pmb{u}}_i = \frac{1}{2}\left(\pmb{o}_i^* + \frac{1}{|\mathcal{U}_i^2|}\sum_{\pmb{u}_i^j\in \mathcal{U}_i^2}\pmb{u}_i^j\right)$ . This fused feature is then input into Equation (3) to obtain an interactive feature prediction  $\hat{\pmb{p}}_2(\bar{\pmb{u}}_i)$ .
+
+However, the aforementioned feature-level fusion can not fully capture the holistic semantics of the responses. To address this limitation, we leverage an LLM to reconstruct the 3D scene from all responses in textual space. More specifically, inspired by Chain-of-Thought (CoT) [47], we decompose this process into object extraction and scene reconstruction for a more stable and comprehensive scene summary. The LLM first identifies the scene objects across multi-round responses and then summarizes an object-centric scene reconstruction text. Finally, the texts are encoded into feature  $s_i$ , from which a interactive semantic prediction  $\hat{p}_3(s_i)$  is computed using Equation (3).
+
+Finally, the initial and interactive predictions are combined through weighted fusion to obtain the final scene retrieval prediction as follows:
+
+$$
+\hat {\boldsymbol {p}} _ {c} (\boldsymbol {u} _ {i}) = \lambda_ {1} \hat {\boldsymbol {p}} _ {1} (\boldsymbol {u} _ {i}) + \lambda_ {2} \hat {\boldsymbol {p}} _ {2} (\bar {\boldsymbol {u}} _ {i}) + \lambda_ {3} \hat {\boldsymbol {p}} _ {3} (\boldsymbol {s} _ {i}), \tag {5}
+$$
+
+where  $\hat{\pmb{p}}_c(\pmb{u}_i)$  is the final retrieval prediction,  $\lambda_{1}$ ,  $\lambda_{2}$ , and  $\lambda_{3}$  are trade-off parameters. Benefiting from the adaptive questioning and the comprehensive retrieval information fusion, our IDEal can alleviate the limitations of initial queries through progressive interaction.
+
+# 3.3 Interaction Adaptation Tuning
+
+Although IRR can exploit interactions to promote retrieval quality, the limited text domain of the retriever remains a bottleneck that restricts further performance improvements. To overcome this limitation, we propose an Interaction Adaptation Tuning strategy (IAT), which enhances texts to approximate the domain of interaction texts.
+
+We begin by integrating information and descriptions of the same scenes from the training data to construct simulated memory for text augmentation. Following IRR, we first provide a training-database-based answerer (i.e., an LLM) with the constructed memory and initial queries. We then simulate the
+
+IRR interaction process by iteratively posing a fixed number of  $\mathcal{Q}_1$  and  $\mathcal{Q}_2$  questions. The response descriptions yield augmented texts that closely approximate the interaction scenario.
+
+After obtaining the enriched augmented texts, inspired by the contrastive domain adaptation paradigm [48, 49], we try to minimize the theoretical risk  $\mathcal{R}(\pmb{\theta})$  associated with our retriever among the augmented text features, as formulated below. It facilitates model adaptation of the retriever to the augmented text domain without requiring access to its implementation details.
+
+$$
+\mathcal {R} (\pmb {\theta}) = \mathcal {R} _ {d i s} (\pmb {\theta}) + \mathcal {R} _ {d i v} (\pmb {\theta}) = \mathbb {E} _ {\tilde {\mathcal {U}}} \left[ \left(- \mathbb {E} _ {\tilde {\mathcal {U}} ^ {+}} \left\{\mathcal {S} (\tilde {\pmb {u}} _ {i} ^ {+}, \tilde {\pmb {u}} _ {i}) \right\}\right) + \left(\mathbb {E} _ {\tilde {\mathcal {U}} ^ {-}} \left\{\mathcal {S} (\tilde {\pmb {u}} _ {i} ^ {-}, \tilde {\pmb {u}} _ {i}) \right\}\right) \right], \quad (6)
+$$
+
+where the two components  $\mathcal{R}_{dis}(\pmb{\theta})$  and  $\mathcal{R}_{div}(\pmb{\theta})$  respectively reflect discriminability and diversity risks,  $\mathbb{E}$  denotes expectation,  $\mathbb{E}_{\tilde{\mathcal{U}}}$  is taken with respect to the distribution for the target domain features (i.e., augmented text features  $\tilde{\mathcal{U}} = \{\tilde{u}_i\}_{i=1}^{n_t}$ ), and  $\tilde{\mathcal{U}}^+$  and  $\tilde{\mathcal{U}}^-$  is the distribution for the corresponding positive features  $\tilde{\pmb{u}}_i^+$  and negative features  $\tilde{\pmb{u}}_i^-$ , respectively.
+
+On the one hand, minimizing the discriminability risk  $\mathcal{R}_{dis}(\theta)$  requires encouraging the augmented text features to align closely with those belonging to the same scenes. However, the scale and complexity of scenes often lead to substantial variability even among features corresponding to the same scenes. This introduces significant uncertainty in the selection of positive samples, complicating the risk optimization process. To handle this, we adopt the corresponding 3D scene features as substitutes for the text features to construct positive pairs. This is based on the assumption that the scene features encoded by the well-trained model are more stably located near the center of the corresponding description distribution. Finally, we mitigate the aforementioned discriminability risk  $\mathcal{R}_{dis}(\theta)$  by minimizing a negative log-based proxy loss term  $\mathcal{L}_{dis}$ , which could be written as follows:
+
+$$
+\mathcal {L} _ {d i s} = - \sum_ {i = 1} ^ {b} \sum_ {j = 1} ^ {n _ {c}} y _ {i j} \log \mathcal {S} (\tilde {\boldsymbol {u}} _ {i}, \boldsymbol {v} _ {j}), \tag {7}
+$$
+
+where  $b$  is the size of the mini-batch. On the other hand, motivated by [50], we attempt to approximate the minimization of the divergence risk  $\mathcal{R}_{div}(\pmb{\theta})$  by minimizing its upper bound, i.e.,
+
+$$
+\sup  \left(\mathcal {R} _ {d i v} (\boldsymbol {\theta})\right) \sim \left\{\mathbb {E} _ {\tilde {\boldsymbol {u}} ^ {-} \sim \tilde {\mathcal {U}} ^ {-}} \left(\mathcal {S} \left(\tilde {\boldsymbol {u}}, \tilde {\boldsymbol {u}} ^ {-}\right)\right); \mathbb {V} _ {\tilde {\boldsymbol {u}} ^ {-} \sim \tilde {\mathcal {U}} ^ {-}} \left(\mathcal {S} \left(\tilde {\boldsymbol {u}}, \tilde {\boldsymbol {u}} ^ {-}\right)\right) \right\}, \tag {8}
+$$
+
+where  $\sup (\cdot)$  means the upper bound and  $\mathbb{V}$  is the variance. We can obviously see that the divergence risk is affected by the mean and variance of the selected negative samples. Yet existing methods [48] usually treat others within the same mini-batch as negative samples for contrastive learning, thereby minimizing the expectation term. Due to the stochasticity of mini-batch sampling, similar samples may be mistakenly chosen as negatives, which increases the variance of negative samples, enlarging the upper bound of the divergence risk.
+
+To tackle it, we propose a weighted complementary contrastive loss as a surrogate objective to achieve divergence risk optimization more robustly, which can be formulated as:
+
+$$
+\mathcal {L} _ {d i v} = \sum_ {i = 1} ^ {b} \sum_ {j \neq i} ^ {b} \underbrace {\exp \left(- \max  \left(0 , \mathcal {S} \left(\tilde {u} _ {i} , \tilde {u} _ {j}\right) - \gamma\right)\right)} _ {\text {W e i g h t i n g t e r m}} \underbrace {\log \left(1 - \mathcal {S} \left(\tilde {u} _ {i} , \tilde {u} _ {j}\right)\right)} _ {\text {C o m p l e m e n t a r y c o n t r a s t i v e t e r m}}, \tag {9}
+$$
+
+where  $\gamma$  is a threshold, above which samples are assigned lower weights. Minimizing the complementary contrastive term optimizes the expectation over negative pairs, while the weighting component can mitigate the impact of high-variance false-negative samples.
+
+Finally, we combine both terms to obtain our loss for domain adaptation tuning, as follows:
+
+$$
+\mathcal {L} = \lambda \mathcal {L} _ {\text {d i s}} + (1 - \lambda) \mathcal {L} _ {\text {d i v}}, \tag {10}
+$$
+
+where  $\lambda$  is a hyperparameter to control the contribution of each component. Minimizing this proxy loss facilitates the reduction of domain adaptation risk, thereby enabling the retriever to better adapt to the domain of interaction text.
+
+Table 1: Performance comparison on ScanRefer, Nr3D, and Sr3D in terms of R@1, R@5, R@10, and their sum (Rsum).  $\dagger$  denotes the use of coarse-grained descriptions as memory.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="4">ScanRefer</td><td colspan="4">Nr3D</td><td colspan="4">Sr3D</td></tr><tr><td>R@1</td><td>R@5</td><td>R@10</td><td>Rsum</td><td>R@1</td><td>R@5</td><td>R@10</td><td>Rsum</td><td>R@1</td><td>R@5</td><td>R@10</td><td>Rsum</td></tr><tr><td colspan="13">w/ coarse-grained descriptions:</td></tr><tr><td>VSE∞ (CVPR&#x27;21)</td><td>9.7</td><td>33.1</td><td>50.2</td><td>93.0</td><td>5.8</td><td>21.5</td><td>32.5</td><td>59.8</td><td>5.5</td><td>18.9</td><td>27.4</td><td>51.8</td></tr><tr><td>CHAN (CVPR&#x27;23)</td><td>9.4</td><td>32.3</td><td>52.1</td><td>93.8</td><td>7.5</td><td>15.0</td><td>32.1</td><td>54.6</td><td>5.4</td><td>21.3</td><td>34.8</td><td>61.5</td></tr><tr><td>HREM (CVPR&#x27;23)</td><td>10.2</td><td>34.0</td><td>51.4</td><td>95.6</td><td>7.3</td><td>18.1</td><td>31.9</td><td>57.3</td><td>5.4</td><td>21.9</td><td>33.6</td><td>60.9</td></tr><tr><td>CRCL (NeurIPS&#x27;23)</td><td>10.3</td><td>32.4</td><td>49.8</td><td>92.5</td><td>8.1</td><td>22.5</td><td>33.2</td><td>63.8</td><td>4.9</td><td>19.5</td><td>31.9</td><td>56.3</td></tr><tr><td>RoMa (TMM&#x27;25)</td><td>11.4</td><td>34.8</td><td>54.4</td><td>100.6</td><td>6.5</td><td>24.8</td><td>37.6</td><td>68.9</td><td>7.8</td><td>27.3</td><td>39.3</td><td>74.4</td></tr><tr><td>IDeal</td><td>16.0</td><td>42.7</td><td>59.8</td><td>118.5</td><td>11.7</td><td>34.8</td><td>50.4</td><td>96.9</td><td>10.3</td><td>30.2</td><td>48.5</td><td>89.0</td></tr><tr><td colspan="13">w/ fine-grained descriptions:</td></tr><tr><td>ChatIR† (NeurIPS&#x27;23)</td><td>21.8</td><td>55.4</td><td>73.1</td><td>150.3</td><td>15.6</td><td>40.3</td><td>58.1</td><td>114.0</td><td>12.1</td><td>35.9</td><td>50.3</td><td>98.3</td></tr><tr><td>Rewrite† (ICMR&#x27;24)</td><td>17.4</td><td>47.0</td><td>63.7</td><td>128.1</td><td>17.1</td><td>31.4</td><td>45.8</td><td>94.3</td><td>12.4</td><td>28.1</td><td>40.4</td><td>80.9</td></tr><tr><td>MERLIN† (EMNLP&#x27;24)</td><td>31.1</td><td>68.8</td><td>83.8</td><td>183.7</td><td>21.8</td><td>55.0</td><td>71.8</td><td>148.6</td><td>14.2</td><td>42.0</td><td>60.3</td><td>116.5</td></tr><tr><td>BASELINE: IR†</td><td>29.9</td><td>68.0</td><td>83.3</td><td>181.2</td><td>18.0</td><td>51.6</td><td>60.2</td><td>129.8</td><td>14.6</td><td>39.2</td><td>55.3</td><td>109.1</td></tr><tr><td>BASELINE SUM†</td><td>34.4</td><td>69.5</td><td>85.1</td><td>189.0</td><td>22.5</td><td>55.2</td><td>67.5</td><td>145.2</td><td>16.4</td><td>41.5</td><td>62.1</td><td>120.0</td></tr><tr><td>IDeal†</td><td>37.8</td><td>71.8</td><td>86.4</td><td>196.0</td><td>26.4</td><td>62.7</td><td>78.7</td><td>167.8</td><td>20.2</td><td>43.1</td><td>63.1</td><td>126.4</td></tr></table>
+
+# 4 Experiments
+
+# 4.1 Experimental Setting
+
+Datasets, baselines, and evaluation metrics: We adopt the ScanNet 3D scene set along with several description sets (i.e., ScanRefer [51], Nr3D [52], Sr3D [52], and SceneDepict-3D2T [6]) to conduct experiments, where ScanRefer, Nr3D, and Sr3D are employed as query sets, and SceneDepict-3D2T is employed to simulate fine-grained memory. To verify the superiority of our IDeal, we introduce eleven comparative baseline methods: five conventional offline cross-modal matching methods (i.e., VSE $\infty$  [45], CHAN [53], HREM [54], CRCL [37], and RoMa [6]), three interactive cross-modal retrieval methods (i.e., ChatIR [8], Rewrite [41], and MERLIN [16]), and two additional strong interactive baselines (IR and SUM). More specifically, the Iterative Reranking (IR) involves multi-round interaction, where the results are iteratively re-ranked based on the response of each round. The Summary reranking (SUM) also involves interaction, but ultimately aggregates all answers into a comprehensive description for matching.
+
+In addition, we follow [55, 56] to report R@1, R@5, R@10, and their summation (Rsum) as the evaluation metrics. Due to the space limitation, more details of datasets, prompts, and additional experiments are provided in the Supplemental Material.
+
+Implementation details: All methods are implemented in PyTorch and carried out on GeForce RTX 3090 GPUs. We adhere to the experimental settings of [6] for all method implementations. We adopt widely-used DGCNN [57] and BERT [58] to obtain fine-grained features for 3D point clouds and texts, respectively. To implement interaction, we explore two approaches to constructing memory:
+
+1) Coarse-grained description: We leverage an LLM to generate rich expansions of queries, serving as memory without introducing any additional information leakage. 2) Fine-grained description: In line with existing interactive methods [43, 13], we simulate the user's memory in real-world scenarios using fine-grained scene descriptions, albeit with access to partial additional information. In our experiments, we utilize Qwen-7B-Instruct [59] as our investigated LLM for the interaction experiments.
+
+Table 2: Performance comparison on ScanRefer and Nr3D in terms of R@1, R@5, R@10, and their sum. +IDEal indicates plugging the model into our IDEal. † denotes the use of fine-grained descriptions as memory.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="4">ScanRefer</td><td colspan="4">Nr3D</td></tr><tr><td>R@1</td><td>R@5</td><td>R@10</td><td>Rsum</td><td>R@1</td><td>R@5</td><td>R@10</td><td>Rsum</td></tr><tr><td>VSE∞</td><td>9.7</td><td>33.1</td><td>50.2</td><td>93.0</td><td>5.8</td><td>21.5</td><td>32.5</td><td>59.8</td></tr><tr><td>+IDEal</td><td>13.3</td><td>38.9</td><td>57.6</td><td>109.8</td><td>8.7</td><td>27.5</td><td>42.1</td><td>78.3</td></tr><tr><td>VSE∞†</td><td>14.9</td><td>42.3</td><td>61.5</td><td>118.7</td><td>16.4</td><td>47.5</td><td>55.2</td><td>119.1</td></tr><tr><td>+IDEal†</td><td>35.8</td><td>70.6</td><td>85.0</td><td>191.4</td><td>21.2</td><td>52.1</td><td>68.4</td><td>141.7</td></tr><tr><td>CRCL</td><td>10.3</td><td>32.4</td><td>49.8</td><td>92.5</td><td>8.1</td><td>22.5</td><td>33.2</td><td>63.8</td></tr><tr><td>+IDEal</td><td>13.4</td><td>35.5</td><td>56.1</td><td>105.0</td><td>7.4</td><td>25.4</td><td>38.3</td><td>71.1</td></tr><tr><td>CRCL†</td><td>17.5</td><td>45.1</td><td>58.3</td><td>120.9</td><td>13.4</td><td>44.5</td><td>51.5</td><td>109.4</td></tr><tr><td>+IDEal†</td><td>31.7</td><td>66.9</td><td>83.5</td><td>182.1</td><td>15.8</td><td>50.4</td><td>64.4</td><td>130.6</td></tr><tr><td>RoMa</td><td>9.7</td><td>33.1</td><td>50.2</td><td>93.0</td><td>8.3</td><td>27.9</td><td>37.2</td><td>73.4</td></tr><tr><td>+IDEal</td><td>16.0</td><td>42.7</td><td>59.8</td><td>118.5</td><td>11.7</td><td>34.8</td><td>50.4</td><td>96.9</td></tr><tr><td>RoMa†</td><td>16.7</td><td>44.8</td><td>61.6</td><td>123.1</td><td>17.4</td><td>48.5</td><td>57.5</td><td>123.4</td></tr><tr><td>+IDEal†</td><td>37.8</td><td>71.8</td><td>86.4</td><td>196.0</td><td>25.4</td><td>60.7</td><td>75.7</td><td>161.8</td></tr></table>
+
+# 4.2 Comparison on Text-3D Scene Retrieval
+
+Table 1 presents a comparison between our IDEal and conventional and interactive cross-modal matching methods under two memory settings. Table 2 further demonstrates the performance gains brought by integrating our interactive framework into conventional single-round methods. These results could yield the following observations: 1) Even without additional fine-grained information, our IDEal achieves competitive performance, highlighting its ability to uncover complementary information implicitly embedded in the queries, gradually alleviating inherent query limitations. 2) Compared to existing interactive methods, our IDEal also achieves superior performance with access to fine-grained descriptions. This demonstrates that our interactive questioning and retrieval strategies enable an ongoing understanding of user requirements and support a comprehensive interpretation of complex scenes. 3) Our IDEal can be seamlessly integrated into conventional cross-modal retrieval methods and yields substantial performance gains under both memory settings. This suggests that, beyond equipping offline models with interactive capabilities, IDEal empowers them to more effectively comprehend complex descriptions through interaction.
+
+# 4.3 Ablation Study
+
+In this section, we conduct an ablation study to evaluate the contribution of each proposed component to our IDEal. Specifically, we first ablate the router in the questioner, restricting it to ask either  $\mathcal{Q}_1$  or  $\mathcal{Q}_2$  continuously. In addition, we remove each of the three retrieval prediction strategies, and we examine removing CoT prompting in the reconstruction in the proposed retriever. Finally, we investigate the effect of not using the IAT strategy for domain adaptation and sequentially ablate its two loss terms. The results in Table 3 lead to the following observation: 1) Removing or replacing any component from IDEal results in performance degradation, highlighting the contribution of each component. Specifically, the adaptive questions generated by our questioner facilitate a meticulous
+
+and comprehensive understanding of scenes. The various feature aggregation strategies in the retriever contribute to precise scene matching. 2) Removing or substituting IAT components consistently leads to performance degradation, underscoring the necessity of text domain alignment and adaptation risk minimization in our IAT.
+
+Table 3: Ablation studies for components of our IDeal on ScanRefer. RSum is the sum of R@1, R@5, R@10. w/o stands for without use.
+
+<table><tr><td rowspan="2" colspan="2">Configurations</td><td colspan="4">ScanRefer</td></tr><tr><td>R@1</td><td>R@5</td><td>R@10</td><td>Rsum</td></tr><tr><td rowspan="2">Questioner</td><td>w/o Q1</td><td>36.0</td><td>71.2</td><td>86.7</td><td>193.9</td></tr><tr><td>w/o Q2</td><td>26.3</td><td>61.5</td><td>77.3</td><td>165.1</td></tr><tr><td rowspan="4">Retriever</td><td>w/o p1(u_i)</td><td>35.2</td><td>70.1</td><td>86.5</td><td>191.8</td></tr><tr><td>w/o p2(u_i)</td><td>28.1</td><td>63.3</td><td>80.6</td><td>172.0</td></tr><tr><td>w/o p3(s_i)</td><td>31.8</td><td>67.8</td><td>84.2</td><td>183.8</td></tr><tr><td>w/o CoT</td><td>35.7</td><td>71.5</td><td>85.9</td><td>193.1</td></tr><tr><td rowspan="3">Adaptation</td><td>w/o IAT</td><td>16.6</td><td>48.4</td><td>64.4</td><td>129.4</td></tr><tr><td>w/o L_dis</td><td>34.9</td><td>69.5</td><td>84.4</td><td>188.8</td></tr><tr><td>w/o L_div</td><td>35.1</td><td>69.4</td><td>84.1</td><td>191.7</td></tr><tr><td>Full</td><td>IDeal</td><td>37.8</td><td>71.8</td><td>86.4</td><td>196.0</td></tr></table>
+
+![](images/6875889acf60167b6a19e56b08d195912219c6d2cf3f818bdbe221de76ed77b1.jpg)  
+(a)  $\alpha$  in retriever.
+
+![](images/eb1002a5918464c321ff518a12b2a6aeb8c6e4dd16f9bf443e9f66480a622a84.jpg)  
+(b)  $\lambda_1, \lambda_2, \lambda_3$  in retriever.
+
+![](images/e1fa700f383f544c980e54ca5f86f7a77a13b651c9ca8dddf267118cb283e690.jpg)  
+Figure 3: PTM performance in terms of  $\mathbf{R}@\mathbf{1}$  versus different values of the parameters of our IDeal on ScanRefer. (a) and (b) display  $\alpha$ ,  $\lambda_1$ ,  $\lambda_2$ , and  $\lambda_3$  in our retriever. (c) shows  $k$  and  $\beta$  in our questioner. (d) shows  $\lambda$  and  $\gamma$  in IAT.
+
+![](images/832a1cd2b71e8d2e34f41aa17f28268d2f242a40208db943ed6c7219b3038e63.jpg)  
+(c)  $k,\beta$  in questioner.  
+(d)  $\lambda, \gamma$  in IAT.
+
+# 4.4 Parameter Analysis
+
+To evaluate the sensitivity of our IDeal to different hyperparameter settings, we plot the retrieval performance versus different values on ScanRefer, as shown in Figure 3. The experimental results lead to the following observation: 1) For our retriever, tuning greater weights to interactive and reconstruction predictions helps achieve a well-balanced trade-off that fully leverages the interactive responses. Additionally, a higher feature fusion weight (e.g.,  $\alpha = 0.75$ ) represents a emphasis on
+
+![](images/70060f1456ae19d08677ef913c38abf3b668cd7db6941c3b604d01b1c59445bd.jpg)  
+(a) ScanRefer
+
+![](images/ab22a1a3d70c5c45906348f3f5671d957f83161bf438dd30553fcad31f92c25a.jpg)  
+(b)  $\mathrm{Nr3D}$  
+Figure 4: Performance (R@1) versus rounds on two datasets. Round 0 indicates the setting without interaction. The green and purple bars represent the cases with coarse-grained and fine-grained memory descriptions, respectively. The lighter bars indicate no performance gain.
+
+the integration of discriminative features from the refined descriptions, leading to more effective interaction feature fusion. 2) For our proposed questioner, using reasonable and moderate settings of  $k$  and  $\beta$  (e.g.,  $k = 20$ ,  $\beta = 2.0$ ) enables accurate identification of informative descriptions, thereby supporting reasonable decisions on question types in the next round. 3) During domain adaptation tuning, a relatively wide range of  $\lambda$  and  $\gamma$  values in IAT (i.e.,  $\lambda \in [0.2, 0.5]$  and  $\gamma \in [0.1, 0.8]$ ) ensures effective contrastive adaptation and mitigates the impact of false negatives.
+
+![](images/4ca03a4961e7e0321e92712791a57cca3e8d8dd06bb968840a85715e39d14d54.jpg)  
+Figure 5: Case illustrations of interactive cross-modal scene matching process of IDEal on ScanRefer. Cases 1-2 and 3-4 are with coarse-grained and fine-grained memory descriptions, respectively.
+
+Initial Query [Rank 16] this is a computer monitor with two others to the left and right. there appears to be other monitors also in front of it and diagonally.
+
+# Round 1 Answer
+
+this is a computer monitor with two others to the left and right. it is surrounded by additional monitors in front and diagonally, there are screens placed side by side providing ample workspace. there seem to be more screens angled forward to offer better visibility ... [Rank 4]
+
+![](images/e229a25dd779bdc14bd9aa174aeeb5a44d26f8d1cc640109bdcd99a97a9ebae2.jpg)
+
+Initial Query [Rank 23] a long brown table. It is located near a single chair which is very close to the wall.
+
+# Round 2 Answer
+
+[Rank 8]
+
+this is a long brown/plain table. it is situated near a single chair which is very close to the wall, the single chair is placed right by the wall. there is no indication of any windows or doors being adjacent to the table or chair. the lengthy beige undororned surface (table) stands alongside the solitary stool.
+
+Case 4 Round 1 Answer [Rank 2] this is a description of the scene where a brown wooden chair is near the gray trash can to the right of the painting. it is also mentioned that there is a gray trash can to the right of the table. there is a brown chair at the end of the table as well. additionally, there are several matching chairs with brown leather surrounding the table...
+
+Round 2 Answer
+... there is a rectangular refrigerator,
+Next to fridge, there is a black
+square TV... [Rank 1]
+Round 5 Answer
+... the brown wooden chair is near the
+trash can to the right of the painting,
+with some empty space between them
+[Rank 1]
+
+Initial Query Ground Truth there is a large painting on the wall . there is a chair at the end of the table close to the painting. we are talking about the other chair that is close to the trash can to the right side of the painting . it is brown and wooden . [Rank 4]
+
+Merrlin Round 1 Answer [Rank 5] The large painting on the wall appears to be representational rather than abstract, as it depicts a realistic-scene. Besides the table and chairs already described...The color scheme of the room seems to consist mainly of neutral tones such as browns,grays,and blacks.
+
+Round 2 Answer [Rank 4]  
+The second chair...painting...first chair.Table...round trashcan...  
+Round 4 Answer [Rank 4]  
+The round trashcan...table...the second chair...  
+Round 5 Answer [Rank 4]  
+The round trashcan.table...the surrounding of second chair ...
+
+# 4.5 Visualization Analysis
+
+To provide a comprehensive analysis of IDEal, we conduct a series of visualization experiments. Specifically, we first present the changes in retrieval performance of IDEal across multiple rounds of interaction, as shown in Figure 4, to analyze the incremental gains brought by each interaction. In addition, we visualize several representative cases, as illustrated in Figure 5. The observations can be drawn from the results: 1) Interaction consistently improves performance over the first several rounds (five rounds with fine-grained and three rounds with coarse-grained descriptions). Although redundant interactions may inevitably cause performance to saturate or even slightly degrade due to LLM hallucinations or excessively long texts, these results indicate that a reasonable number of interaction steps can effectively enhance the query and improve retrieval performance. 2) Under the setting with coarse-grained memory texts, IDEal can infer and decompose object attributes and relationships within the queries through interaction, leading to improved retrieval performance. Moreover, with the integration of fine-grained memory, IDEal leverages targeted and switchable questioning to elicit informative responses, continually improving retrieval precision. In contrast, MERLIN [16] frequently generates redundant descriptions confined to local details of scenes.
+
+# 5 Conclusion
+
+In this paper, we propose a novel Interactive Text-3D Scene Retrieval Method, namely IDEal, to address the Text-3D Scene Retrieval (T3SR). Our IDEal integrates two components: the Interactive Retrieval Refinement Framework (IRR) and the Interaction Adaptation Tuning strategy (IAT). Specifically, IRR continuously conducts adaptive questioning and comprehensive response fusion,
+
+enabling holistic exploration of 3D scenes for more precise retrieval. IAT performs contrastive domain adaptation for the retriever toward realistic texts, overcoming the performance bottleneck during interaction. Extensive experiments demonstrate the superiority of our IDEal in T3SR task.
+
+Limitations and Potential Impact Statement: Although our work has taken the initial step forward in interactive T3SR, there are some limitations and potential impacts that should be acknowledged. First, the performance of the methods is relatively low. Second, we employ LLMs, and more stable and unbiased LLMs and interaction approaches merit further exploration in the future.
+
+# Acknowledgments
+
+This work was supported in part by NSFC under Grant 62472295 and 62372315; in part by the Fundamental Research Funds for the Central Universities under Grant CJ202403; in part by Sichuan Science and Technology Planning Project under Grant 24NSFTD0130, 2024ZDZX0004, 2024NSFTD0049, and in part by the Chengdu Science and Technology Project under Grant 2023-XT00-00004-GX.
+
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+
+# NeurIPS Paper Checklist
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+# Guidelines:
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+- The answer NA means that the paper does not include experiments.  
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+# 7. Experiment statistical significance
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+
+# Guidelines:
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+- The answer NA means that the paper does not include experiments.  
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+
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: The compute resources are reported in the experiment settings.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
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+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
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+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
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+Justification: The broader impacts are discussed with the limitations.
+
+Guidelines:
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+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
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+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
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+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
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+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
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+Answer: [NA]
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+Justification: This work does not involve crowdsourcing nor research with human subjects.
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+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
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+
+# Answer: [Yes]
+
+Justification: We have fully disclosed the details of the use of the adopted LLMs in our supplementary material.
+
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+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.

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2025/Intermediate Domain Alignment and Morphology Analogy for Patent-Product Image Retrieval/full.md

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+# Intermediate Domain Alignment and Morphology  
+Analogy for Patent-Product Image Retrieval
+
+Haifan Gong $^{1,2\dagger}$ , Xuanye Zhang $^{1,2\dagger}$ , Ruifei Zhang $^{1,2}$ , Yun Su $^{3}$ , Zhuo Li $^{1,2}$ , Yuhao Du $^{1,2}$ , Anningzhe Gao $^{2}$ , Xiang Wan $^{1,2*}$ , Haofeng Li $^{4,2*}$
+
+<sup>1</sup>The Chinese University of Hong Kong, Shenzhen,
+
+2Shenzhen Research Institution of Big Data,
+
+<sup>3</sup>University of Waterloo,
+
+$^{4}$ School of Systems Science and Engineering, Sun Yat-sen University
+
+# Abstract
+
+Recent advances in artificial intelligence have significantly impacted image retrieval tasks, yet Patent-Product Image Retrieval (PPIR) has received limited attention. PPIR, which retrieves patent images based on product images to identify potential infringements, presents unique challenges: (1) both product and patent images often contain numerous categories of artificial objects, but models pre-trained on standard datasets exhibit limited discriminative power to recognize some of those unseen objects; and (2) the significant domain gap between binary patent line drawings and colorful RGB product images further complicates similarity comparisons for product-patent pairs. To address these challenges, we formulate it as an open-set image retrieval task and introduce a comprehensive Patent-Product Image Retrieval Dataset (PPIRD) including a test set with 439 product-patent pairs, a retrieval pool of 727,921 patents, and an unlabeled pre-training set of 3,799,695 images. We further propose a novel Intermediate Domain Alignment and Morphology Analogy (IDAMA) strategy. IDAMA maps both image types to an intermediate sketch domain using edge detection to minimize the domain discrepancy, and employs a Morphology Analogy Filter to select discriminative patent images based on visual features via analogical reasoning. Extensive experiments on PPIRD demonstrate that IDAMA significantly outperforms baseline methods (+7.58 mAR) and offers valuable insights into domain mapping and representation learning for PPIR. (The PPIRD dataset is available at: https://loslorien.github.io/idama-project/)
+
+# 1 Introduction
+
+Patent systems play a critical role in protecting intellectual property and fostering innovation. The rapid advancement of artificial intelligence (AI) has revolutionized numerous domains, including patent retrieval [1, 2, 3, 4, 5]. Specifically, in the NLP area, significant progress has been made in both patent-patent retrieval [1, 2] and patent-product retrieval [3, 4], leveraging textual descriptions to identify similarities and potential infringements. In contrast, the CV community has primarily focused on patent-patent retrieval [5], leaving Patent-Product Image Retrieval (PPIR) unexplored. Considering the increasing complexity of e-commerce platforms (a product may have an incomplete, confusing name or descriptions, but its images are always clear) and the critical need to detect patent infringements efficiently, there is an urgent demand for research on PPIR.
+
+PPIR faces two primary challenges: 1) product and patent images often contain numerous categories (more than 250,000) of artificial objects, such as mechanical parts or electronic devices, but most
+
+![](images/fde3e113762c06c7b64a21be3127925186a0a985f13550d7c8b36d36c2932309.jpg)  
+Figure 1: Characteristics of Patent-Product Image Retrieval (PPIR) task: (Upper: Product Images, Lower: Patent Images) 1) PPIR task has large-scale artificial categories, which are significantly different from those in general vision datasets; 2) Patent binary line-drawings images and Product colorful RGB images have huge domain gap, casting negative effect for PPIR tasks.
+
+of these categories are distinct from those in general pre-training datasets like ImageNet [6]. 2) There is a significant domain gap between patent images (typically binary line drawings) and product images (typically colorful RGB images). This domain discrepancy further exacerbates the difficulty of accurate retrieval. (See Fig. 1 for details.)
+
+Due to the unique characteristics of PPIR, existing general image retrieval methods [7] exhibit suboptimal performance for two reasons: 1) General image retrieval approaches often struggle to discriminate a large number of unseen artificial-category objects, which makes it challenging to grasp distinctive visual features of patents/products for accurate retrieval; 2) The domain gap causes distribution shift for specific categories, which further aggravate the retrieval performance, especially for some similar artificial objects.
+
+We formulate PPIR as an open-set image retrieval task to simulate real-world scenarios. Moreover, to address the lack of suitable data for PPIR, we introduce the Patent-Product Image Retrieval Dataset (PPIRD), a large-scale dataset designed to facilitate research in this area. PPIRD comprises two components: (1) a testing set containing 439 product-patent pairs alongside a retrieval pool of 727,921 patent images and (2) an unlabeled pre-training set with 3,799,695 product/patent images. We offer highly detailed product descriptions for each product and the corresponding potential patent infringement pair to verify potential patent-patent infringement. (The extremely detailed information required and the difficulties of ascertaining potential patent-patent infringement both limit the scale of the infringed product-patent pair.)
+
+Furthermore, we propose an extremely simple but effective Intermediate Domain Alignment and Morphology Analogy (IDAMA) strategy for PPIR. IDAMA contains an Intermediate Domain Mapping (IDM) strategy and a Morphology Analogy Filter (MAF) strategy: 1) IDM aligns binary line drawing patent images and colorful RGB product images by mapping them into an intermediate sketch domain using an edge detector. We provide a theoretical analysis to prove that this alignment effectively mitigates the domain discrepancy, enabling more accurate retrieval. 2) MAF leverages a cognitive principle of morphology analogy. An unknown object can be described by analogy to a known object—by using high classification confidence (regardless of the label) to select a discriminative image for artificial patent images for retrieval similarity comparison. By selecting discriminative images for unseen artificial patents, MAF can grasp distinctive visual features of the patents for PPIR.
+
+Experiments on PPIRD demonstrate that IDAMA achieves significant improvements over baseline methods  $(+7.58\mathrm{mAR})$ , proving the effectiveness of IDM and MAF in addressing the unique challenges of PPIR. Furthermore, we systematically compare various domain alignment and representation learning methods on PPIRD, providing insights into their strengths and limitations in coping with unseen artificial objects in cross-domain scenarios. These findings offer valuable guidance for future research in PPIR and related areas.
+
+In summary, this work makes three key contributions:
+
+1. We formulate PPIR as an open-set image retrieval task and introduce PPIRD, a large-scale dataset for Patent-Product Image Retrieval;  
+2. We propose IDAMA, a novel strategy that combines intermediate domain alignment and morphology analogy to address challenges of PPIR; We further provide theoretical analysis to demonstrate the advances of intermediate domain alignment via compressive sensing.  
+3. We comprehensively evaluate domain alignment and representation learning methods, providing insights for efficient cross-domain learning.
+
+# 2 Related Works
+
+# 2.1 Image-based Patent/Sketch Datasets and Benchmarks
+
+The availability of large-scale, well-structured datasets is essential for advancing AI research in patent analysis. [8] presented PatentMatch, a dataset specifically created for matching patent claims with prior art, facilitating studies in patent infringement detection and novelty assessment. For image-focused research, [5] developed DeepPatent, a large-scale benchmarking corpus aimed at patent drawing recognition and retrieval. [9] extended this work with DeepPatent2, focusing on technical drawing understanding. [10] introduced the Harvard USPTO Patent Dataset, a comprehensive corpus of patent applications designed to support diverse AI research tasks. However, the research on image-based patent retrieval mainly focuses on image retrieval between patents and solving patent infringement problems. Product infringement detection and corresponding patent-product image retrieval (PPIR) are under-explored. This paper introduces a large-scale patent-product image retrieval dataset to satisfy the requirements.
+
+On the other hand, Im4Sketch [11] proposes a large-scale natural RGB/sketch image retrieval dataset together with a method to address the RGB-sketch retrieval challenges. However, the images of Im4Sketch dataset are collected from ImageNet-1K [6] and adopt the closed-set image retrieval formulation, which exists two main differences from PPIR task: 1) PPIR contains large-scale artificial categories that do not exist in ImageNet-1K. 2) The huge amounts of artificial categories and the continuously appearing products make PPIR not suitable to be formulated as a closed-set image retrieval task.
+
+# 2.2 Patent Retrieval in NLP areas
+
+The analysis and retrieval of patents have been pivotal research areas in the application of artificial intelligence (AI) across multiple domains, including classification and retrieval [12, 13]. Accurate patent classification is critical for efficiently organizing and accessing patent information. Several studies have explored AI-driven methods to automate patent classification, with notable contributions from [14, 15, 16, 17]. These works investigate using machine learning models to handle the complexity of patent data. For example, [18] examined the potential of AI in solving the patent classification problem, analyzing various machine learning models and their effectiveness in handling the diverse and intricate nature of patent data.
+
+In patent retrieval, AI techniques have been widely used to enhance the accuracy and efficiency of retrieving relevant patents. Traditional text-based methods have seen significant advancements by applying deep learning models. [1] proposed an advanced patent prior art search method using deep learning, which relies on semantic embeddings to capture the contextual meaning of patent documents, offering improvements over keyword-based retrieval systems. Similarly, [19] integrated text embeddings with knowledge graph embeddings to further refine patent retrieval, highlighting the importance of external knowledge sources in improving search performance. More recently, [20] explored the use of large language models (LLMs) [21, 22] for patent retrieval, demonstrating their potential to improve retrieval accuracy. To satisfy the real-world requirements of product infringement detection, [23, 14, 15, 16, 17, 10] propose and explore patent-product retrieval. Patent-product retrieval is formulated as an unsupervised retrieval task to satisfy the continuously emerging new products.
+
+In summary, the formulation of patent retrieval in NLP areas has developed from classification or supervised retrieval to open-set unsupervised retrieval using a retrieval pool. Considering that open-set unsupervised retrieval from a retrieval pool is closer to real-world scenarios, which can
+
+![](images/33f3ff382f8d678cf7125a1b117a74be6d300cd61fd919be571c1c09a2ff1159.jpg)  
+Figure 2: Pipeline of IDAMA: IDAMA consists of Intermediate Domain Mapping and Morphology Analogy Filter methods. The pipeline of IDAMA is as follows: 1) Using IDM to align product/patent images by mapping them into intermediate sketch domains; 2) Using MAF to filter discriminative mapped patent images; 3) Comparing product-patent similarity by obtaining product-patent similarity score from product-patent similarity matrix between filtered patent images and mapped product images for infringement detection.
+
+address the issue of conducting infringement detection for continuously emerging products, the proposed PPIR tasks also adopt such formulations to satisfy real-world requirements.
+
+# 2.3 Patent Retrieval with Images
+
+Patent images often include technical drawings and diagrams, and image retrieval in this domain has emerged as an important area of study. [3] focused on improving the quality of convolutional neural network (CNN) [24] training datasets for patent image retrieval, emphasizing the importance of dataset quality in model performance. Building on this, [4] introduced methods using cross-entropy-based metric learning to enhance the accuracy of patent image retrieval systems. In another study, [2] proposed transformer-based deep metric learning for patent image retrieval, showcasing the effectiveness of transformer architectures in this domain. Some works achieve pure patent retrieval by transferring the binary line-draw patent images [25, 26, 27] into sketches to enhance performance.
+
+However, the huge domain gap between patent binary line-drawing images and product colorful RGB images makes it extremely challenging for unsupervised open-set PPIR compared to patent retrieval tasks. Although there are some works that aims to address the domain gap [28, 29, 30, 31, 32], especially in cross-domain image retrieval [33, 34]. However, these methods usually focus on close-set with options rather than the challenging open-world setting. Therefore, this paper proposes domain mapping methods to bridge the domain gaps between patent and product through an intermediate domain to alleviate the challenges.
+
+# 3 Methodology
+
+The Intermediate Domain Alignment and Morphology Analogy (IDAMA) strategy for PPIR consists of Intermediate Domain Mapping (IDM) (Sec. 3.1) and Morphology Analogy Filter (MAF) (Sec. 3.2). We also prove a brief for IDM's effectiveness (Sec. 3.3). The pipeline of IDAMA is demonstrated in Fig. 2: 1) Using IDM to align product/patent images by mapping them into intermediate sketch domains; 2) Using MAF to filter discriminative mapped patent images; 3) Comparing product-patent similarity by obtaining product-patent similarity score from product-patent similarity matrix between filtered patent images and mapped product images for infringement detection.
+
+# 3.1 Intermediate Domain Mapping
+
+The PPIR faces a primary challenge: the domain gap [35, 36, 37] between product and patent images. Specifically, product images are RGB images containing rich color, texture, and other information [38], while patent images are binary line drawings that abstractly represent products [39, 40]. Resolving the domain gaps between product and patent images thus becomes our primary task.
+
+To address the challenge, we propose the Intermediate Domain Mapping (IDM) (See Fig. 2 for details) to align binary drawings, patent images, and colorful RGB product images by mapping both of them to the intermediate sketch domain.
+
+Specifically, we denote a sketch mapper as  $M_{\mathrm{sketch}}(\cdot)$  [41], which can map both product images  $I_r$  and patent images  $I_p$  to the intermediate sketch domain. In the PPIR process, IDM for PPIR first maps raw product images  $I_r$  and patent images  $I_p$  to the intermediate sketch domain and obtains mapped sketch images  $I_r'$  and  $I_p'$ , which can be expressed as:
+
+$$
+I _ {r} ^ {\prime} = M _ {\text {s k e t c h}} \left(I _ {r}\right), \tag {1}
+$$
+
+$$
+I _ {p} ^ {\prime} = M _ {\text {s k e t c h}} \left(I _ {p}\right). \tag {2}
+$$
+
+With the above-defined sketch-based intermediate domain mapping method, we can mitigate the domain gap between patents and products. Analysis in Fig. 4 further confirms the effectiveness of our process.
+
+# 3.2 Morphology Analogy Filter
+
+PPIRD needs to handle a large number of unseen man-made objects. Because general pretraining datasets contain limited man-made categories, extracting features with backbones pre-trained on these datasets is less discriminative for efficient product-patent retrieval.
+
+To overcome the challenge, we observe how human beings grasp the distinct visual feature of an unseen object: human beings analogize the morphological feature of the unseen object to some familiar objects [42]. By doing this, even those who do not see the object can recognize the object through the morphology analogy. Inspired by this cognitive principle, we propose a Morphology Analogy Filter (MAF) for filtering patent images before mapping to the edge domain.
+
+Specifically, given a Filter  $F$  (a classification network) and mapped patent images  $I_{p}$ , MAF first obtain the classification vector  $c_{p}$  by
+
+$$
+c _ {p} = F \left(I _ {p}\right) \tag {3}
+$$
+
+Then, MAF obtains discriminative patents  $I_{pd}$  image using the max classification scores regardless of the label itself (class-agnostic scores), which can be expressed as:
+
+$$
+I _ {p d} = \operatorname {a r g w h e r e} \left(\operatorname {a r g m a x} \left(c _ {p _ {j}}\right) > \tau_ {m a f}\right) \tag {4}
+$$
+
+where  $c_{p_j}$  is the classification score in  $c_p$  and  $\tau_{maf}$  is the MAF threshold. If all the image classification scores are less than  $\tau_{maf}$ , we preserve the image of max  $c_{p_j}$  for the patent.
+
+We only conduct MAF for patent images as follows: The motivation of PPIR is to discover whether a product infringes any patent. Each image contains valuable visual features. It'd be better to follow the 'Better to have more than to miss out' philosophy for products. Because morphology similarity may indicate potential infringement for patent images, it'd be better to follow the rule 'Better to have less than to have something of poor quality' to select distance visual features of patents to avoid the negative influence. Therefore, we only conduct MAF for patent images.
+
+Similarity Comparison: Then, a feature extractor  $E(\cdot)$  are adopted to extract feature  $f_{r}$  and  $f_{p}$  from  $I_{r}$  and  $I_{pd}$ :
+
+$$
+f _ {r} ^ {\prime} = E _ {(.)} \left(I _ {r}\right) \tag {5}
+$$
+
+$$
+f _ {p} ^ {\prime} = E _ {(\cdot)} \left(I _ {p d}\right) \tag {6}
+$$
+
+After obtaining the intermediate sketch-domain feature  $f_r'$  and  $f_p'$ , PPIRD calculates the cosine similarity score for them to conduct PPIR:
+
+$$
+s _ {r, p} ^ {\prime} = \operatorname {s i m} \left(f _ {r} ^ {\prime}, f _ {p} ^ {\prime}\right) \tag {7}
+$$
+
+where  $sim(\cdot)$  denotes the cosine similarity function.
+
+For PPIR tasks, each patent  $p$  and product  $r$  may contain more than one image, which can be expressed as  $r = \{I_{p1}, I_{p2}, \ldots\}$  and  $p = \{I_{r1}, I_{r2}, \ldots\}$ . IDM uses a maximum of image-level similarity scores (formulating product-patent similarity matrix) in the intermediate domain as product-patent similarity score  $s_{p,r}$ :
+
+$$
+s _ {p, r} = \max  \left(s _ {r 1, p 1} ^ {\prime}, s _ {r 1, p 2} ^ {\prime}, \dots , s _ {r 2, p 2} ^ {\prime}, \dots\right) \tag {8}
+$$
+
+IDM for PPIR compares each product  $r$  with all the patents in the retrieval pools, denoted as  $P = \{p_1, p_2, \ldots\}$ , obtains their similarity score set, denoted as  $S_r = \{s_{p1,r}, s_{p2,r}, \ldots\}$ , and then sort  $S_r$  for patent-product retrieval.
+
+# 3.3 Understanding Intermediate Domain Mapping via Compressive Sensing
+
+Preliminaries. Let  $I \in \mathbb{R}^n$  denote a vectorised image that can be decomposed as
+
+$$
+I = s + \eta , \tag {9}
+$$
+
+where  $s$  encodes the geometrical structure (edges, contours) while the term  $\eta$  contains texture, colour and background clutter. The mapper  $M$  used in intermediate domain mapping (IDM) has a linear front-end  $\mathbf{A} \in \mathbb{R}^{m \times n}$ ,  $m \ll n$ , whose entries are i.i.d.  $\mathcal{N}(0,1 / m)$  random variables. Such a matrix acts as a Johnson-Lindenstrauss embedding and, with overwhelming probability, satisfies the Restricted Isometry Property (RIP) required by compressed-sensing theory [43, 44, 45, 46, 47].
+
+Model assumptions. 1)  $s$  is  $k$ -sparse in a known orthonormal basis  $\Psi$ ; 2) the embedding dimension obeys  $m \geq C k \log (n / k)$  for a universal constant  $C > 0$ ; 3) the backbone network  $E$  is  $L$ -Lipschitz, i.e.  $\| E(x) - E(y)\|_2 \leq L \| x - y \|_2$  for all  $x, y$  (see, e.g., [48, 49]).
+
+Definition 1 (Restricted Isometry Property [50]). A matrix  $\mathbf{A}$  satisfies RIP  $(2k,\delta)$  if, for every  $2k$ -sparse vector  $x\in \mathbb{R}^n$
+
+$$
+(1 - \delta) \| x \| _ {2} ^ {2} \leq \| \mathbf {A} x \| _ {2} ^ {2} \leq (1 + \delta) \| x \| _ {2} ^ {2}. \tag {10}
+$$
+
+Claim 2 (Distance preservation). Consider a patent-product pair  $(I_p, I_{pr})$  whose structural components  $s_p, s_{pr}$  satisfy  $\| s_p - s_{pr} \|_2 \leq \varepsilon$ . If  $\mathbf{A}$  fulfils RIP  $(2k, \delta)$ , then
+
+$$
+\left\| \mathbf {A} s _ {p} - \mathbf {A} s _ {p r} \right\| _ {2} \leq (1 + \delta) \varepsilon . \tag {11}
+$$
+
+Theorem 1 (IDM feature-space contraction). Under Assumptions 1-3 and Definition 1, the feature distance of the mapped images satisfies
+
+$$
+\left\| \tilde {z} _ {p} - \tilde {z} _ {p r} \right\| _ {2} \leq L \left(\left(1 + \delta\right) \varepsilon + \frac {m}{n} \left(\left\| \eta_ {p} \right\| _ {2} + \left\| \eta_ {p r} \right\| _ {2}\right)\right), \tag {12}
+$$
+
+where  $\tilde{z}_p = E(\mathbf{A}I_p)$  and  $\tilde{z}_{pr} = E(\mathbf{A}I_{pr})$
+
+Remark 3. Because  $m \ll n$ , the nuisance term is suppressed by the factor  $m / n$ , whereas the structural term is almost isometrically preserved by RIP. Consequently, with high probability,  $\| \tilde{z}_p - \tilde{z}_{pr} \|_2 \ll \| z_p - z_{pr} \|_2$ , which explains the empirical robustness of Intermediate Domain Mapping.
+
+The proofs of Claim 2 and Theorem 1 are deferred to Appendix A.
+
+# 4 Experiments
+
+# 4.1 PPIRD: A Patent-Product Image Retrieval Dataset
+
+Retrieving patents for infringing products is complex and time-consuming, heavily relying on patent experts to search massive patent databases to identify potential infringing patents [13]. This reliance makes it challenging to obtain a large-scale paired dataset of product-patent images to train models in a supervised manner. To address the challenge, we proposes PPIRD, consisting of PPIRD-testing and PPIRD-unlabeled, and formulates it as a retrieval task. To the best of knowledge, we provide the first benchmark for image-based patent-product retrieval at near million data level.
+
+PPIRD-testing set: The PPIRD-testing set contains 439 product-patent pairs and a retrieval pool of 727,921 patent images. Experts annotate and validate the product-patent infringement pairs (Expert can refer the detailed descriptions of 439 products for more accurate validation). For the infringed patents, we use Google Patents to search 9,999 patents for each product (using their name as a keyword) and download their patent images to formulate a retrieval pool. We formulate PPIRD-testing as product-patent-level retrieval rather than image-level retrieval like DeepPatent [5] to simulate real-world scenarios better.
+
+The formulation of PPIRD-testing is as follows: let the product dataset be  $D_r$  and the patent dataset be  $D_p$ . For a product  $x_i$  from the product dataset  $D_r$ , we aim to find its corresponding patent  $y_j$  in
+
+Table 1: Qualitative Results of IDAMA: 1) IDAMA can bring significant performance enhancement compared with baseline methods, and both IDM and MAF can contribute to the improvement. 2) Comparison between DeepPatent [5] and MAF ('IDM+DeepPatent' and 'IDAMA') proves the intuitive idea of MAF can bring more performance enhancement even without extra pre-training. 3) Comparison between UCDIR [33] and UCDIR+IDM/IDAMA also demonstrates that our method can consistently boosts the performance. 4) Contrastive Learning ('IBOT') may be more suitable for PPIR in intermediate sketch domain.  
+
+<table><tr><td>Method</td><td>Backbone</td><td>Pre-train</td><td>mAR</td><td>R@100</td><td>R@500</td><td>R@1000</td><td>R@2000</td></tr><tr><td>Product-Patent</td><td>ResNet-18</td><td>Supervised</td><td>13.50</td><td>1.59</td><td>11.62</td><td>14.12</td><td>26.65</td></tr><tr><td>IDM</td><td>ResNet-18</td><td>Supervised</td><td>21.70</td><td>3.42</td><td>14.12</td><td>26.20</td><td>43.05</td></tr><tr><td>IDM+DeepPatent</td><td>ResNet-18</td><td>Supervised</td><td>21.98</td><td>3.19</td><td>13.90</td><td>25.74</td><td>45.10</td></tr><tr><td>IDAMA</td><td>ResNet-18</td><td>Supervised</td><td>22.84</td><td>4.10</td><td>15.03</td><td>26.42</td><td>45.79</td></tr><tr><td>Product-Patent</td><td>ResNet-50</td><td>Supervised</td><td>18.05</td><td>2.73</td><td>13.90</td><td>19.82</td><td>35.76</td></tr><tr><td>UCDIR</td><td>ResNet-50</td><td>UCDIR</td><td>18.64</td><td>2.87</td><td>14.03</td><td>20.17</td><td>35.81</td></tr><tr><td>UCDIR+IDM</td><td>ResNet-50</td><td>UCDIR</td><td>24.94</td><td>4.96</td><td>18.13</td><td>28.42</td><td>46.91</td></tr><tr><td>UCDIR+IDAMA</td><td>ResNet-50</td><td>UCDIR</td><td>25.36</td><td>5.31</td><td>18.47</td><td>28.97</td><td>47.28</td></tr><tr><td>IDM</td><td>ResNet-50</td><td>Supervised</td><td>25.06</td><td>5.24</td><td>18.45</td><td>28.93</td><td>47.61</td></tr><tr><td>IDM+DeepPatent</td><td>ResNet-50</td><td>Supervised</td><td>25.12</td><td>5.47</td><td>18.68</td><td>28.25</td><td>48.06</td></tr><tr><td>IDAMA</td><td>ResNet-50</td><td>Supervised</td><td>25.63 (+7.58)</td><td>5.47</td><td>19.13</td><td>29.84</td><td>48.06</td></tr><tr><td>IDAMA</td><td>Swin-B</td><td>Supervised</td><td>26.20</td><td>6.38</td><td>19.36</td><td>29.84</td><td>49.20</td></tr><tr><td>IDAMA</td><td>ViT-B</td><td>Supervised</td><td>26.43</td><td>6.83</td><td>20.05</td><td>30.30</td><td>48.52</td></tr><tr><td>IDAMA</td><td>Swin-L</td><td>Supervised</td><td>28.02</td><td>7.52</td><td>22.32</td><td>33.94</td><td>48.29</td></tr><tr><td>IDM</td><td>ViT-L</td><td>Supervised</td><td>26.99</td><td>6.83</td><td>22.55</td><td>31.44</td><td>47.15</td></tr><tr><td>IDAMA</td><td>ViT-L</td><td>Supervised</td><td>28.30</td><td>7.97</td><td>23.23</td><td>33.48</td><td>48.52</td></tr><tr><td>IDAMA</td><td>ViT-L</td><td>MAE [52]</td><td>23.35</td><td>5.24</td><td>17.31</td><td>28.02</td><td>42.82</td></tr><tr><td>IDAMA</td><td>ViT-L</td><td>IBOT [53]</td><td>31.61</td><td>9.34</td><td>28.02</td><td>36.21</td><td>52.85</td></tr></table>
+
+the patent dataset  $D_p$ . We represent the labels as the correspondence between infringing products and patent images, forming a set  $R = \{(x_i, y_{ij}) \mid i = 1, \dots, I = 439\}$ . This work collected 439 product-patent image pairs to evaluate product-patent retrieval.
+
+Evaluation metrics: Given the unsupervised formulation of PPIR tasks, we assess the retrieval performance by calculating the cosine similarity between patent and product image features. In PPIR tasks, one product or patent may contain more than one image, and choose to use the average similarity of all the product-image and patent-image pairs of the product or patent to denote the retrieval similarity in this pair. Moreover, PPIR contains 1-to-N situations, indicating that one product may infringe several patents, and we regard retrieving at least one patent as a successful retrieval.
+
+We sort the patents in the retrieval pool according to their similarity scores and two metrics to systemically evaluate PPIR performance [51]: the mean Average Recall (mAR) and top-k recall at specific thresholds ( $k = 100, 500, 1000, 2000$ ) (R@K) for each patent-product pair. mAR calculates the average recall rate of different top-k thresholds. AR considers the PPIR performance of different thresholds and can reflect the overall PPIR performance among other difficulties. Considering the 1-to-N matching situations, we use recall metric for evaluation. The R@K can demonstrate the model discrimination capability in specific scenarios.
+
+# 4.2 Pre-training in Intermediate Domain
+
+The capability of feature extractor  $E_{(\cdot)}(\cdot)$  can seriously influence PPIR performance. Therefore, pre-training in the intermediate sketch-image domain may contribute to enhancing the representation and discrimination capability of the model, obtaining  $E_{\mathrm{sketch}}(\cdot)$  and improving PPIR performance. In this section, We compare the performance between supervised and unsupervised methods for feature learning in the intermediate domain. In this section, all the images, including images of both PPIRD unlabeled  $\mathcal{D}_{\mathrm{unlabeled}}$  and ImageNet  $\mathcal{D}_{\mathrm{ImageNet}}$ , are mapped to the intermediate domain using  $M_{\mathrm{sketch}}(\cdot)$  to extract sketch images for pre-training:
+
+$$
+I _ {p r e} ^ {\prime} = M _ {s k e t c h} \left(I _ {p r e}\right), \quad \forall I _ {p r e} \in \mathcal {D} _ {u n l a b e l e d}, \mathcal {D} _ {I m a g e N e t}, \tag {13}
+$$
+
+where  $I_{pre}^{\prime}$  and  $I_{pre}$  denote the images of intermediate and raw domains, respectively.
+
+Unsupervised Pre-training: Unsupervised pre-training has played a significant role in computer vision recently [54, 53, 55, 56, 57, 58, 52], enabling efficient visual representation encoders without
+
+Table 2: Qualitative Results of different domain mapping methods: Compared with other mapping methods, IDM ('Product(Edge)-Patent(Edge)' is the more suitable mapping method for PPIRD and can bring more performance enhancement.  
+
+<table><tr><td>Method</td><td>mAR</td><td>R@100</td><td>R@500</td><td>R@1000</td><td>R@2000</td></tr><tr><td>Product-Patent</td><td>18.05</td><td>2.73</td><td>13.90</td><td>19.82</td><td>35.76</td></tr><tr><td>Product-Patent (Colorized by [27])</td><td>20.44</td><td>3.19</td><td>15.49</td><td>23.01</td><td>40.09</td></tr><tr><td>Product (Binary Line)-Patent</td><td>22.67</td><td>3.64</td><td>17.77</td><td>25.51</td><td>43.74</td></tr><tr><td>Product (Edge)-Patent</td><td>23.80</td><td>4.33</td><td>18.45</td><td>27.33</td><td>45.10</td></tr><tr><td>Product (Edge)-Patent (Edge)</td><td>25.63</td><td>5.47</td><td>19.13</td><td>29.84</td><td>48.06</td></tr></table>
+
+![](images/96ebe876bad18ce5f0280e6e9d1bd590ac79826b426309ec8790d96b50b790c4.jpg)  
+(a) Self-supervised & unsupervised pre-training  
+Figure 3: Comparison between self-/unsupervised pretraining and supervised pretraining strategies. Subplot (a): self-/unsupervised contrastive learning method 'IBOT' pre-trained on PPIRD unlabeled, Subplot (b): Supervised pretraining on ImageNet1k-Edge. Matched product-patent image pairs are depicted using the same color. The visualization demonstrates that the unsupervised pretraining method effectively brings matched pairs closer in the feature space, enhancing their alignment.
+
+![](images/67ba319f5ee801eca9b394190ffc32d0c9155cb4cefeb652b3c231fdc4e9dfba.jpg)
+
+![](images/7c49116d2452a5c5007c163c81211b2d90bc1aba0b6835ef8441fffc96d50eb9.jpg)  
+(b) Supervised pre-training
+
+![](images/3305940929b0aad4b7e3b4cb057f4d8b1a22d83fb58c5d6d61c7a1a65af631c0.jpg)
+
+the need for labeled training data. We utilize unsupervised pre-training methods of both Masked Image Modeling (MAE [52]) and Contrastive Learning (IBOT [53]) to obtain feature encoders for the intermediate domain by minimizing the unsupervised loss function  $\mathcal{L}_{\mathrm{unsup}}$ :
+
+$$
+\min  _ {E} \sum_ {i = 1} ^ {N} \mathcal {L} _ {\text {u n s u p}} \left(E \left(I _ {p r e} ^ {\prime}\right)\right), \tag {14}
+$$
+
+where  $E(\cdot)$  is the feature extractor and  $\mathcal{L}_{\mathrm{unsup}}$  is the unsupervised loss function specific to the pretraining method used. Images of PPIRD unlabeled  $D_{\mathrm{unlabeled}}$  are used for unsupervised pre-training.
+
+Supervised Pre-training: According to the conclusion of [59], supervised pre-training may be more suitable for learning representative and distinguishable features in the intermediate sketch-image domain due to the sparsity of this domain. Specifically, Masked Image Modeling or Contrastive learning may be challenging to reconstruct those large blank areas or discriminate blank areas from others, hindering the model's ability to learn representative and distinguishable feature representations. Considering this possibility, we conduct a simple yet effective supervised pre-training method by mapping the images in ImageNet and using the mapped well-annotated data for backbone pre-training. Based on these mapped sketch images, we train a classifier to obtain feature extractor  $E(\cdot)$  using the original labels of these images. The training objective is to minimize the cross-entropy loss:
+
+$$
+\mathcal {L} = - \sum_ {i} y _ {i} \log \hat {y} _ {i}, \quad \text {w i t h} \quad \hat {y} _ {i} = \operatorname {s o f t m a x} \left(F \left(I _ {\text {s k e t c h}, i}\right)\right), \tag {15}
+$$
+
+where  $y_{i}$  is the ground truth label, and  $I_{\mathrm{sketch}}^{\prime}$  is the sketch images. The advantage of this approach is that our sketch feature extractor has an explicit learning objective and does not rely on negative samples in contrastive learning or masks in self-supervised learning. This endows feature extractor with excellent discriminative ability. More implemental details are provided in Sec. B.
+
+# 4.3 Main Results
+
+We compare IDAMA with raw ResNet [60] (denoted as 'Product-Patent') and the current SOTA patent retrieval method DeepPatent [5]. For a fair comparison, we add IDM before DeepPatent [5] since it is designed for patent image retrieval. Tab. 1 demonstrates the quantitative results on PPIRD. As is shown, IDAMA can bring significant performance enhancement compared with baseline methods, and both IDM and MAF can contribute to the improvement.
+
+![](images/72349558a76fda9c25e87938f812717f7bae7cc4926872ce5c4b45dd151229dc.jpg)  
+(a) Directly extract original product and patent feature
+
+![](images/358f317fecbeeff7898fb9422b4ee6600914b4829c60ccc93a26de855c3ed051.jpg)
+
+![](images/79afbee41b7960a4d44eb6f456b671a939c5196d406b2ee6fa7ae62ecdbda7b6.jpg)  
+(b) Colorized patent to align product
+
+![](images/8ca6d3925a41dfeba0fba1a03b8a8e4941c0973b3002cacb566daa7c0eaa37a1.jpg)
+
+![](images/f738a4175c3be15ec40117d614675dbaf142c65c6c2bf824e675706b232a31d6.jpg)  
+(c) Extract product sketch to align patent
+
+![](images/96fee47a50bc0cadf651d3760e1ce8501fc56516865e82e9ab4194d8f0bfa2ad.jpg)  
+(d) Joint mapping of product and patent to edge map  
+Figure 4: t-SNE results for different domain mapping methods: (a) Product-Patent: Directly extract original product and patent feature; (b) Product-Patent (Colorized): Colorized patent to align product; (c) Product (Binary Line)-Patent: Extract product edge to align patent; (d) IDM: Extract both product images and patent images to sketch images. In each subplot, closer interleaving of the two colored points indicates a greater reduction in the domain gap, indicating better patent/product image alignment. IDM can best achieve the goal.
+
+Comparison with DeepPatent: The comparison with DeepPatent [5] proves that PPIR (product-patent level retrieval) is quite different from Patent Image Retrieval (image level retrieval) tasks. PPIR is more challenging because the product and the patent are not the same object. When comparing DeepPatent with MAF ('IDM+DeepPatent' and 'IDAMA' in Tab. 1), though the idea of MAF is more intuitive and even without requiring extra pre-training, it brings more performance enhancement, indicating the importance of integrating human beings' philosophy for challenging tasks.
+
+Comparison of Different Pre-training methods: When comparing with different types of pretraining methods, the contrastive learning methods ('IBOT' in Tab. 1) may be the most suitable pre-training methods for PPIR tasks. We can notice that Masked Image Modeling methods ('MAE' in Tab. 1) perform worse than supervised learning, indicating these methods may not be suitable for such sketch domain lack of texture or color information. Fig. 3 reaches similar conclusion.
+
+# 4.4 Comparison with Other Domain Mapping Approach
+
+We compare IDM with some naive domain mapping approaches for PPIR:
+
+Mapping colorful RGB product images into binary line drawings through Edge Detection (denoted as 'Product(Binary Line)-Patent'): We employ an Edge Detection Mapper  $M_{\mathrm{edge}}(\cdot)$  to extract edge from product images  $I_r$ , converting them into line drawings  $I_{re} = M_{\mathrm{edge}}(I_r)$ . The advantage of this method is its computational efficiency and ease of implementation. However, the accuracy of the edge feature extraction algorithm is heavily dependent on it. Under varying lighting and texture conditions, the extracted edge maps from product images may exhibit discontinuities, affecting the performance of PPIR. Additionally, the feature extractor demonstrates in-robustness in extracting features from binary line-drawing images.
+
+Converting patent images into RGB colorful images through diffusion (denoted as 'Product-Patent (Colorized)'): Recently, generative models [61, 62, 63] have gained increasing attention, with many models capable of efficiently generating RGB images based on line drawings [25, 26, 27]. We utilize a generative model  $M_{\mathrm{colorize}}(\cdot)$  [25] to transform patent images into pseudo-product images,  $I_{pg} = M_{\mathrm{colorize}}(I_p)$ . The advantage of this method is that the generated images are in RGB format, allowing direct use of feature encoders pre-trained on natural images for feature extraction. However, challenges include the high computational cost of generative models and the lack of specific domain knowledge, which leads to the color distribution and texture of generated images may differ substantially from natural images.
+
+We conduct extensive evaluations of the above three approaches using ResNet-50 [60], pre-trained on ImageNet-1K [6] (or ImageNet-1K-Edge), for feature extraction. According to the t-SNE [64] results in Fig. 4, IDM (denoted as 'Product (Edge)-Patent (Edge)'), mapping patent images and product images into the intermediate sketch domain is the best method to align patent/product images and alleviate the negative effects of domain gaps. Tab. 2 reaches a similar conclusion. We also conduct experiments by only mapping product images to directly sketch and compare their similarity with patent images. Experiment result 'Product (Edge)-Patent' in Tab. 2 proves the domain matching method is suboptimal for the tasks, indicating a domain gap between sketch and binary line drawings.
+
+# 5 Conclusion
+
+In summary, we formulate Patent-Product Image Retrieval (PPIR) as an open-set image retrieval task and propose a large-scale Patent-Product Image Retrieval Dataset (PPIRD) comprising (1) a testing set with 439 product-patent pairs (annotated and validated by experts with detailed descriptions) and a retrieval pool of 727,921 patents, and (2) an unlabeled pre-training set with 3,799,695 product/patent images. We propose a novel Intermediate Domain Alignment and Morphology Analogy (IDAMA) strategy tailored for PPIR. IDAMA contains an Intermediate Domain Mapping method to align binary line drawing patent images and colorful RGB product images by mapping them into an intermediate sketch domain using an edge detector to effectively mitigate the domain discrepancy and a Morphology Analogy Filter to select discriminative patent images containing distinctive visual feature of patents for efficient similarity comparison (inspired the cognitive principle—an unknown object can be described by analogy to a known object (patent image with high classification score regardless of label). Extensive experiments on PPIRD demonstrate that the intermediate domain is more suitable for aligning patent/product images and improving performance.
+
+# Acknowledgements
+
+This work is supported in part by the Shenzhen Science and Technology Program (JCYJ20220818103001002), the Guangdong Provincial Key Laboratory of Big Data Computing, The Chinese University of Hong Kong, Shenzhen, the Longgang District Special Funds for Science and Technology Innovation (LGKCSDPT2023002), the Project (No. 20232ABC03A25), the Scientific Research Program of Wuxi Health Commission (Project NO. Z202309).
+
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+
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+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
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+
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+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [Yes]
+
+Justification: We have provide the full set of assumptions and a complete proof.
+
+Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
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+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
+
+Justification: We have provide all the information needed to reproduce the results in this paper.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.
+
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.  
+- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.  
+- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.  
+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example  
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: The data are publicly available, while the code is available in the submission system.
+
+Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
+- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).  
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+- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.  
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
+
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: All these details are available in the paper.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [Yes]
+
+Justification: We have conducted the t-test in the comparison of our method and the previous methods.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
+- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)  
+- The assumptions made should be given (e.g., Normally distributed errors).  
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We have provided all these information in this paper.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
+
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more computer than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: This work follows the NeurIPS code of ethics.
+
+Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: We have discuss the broader impact in the discussion section.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: The paper poses no such risks.
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.
+
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: We have cited all related paper in this work.
+
+Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [Yes]
+
+Justification: The new assets introduced in the paper are well documented.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: We use publicity available dataset.
+
+# Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: This work uses the public available dataset as they already received the IRB approvals.
+
+# Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: The core method development in this research does not involve LLMs as any important.
+
+# Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# A Proofs for Section 3.3
+
+# A.1 Proof of Claim 2
+
+Let  $x \coloneqq s_p - s_{pr}$ . By assumption,  $x$  is  $2k$ -sparse and  $\| x \|_2 \leq \varepsilon$ . Because  $\mathbf{A}$  satisfies RIP(2k, δ),
+
+$$
+\| \mathbf {A} x \| _ {2} \leq \sqrt {1 + \delta} \| x \| _ {2} \leq (1 + \delta) \varepsilon . \tag {16}
+$$
+
+Since  $\mathbf{A}$  is linear,  $\mathbf{A}x = \mathbf{A}s_p - \mathbf{A}s_{pr}$ , which concludes the proof.
+
+# A.2 Proof of Theorem 1
+
+Decompose each image as  $I = s + \eta$  and observe
+
+$$
+\mathbf {A} I _ {p} - \mathbf {A} I _ {p r} = (\mathbf {A} s _ {p} - \mathbf {A} s _ {p r}) + (\mathbf {A} \eta_ {p} - \mathbf {A} \eta_ {p r}). \tag {17}
+$$
+
+Applying the backbone  $E$  and using its  $L$ -Lipschitz continuity [48],
+
+$$
+\begin{array}{l} \| \tilde {z} _ {p} - \tilde {z} _ {p r} \| _ {2} = \| E (\mathbf {A} I _ {p}) - E (\mathbf {A} I _ {p r}) \| _ {2} \\ \leq L \left\| \mathbf {A} I _ {p} - \mathbf {A} I _ {p r} \right\| _ {2} \tag {18} \\ \leq L \left(\| \mathbf {A} s _ {p} - \mathbf {A} s _ {p r} \| _ {2} + \| \mathbf {A} \eta_ {p} \| _ {2} + \| \mathbf {A} \eta_ {p r} \| _ {2}\right). \\ \end{array}
+$$
+
+The first term is bounded by Claim 2. For the nuisance terms we appeal to the concentration of the  $\chi^2$  distribution of Gaussian projections [46, Thm. 7.5]: for any fixed vector  $\eta$  and all  $t\in (0,1)$
+
+$$
+\Pr \left(\left| \| \mathbf {A} \eta \| _ {2} ^ {2} - \frac {m}{n} \| \eta \| _ {2} ^ {2} \right| \geq t \frac {m}{n} \| \eta \| _ {2} ^ {2}\right) \leq 2 \exp (- c t ^ {2} m), \tag {19}
+$$
+
+where  $c > 0$  is an absolute constant. Selecting  $t = \frac{1}{2}$  and using  $\sqrt{a \pm b} \leq \sqrt{a} \pm \frac{b}{2\sqrt{a}}$  gives, with probability at least  $1 - 2e^{-cm / 4}$ ,
+
+$$
+\left\| \mathbf {A} \eta \right\| _ {2} \leq \frac {m}{n} \| \eta \| _ {2}. \tag {20}
+$$
+
+Applying this bound to both  $\eta_p$  and  $\eta_{pr}$  and combining all inequalities establishes the asserted result.
+
+# B Implemental Details
+
+The  $\tau_{maf}$  is set to 0.4. We directly use the Supervised Pre-trained ResNet-50 on ImageNet-1K as the Morphology Analogy Filter. For all the supervised pre-training models (including ResNet [60], SwinTransformer [65], and ViT [66]), the pre-training dataset is ImageNet-1K-Edge (Edge is extracted using [25]) the optimizer is AdamW, the learning rate is set to 0.002, the beta of the optimizer is set to (0.9, 0.95), and the weight decay is set to 0.001. We train all the models for 100 epochs. For all the self-supervised/unsupervised pretraining, the pre-training dataset is PPIR unlabeled-Edge (Edge is extracted using [25]), the batch size is 256, the optimizer is AdamW, the learning rate is 0.002. We pre-train all the models for 100 epochs. The batch size is set from 256 to 1024, with more details are available in the supplementary material.
+
+# C Appendix on Training Details
+
+Table 3: Hyper-parameters for supervised training. Values separated by “/” correspond to the successive model sizes listed in the Model row.  
+
+<table><tr><td>Hyper-parameter</td><td>ResNet</td><td>Swin</td><td>ViT</td></tr><tr><td>Model</td><td>ResNet-18 / 50 / 101</td><td>Swin-T / S / B</td><td>ViT-B-16 / L-16</td></tr><tr><td>Dataset</td><td>ImageNet1K-edge</td><td>ImageNet1K-edge</td><td>ImageNet1K-edge</td></tr><tr><td>Batch size</td><td>4096 / 2048 / 1024</td><td>2048 / 1024 / 512</td><td>512 / 256</td></tr><tr><td>Optimizer</td><td>AdamW</td><td>AdamW</td><td>AdamW</td></tr><tr><td>Learning rate</td><td>0.002</td><td>0.002</td><td>0.002</td></tr><tr><td>β (AdamW)</td><td>(0.9, 0.95)</td><td>(0.9, 0.95)</td><td>(0.9, 0.95)</td></tr><tr><td>Epochs</td><td>100</td><td>100</td><td>100</td></tr><tr><td>Weight decay</td><td>0.001</td><td>0.001</td><td>0.001</td></tr><tr><td>Training time (h)</td><td>5.3 / 9.5 / 18.7</td><td>7.9 / 15.7 / 21.6</td><td>19.8 / 34.7</td></tr></table>
+
+Table 4: Hyper-parameters for self-supervised learning.  
+
+<table><tr><td>Hyper-parameter</td><td>iBOT</td><td>MAE</td></tr><tr><td>Method</td><td>iBOT</td><td>MAE</td></tr><tr><td>Model</td><td>ViT-L-16</td><td>ViT-L-16</td></tr><tr><td>Dataset</td><td>PPIR-unlabeled</td><td>PPIR-unlabeled</td></tr><tr><td>Batch size</td><td>256</td><td>256</td></tr><tr><td>Optimizer</td><td>AdamW</td><td>AdamW</td></tr><tr><td>Learning rate</td><td>0.002</td><td>0.002</td></tr><tr><td>β (AdamW)</td><td>(0.9, 0.95)</td><td>(0.9, 0.95)</td></tr><tr><td>Epochs</td><td>100</td><td>100</td></tr><tr><td>Training time (h)</td><td>97.4</td><td>95.3</td></tr></table>
+
+# D Visualization Results
+
+![](images/22a41eb5b369b4e1c062ac0dca858fa6ac251b373321b4f8ed686cbe36080b02.jpg)
+
+![](images/90d851c8b6cbd255d476393799a32a0d7d7c0a5190edb21d2d50a998da91ff4a.jpg)
+
+![](images/13d56537d642b3284f1c59e729ee6185798871b550c8928091a7b3d126eb5542.jpg)
+
+Figure 5: Upper: Diffusion generated colorful patent images  
+![](images/7de2952010fe1acf719fcf6f318112a97c183e58b7a0341c5ea5b4798fa30720.jpg)  
+Lower: Sketches extracted from raw images
+
+![](images/c1b4c4567fb3a3272276021d872e67709053e1119d81626a4c1f67d1817aadcf.jpg)
+
+![](images/7e5759ceac4ff8d6f9ec850384496c20edcdef50bc8b1f7abae0b89a7e2c6835.jpg)
+
+![](images/9628c25ea0ccfa2f10fe52b92c750ac0fb9e127aff3e3a3e1049a2730bccdf43.jpg)
+
+# E Broader impact, Advantages, and Limitations
+
+Broader Impact. Our work lowers the barrier for large-scale, automated monitoring of potential design infringements, thereby helping both innovators and regulators to protect intellectual property more efficiently and at earlier stages of the product life-cycle. The publicly released PPIRD dataset also creates a new benchmark that can stimulate research on cross-domain retrieval, representation learning under distribution shifts, and open-set recognition. Nevertheless, any technology that simplifies infringement search can also be misused: malicious actors might automate "design-around" strategies or target small businesses with aggressive litigation. We therefore encourage downstream users to combine PPIR systems with human expert review, publish detailed audit logs, and follow responsible-AI frameworks to mitigate unintended consequences.
+
+Advantages. (1) Comprehensive benchmark. PPIRD is, to our knowledge, the largest and most realistic testbed for patent-product image retrieval, containing nearly one million patent drawings and millions of unlabeled images for pre-training. (2) Effective domain bridging. The proposed IDAMA pipeline explicitly maps both binary line drawings and RGB product photos into an intermediate sketch space, reducing domain discrepancy without requiring paired supervision. (3) Open-set readiness. By framing PPIR as an open-set problem and adding a morphology-analogy filter, our method retains high recall on novel object categories that are unseen during conventional pre-training. (4) Plug-and-play. IDAMA is architecture-agnostic; it can be integrated into existing CNN or vision-transformer backbones with minimal code changes and no task-specific annotations.
+
+Limitations. The morphology-analogy filter relies on hand-crafted similarity thresholds; tuning them across industries or jurisdictions may require domain expertise. In addition, our dataset—while large—covers only 439 manually verified product-patent pairs, limiting fine-grained evaluation of failure modes such as functional equivalence or partial design overlap.

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+# Interpretable and Parameter Efficient Graph Neural Additive Models with Random Fourier Features
+
+Thummaluru Siddartha Reddy, Vempalli Naga Sai Saketh, Mahesh Chandran {Thummaluru.Siddarthareddy, nagasaisaketh.vempalli, Mahesh.Chandran} @fujitsu.com Fujitsu Research of India, Bangalore.
+
+# Abstract
+
+Graph Neural Networks (GNNs) excel at jointly modeling node features and topology, yet their black-box nature limits their adoption in real-world applications where interpretability is desired. Inspired by the success of interpretable Neural Additive Models (NAM) for tabular data, Graph Neural Additive Network (GNAN) extends the additive modeling approach to graph data to overcome limitations of GNNs. While being interpretable, GNAN representation learning overlooks the importance of local aggregation and more importantly suffers from parameter complexity. To mitigate the above challenges, we introduce Graph Neural Additive Model with Random Fourier Features (G-NAMRFF), a lightweight, self-interpretable graph additive architecture. G-NAMRFF represents each node embedding as the sum of feature-wise contributions where contributions are modeled via a Gaussian process (GP) with a graph- and feature-aware kernel. Specifically, we construct a kernel using Radial Basis Function (RBF) with graph structure induced by Laplacian and learnable Finite Impulse Response (FIR) filter. We approximate the kernel using Random Fourier Features (RFFs) which transforms the GP prior to a Bayesian formulation, which are subsequently learnt using a single layer neural network with size equal to number of RFF features. G-NAMRFF is light weight with  $168 \times$  fewer parameters compared to GNAN. Despite its compact size, G-NAMRFF matches or outperforms state-of-the-art GNNs and GNAN on node and graph classification tasks, delivering real-time interpretability without sacrificing accuracy<sup>1</sup>.
+
+# 1 Introduction
+
+Graph Neural Networks (GNNs) are powerful and topology-aware deep representation learning architectures that leverage the structure of graphs as an inductive bias [12, 20, 32]. While GNNs have demonstrated impressive performance on various downstream machine learning tasks such as node and graph classification, they remain largely as a black-box models [5, 3]. As a result, the learned representations often lack interpretability which limits their adoption in regulated domains like healthcare, finance, and defense [31, 5]. Interpretability in machine learning focuses on designing models whose representations directly reveal how each feature contributes to the output [28]. This stands in contrast to post hoc explainable models which seeks to generate explanations for the model inference after the model has been trained [3, 13].
+
+Additive models are the cornerstone for interpretable representation learning for tabular data. Notable examples of such models include Neural Additive Model (NAM) [2], Neural Basis Model (NBM) [24] and Gaussian Process Additive Model (GPAM) [41]. The key innovation of this additive model lies in its ability to handle each feature independently by learning a separate univariate function for every feature and then summing these contributions to obtain the final prediction. Although all these models share a common trait i.e., additive structure, they differ in how each feature's
+
+univariate function is represented and learned. For example, NAM approximates each feature's effect with a dedicated deep neural network (DNN), yielding high flexibility but also posing significant computational overhead. In contrast, NBM and GPAM represent each feature using a fixed dictionary of basis functions (e.g., splines, Fourier bases, or Gaussian process kernels) and learn only the linear combination of weights, thereby retaining interpretability while dramatically reducing training time.
+
+Building on the success of additive models for tabular data, efforts have been made to extend it to graph data. GNAN is the first graph neural additive model that incorporates graph structure as an inductive bias into a neural additive models [5], thus making it a self-interpretable glass-box model by design, in contrast to the conventional and well-studied post-hoc models designed to explain black-box GNNs outputs [31, 3, 37, 22, 17]. GNAN disentangles feature effects and graph topology by learning separate univariate functions, one per each feature and one for neighborhood weighting via DNNs, respectively. The neighborhood weighting is done through a distance measure that spans the entire graph. However, this design has two key drawbacks. Firstly, modeling each feature and neighbourhood with separate neural networks with multiple layers leads to a large trainable parameters with a significant training time. Secondly, for a graph with  $N$  nodes and  $E$ -edges,
+
+![](images/3834f8fe865cc76bc81fc07f5130f7611c7ebe583ef0b349d1c8755a2d26fe83.jpg)  
+Figure 1: Parameter complexity.
+
+GNAN aggregation over all nodes requires  $\mathcal{O}(N + E)$  distance computations per node, leading to quadratic scaling of the neighbourhood weighting function with graph size. Additionally, distant, non-neighbouring nodes influence the embeddings through the weighting function in a non-trivial way, introducing opacity to the interpretation unlike localized aggregation.
+
+In this work, we introduce the Graph-aware Neural Additive Model with Random Fourier Features (G-NAMRFF), a lightweight and inherently interpretable glass-box model designed for representation learning with graphs. Similar to NAM, G-NAMRFF represents each node prediction as the sum of individual feature contributions; however, it replaces deep neural networks with a Gaussian Process (GP) prior characterized by a kernel explicitly designed to be both graph- and feature-aware. Specifically, we propose a smoothness-promoting kernel that encourages neighboring nodes with similar attributes to yield similar outputs. In particular, the proposed kernel can be expressed as a Radial Basis Function (RBF) where inputs are projected features that capture one-hop neighbourhood in the graphs. Based on Bochner's theorem, we approximate the kernel effectively using RFFs, transforming the GP priors into a linear Bayesian formulation. To effectively capture multi-hop graph information, we embed the node features using Finite Impulse Response (FIR) filters with trainable coefficients. These coefficients, trained jointly with univariate functions help uncover critical graph structural information by highlighting which hops significantly influence the node predictions. With the equivalent Bayesian linear model established, we learn the model parameters using a single-layer network whose complexity depends solely on the number of RFFs, thereby ensuring G-NAMRFF's computational efficiency and lightweight nature. Figure 1, illustrates the parameter complexity comparison between our proposed model and existing method, clearly demonstrating the reduced parameter requirements of our approach. More importantly, by feeding RFFs with structure-aware features computed via learnable filter coefficients, our model seamlessly integrates graph topology and feature mapping into a single module unlike GNAN.
+
+We summarize the main contributions of this paper as follows:
+
+- We propose G-NAMRFF, a light-weight, interpretable graph neural additive model with RFFs and minimal parameter overhead. G-NAMRFF is a unified approach that seamlessly integrates topology and feature learning in one framework providing feature level interpretability through univariate functions and clear hop-level interpretability through the learned FIR coefficients.  
+- We provide theoretical guarantees that establish the robustness and permutation equivariance properties of G-NAMRFF.  
+- The proposed G-NAMRFF is parameter-efficient with  $168 \times$  fewer parameters compared to GNAN and consistently outperforms it on both node- and graph-classification tasks, and also
+
+matches or outperforms state-of-the-art black-box GNNs, demonstrating its suitability for sensitive and performance critical applications.
+
+Notation: Throughout this paper, we represent matrices and vectors as  $\mathbf{X}$  (boldface capital letters) and  $\mathbf{x}$  (boldface lowercase letters), respectively. The  $(i,j)$ -th element of a matrix is indexed by  $x_{i,j}$ . Identity matrix of dimension  $M$  is denoted by  $\mathbf{I}_M$ .
+
+# 2 Preliminaries
+
+In this section, we set up the mathematical background required for the development of the proposed model.
+
+# 2.1 Graph Neural Additive Network
+
+Consider a graph  $\mathcal{G}$  with  $N$  nodes, with the feature vector associated with each node being  $\mathbf{x} \in \mathbb{R}^D$ ; the feature matrix as  $\mathbf{X} \in \mathbb{R}^{N \times D}$ . The GNAN models each feature independently using a DNN and weights it with a learnable function that captures underlying graph topology. Formally, the embedding of  $k$ -th component of node  $i$  is given by,
+
+$$
+z _ {i, k} = \sum_ {j = 1} ^ {N} \frac {1}{\left| \operatorname {d i s t} _ {i} (j , i) \right|} \rho (\operatorname {d i s t} (i, j)) f _ {k} \left(x _ {j, k}\right), \tag {1}
+$$
+
+Here  $f(.), \rho(.)$  are the learnable non-linear functions.  $\mathrm{dist}(i,j)$  captures the distance between the nodes and  $|\mathrm{dist}_i(i,j)|$  is the number of nodes at  $\mathrm{dist}(i,j)$  from node  $i$ . From (1), it is clear that embedding of the  $i$ -th node requires computing the distance of the  $i$ -th node to all the nodes of  $\mathcal{G}$ , which makes (1) ineffective to handle large graphs.
+
+# 2.2 Gaussian Process
+
+A Gaussian Process (GP) [26] is a probabilistic framework that defines a distribution over functions, where any finite set of function values follows a multivariate Gaussian distribution. It is completely specified by a mean function,  $\mathbf{m}(\mathbf{x})$  and a covariance kernel  $\mathbf{K}_{\theta}(\mathbf{x},\mathbf{x}^{\prime})$ , where  $\mathbf{x}$  and  $\mathbf{x}^{\prime}$  are data points on  $\mathbb{R}^{D}$ . We denote the GP prior as  $f(\mathbf{x}) \sim \mathcal{GP}(\mathbf{m}(\mathbf{x}),\mathbf{K}_{\theta}(\mathbf{x},\mathbf{x}^{\prime}))$ , where  $\mathbf{K}_{\theta}(\mathbf{x},\mathbf{x}^{\prime}) = \exp \left(-\frac{\|\mathbf{x} - \mathbf{x}^{\prime}\|^{2}}{2\theta^{2}}\right)$  with  $\theta$  being the bandwidth.
+
+# 2.3 Kernel Approximation with Random Fourier Features
+
+Random Fourier Features (RFFs) based kernel approximation is a popular technique for addressing the scalability of kernel-based methods such as Gaussian Processes [25]. The key idea behind RFFs is to approximate a shift-invariant kernel by expressing it as an inner product of low-dimensional feature mappings and is built on Bochner's theorem:
+
+Theorem 2.1 (Bochner's Theorem and RFF Approximation [25]). Let  $\mathbf{K}_{\theta} \in \mathbb{R}^{N \times N}$  denote a bounded, shift-invariant and positive definite kernel (i.e.,  $\mathbf{K}_{\theta}$  is a Fourier transform of non negative measure).  $\mathbf{K}_{\theta}$  when properly scaled can be approximated using RFF as
+
+$$
+\mathbf {K} _ {\theta} (\mathbf {x}, \mathbf {x} ^ {\prime}) \approx \boldsymbol {\Phi} _ {\mathbf {a}} ^ {T} (\mathbf {x}) \boldsymbol {\Phi} _ {\mathbf {a}} (\mathbf {x} ^ {\prime}), \quad \boldsymbol {\Phi} _ {\mathbf {a}} (\mathbf {x}) = \left\{\sqrt {\frac {2}{M}} \cos (2 \pi \mathbf {a} _ {\mathrm {m}} ^ {\mathrm {T}} \mathbf {x} + b _ {\mathrm {m}}) \right\} _ {m = 1} ^ {M}, \qquad (2)
+$$
+
+where  $\Phi_{\mathbf{a}}$  is a feature map termed as random Fourier features with  $M$  being the number of RFF features,  $\mathbf{a}_m \sim \mathcal{N}(\mathbf{0}, \mathbf{I}_D)$  and  $b_m \in \mathrm{Unif}(0, 2\pi)$  are drawn from normal and uniform distributions.
+
+This approximation enables a scalable reformulation of the GP prior into linear Bayesian model. Specifically, prior can be modeled as  $f(\mathbf{x}) = \Phi_{\mathbf{a}}^{T}(\mathbf{x})\mathbf{w}$ , where  $\mathbf{w} \in \mathbb{R}^{M \times 1} \sim \mathcal{N}(\mathbf{0}, \mathbf{I}_{M})$ .
+
+# 3 Graph-aware Neural Additive Model with Random Fourier Features
+
+In this section, we present G-NAMRFF, a lightweight, self-interpretable graph neural additive model that integrates graph topology through GP priors and RFF.
+
+Let  $\mathcal{G} = (\mathcal{V},\mathcal{E})$  be weighted and undirected graph with  $N$  nodes and the associated graph Laplacian as  $\mathbf{L}_{\mathcal{G}}\in \mathbb{R}^{N\times N}$ . We assume that each node  $i\in \mathcal{V}$  is associated with a  $D$  dimensional feature  $\mathbf{x}_i\in \mathbb{R}^{D\times 1}$  and represent the feature matrix as  $\mathbf{X}\in \mathbb{R}^{N\times D}$ .
+
+Given  $\mathbf{X}$  and  $\mathbf{L}_{\mathcal{G}}$ , G-NAMRFF models each feature contribution  $f_{k}:\mathbb{R}\to \mathbb{R}$  through a Gaussian process whose covariance kernel enforces smoothness with respect to both the feature values and the underlying graph topology. In particular, G-NAMRFF obtains the contribution of each feature to node prediction  $y_{i}$  as
+
+$$
+y _ {i} = \sum_ {k = 1} ^ {D} f _ {k} (x _ {i, k}), \quad f _ {k} \sim \mathcal {G P} \left(0, \mathbf {K} _ {\mathcal {G}} \left(x _ {i, k}, x _ {j, k}\right)\right) \tag {3}
+$$
+
+where  $\mathbf{K}_{\mathcal{G}}(.,.)$  is a graph covariance kernel. The embedding of node  $i$  i.e.,  $\mathbf{z}_i \in \mathbb{R}^D$  is obtained by stacking  $[f_1(x_{i,1}), f_2(x_{i,2}), \ldots, f_D(x_{i,D})]$ . In what follows, we first define the graph covariance kernels that enforce smoothness assumption, which are shift invariant and can be approximated using RFFs. Then, leveraging this approximation we propose the light-weight model.
+
+# 3.1 Gaussian Process Priors with Graph-aware Kernel
+
+To encode the prior knowledge that adjacent nodes with similar feature values should output similar function values i.e., function should vary smoothly across the nodes, we impose the GP prior with the graph aware covariance kernel as
+
+$$
+\mathbf {K} _ {\mathcal {G}} \left(x _ {i, k}, x _ {j, k}\right) = \exp \left(\frac {- a _ {i , j} \left(x _ {i , k} - x _ {j , k}\right) ^ {2}}{2 \Theta^ {2}}\right), \tag {4}
+$$
+
+where  $\Theta > 0$  is the hyperparameter that controls the width of the function and  $a_{i,j} > 0$  corresponds to  $i$ ,  $j$ -element of weight matrix  $\mathbf{A}$  with weights assigned based on the distance/similarity between the nodes. Observe that if the nodes are close in the graph and features are similar the kernel outputs high covariance, whereas for distant nodes with features being dissimilar  $\mathbf{K}_{\mathcal{G}}$  yields low covariance. Thus, the proposed kernel  $\mathbf{K}_{\mathcal{G}}$  is both feature and graph-aware. Imposing such a prior enforces  $f_{k}(x_{i,k})$  to vary smoothly across the connected nodes. Note that for a fixed  $a_{i,j}$ , kernel defined in (4) is shift invariant with respect to nodes  $i$ ,  $j$  locally, i.e.,  $\mathbf{K}_{\mathcal{G}}(x,y) = \mathbf{G}_{\mathcal{G}}(x - y)$ , with  $\mathbf{G}_{\mathcal{G}}(t) = \exp \left(-\frac{a_{i,j}t^2}{2\Theta^2}\right)$ . Shift invariance of the kernel (Stationarity is required for the approximation of kernels by Bochner's Theorem (see Theorem 2.1) which in turn aids in formulating the light-weight model.
+
+Observe that Eq.(4) can be equivalently expressed as  $\mathbf{K}_{\mathcal{G}}(x_{i,k}, x_{j,k}) = \exp \left(\frac{-(\sqrt{a_{i,j}} x_{i,k} - \sqrt{a_{i,j}} x_{j,k})^2}{2\Theta^2}\right)$ . Let us call projected features as  $\tilde{x}_{i,k} = \sqrt{a_{i,j}} x_{i,k}$  and  $\tilde{x}_{j,k} = \sqrt{a_{i,j}} x_{j,k}$ . By rewriting, we obtain kernel function with graph-aware inputs
+
+$$
+\mathbf {K} _ {\mathcal {G}} \left(x _ {i, k}, x _ {j, k}\right) = \exp \left(\frac {- \left(\tilde {x} _ {i , k} - \tilde {x} _ {j , k}\right) ^ {2}}{2 \Theta^ {2}}\right), \tag {5}
+$$
+
+In other words, the  $\mathbf{K}_{\mathcal{G}}$  is modified RBF with inputs as filtered features based on one-hop neighbourhood since the projected variables use weighted graph Adjacency. From Theorem 2.1, we can approximate  $\mathbf{K}_{\mathcal{G}}$  as  $\mathbf{K}_{\mathcal{G}}(x_{i,k}, x_{j,k}) = \boldsymbol{\Phi}_{\mathbf{a}}^{T}(\tilde{x}_{i,k}) \boldsymbol{\Phi}_{\mathbf{a}}(\tilde{x}_{j,k})$ .
+
+# Remarks:
+
+- Although the proposed kernel function  $\mathbf{K}_{\mathcal{G}}$  is shift invariant and graph-aware, it is apparent that it only captures the information from one hop and may fail to capture the information from larger depths (see the importance in section 6).  
+- More importantly, to study the interpretability, it is also important to analyze for a fixed node  $i$  which neighbors are influencing the predictions.
+
+To address the aforementioned challenges, we reformulate the kernel to incorporate information from deeper neighborhoods. Specifically, we make the kernel inputs topology-aware by replacing the original features with graph-filtered representations. This yields an RBF kernel whose inputs are graph-aware feature embeddings. Notably, this formulation preserves both stationarity globally and positive definiteness, ensuring that the resulting covariance function remains valid under the Gaussian process framework and RFF approximation.
+
+![](images/51ee0bf1180a0c9d7c999de1b0048b632544939c0ee6a13759d5e044a021bce4.jpg)  
+Figure 2: Architecture of G-NAMRFF.
+
+# 3.1.1 Topology-Aware Random Fourier Features with Learnable FIR Filters
+
+We aggregate multi-hop information through learnable FIR filters, represented as polynomial functions of the graph shift operator—either the adjacency matrix or the graph Laplacian. Specifically, when employing the graph Laplacian as the shift operator, the formulation is given by:
+
+$$
+\mathbf {H} = \sum_ {h = 0} ^ {R} \alpha_ {h} \tilde {\mathbf {L}} _ {\mathcal {G}} ^ {h}, \tag {6}
+$$
+
+where  $\tilde{\mathbf{L}}_{\mathcal{G}} = \mathbf{I} - \mathbf{D}^{-1 / 2}\mathbf{A}\mathbf{D}^{-1 / 2}$  is the normalized graph Laplacian and  $R$  is the filter order which is a hyperparameter that controls how multi-hop information is aggregated. In our framework, we set the filter parameters as learnable. We emphasize that instead of fixing the filter parameters, making them learnable enhances the interpretability (see Section 6). In particular, one can inspect the filter coefficients and infer information such as the hops that are contributing to the node embedding. Using Eq.(6), the filtered response is given by  $\tilde{\mathbf{X}} = \mathbf{H}\mathbf{X}\in \mathbb{R}^{N\times D}$ , with  $\tilde{\mathbf{X}}$  is now the smoothened graph features encoding much richer neighborhood beyond one-hop. Note that modifying the kernel input by the filtered response transforms  $\mathbf{K}_{\mathcal{G}}$  to be structure aware RBF that is shift invariant and also non negative. Following earlier discussion, it can be expressed as  $\mathbf{K}_{\mathcal{G}}(x_{i,k},x_{j,k}) = \boldsymbol{\Phi}_{\mathbf{a}}^T (\tilde{x}_{i,k})\boldsymbol{\Phi}_{\mathbf{a}}(\tilde{x}_{j,k})$ , where  $\tilde{x}_{i,k}$  corresponds to  $(i,k)$ -th element of  $\tilde{\mathbf{X}}$ .
+
+Leveraging the kernel approximation technique (Bochner's theorem), the GP prior defined in 3 can be equivalently expressed in linear Bayesian formulation as
+
+$$
+f _ {k} \left(x _ {i, k}\right) = \boldsymbol {\Phi} _ {\mathbf {a}} ^ {T} \left(\tilde {x} _ {i, k}\right) \mathbf {w} _ {k}, \quad \mathbf {w} _ {k} \sim \mathcal {N} \left(\mathbf {0}, \mathbf {I} _ {M}\right) \tag {7}
+$$
+
+where  $\Phi_{\mathbf{a}}(\tilde{x}_{i,k}) = \left\{\sqrt{\frac{2}{M}}\cos (2\pi a_m\tilde{x}_{i,k} + b_m)\right\}_{m = 1}^M$  with  $\Phi_{\mathbf{a}}(.)$  is a feature map, which we refer to as graph-aware Random Fourier Features (G-RFFs),  $a_{m}\sim \mathcal{N}(0,1)$  and  $b_{m}\in \mathrm{Unif}(0,2\pi)$  are drawn from normal and uniform distributions. To summarize, G-NAMRFF, learns contribution of features to node-  $i$  prediction as
+
+$$
+y _ {i} = \sum_ {k = 1} ^ {D} \boldsymbol {\Phi} _ {\mathbf {a}} ^ {T} \left(\tilde {x} _ {i, k}\right) \mathbf {w} _ {k}, \quad \mathbf {w} _ {k} \sim \mathcal {N} (\mathbf {0}, \mathbf {I} _ {M}). \tag {8}
+$$
+
+# RFF Linear Layer
+
+Note that under modified prior, each  $f_{k}$  is a linear transformation whose weights can be obtained by single-layer neural network with exactly  $M$  learnable parameters per feature. Also, embedding the structural information into features and representing the univariate function using graph-aware RFFs results in a light-weight model with number of trainable parameters dependent on number of RFFs and filter order (more discussion in section 4). We conclude this section with some important observations: (a) for  $R = 1$  and  $\alpha_0 = 0$ , the present approach resembles the first approach where features are transformed based on one-hop neighbourhood. (b) The filter order and number of RFF features required are less compared to hidden units for all the graph datasets (more details in Appendix C.3).
+
+# 3.2 Implementation Details
+
+In Fig. 2, we present end to end architecture of G-NAMRFF for obtaining the prediction of  $i$ -th node as a summation of univariate function outputs. Considering  $\mathbf{X} \in \mathbb{R}^{N \times D}$ , we pass each feature
+
+# Algorithm 1 Algorithm for G-NAMRFF
+
+1: Inputs: Feature matrix  $\mathbf{X} \in \mathbb{R}^{N \times D}$ , normalized Laplacian  $\tilde{\mathbf{L}}_{\mathcal{G}}$ , RFF dimension  $M$ , bandwidth  $\Theta$ , filter order  $R$ .  
+2: Trainable parameters:  $\{\alpha_h\}_{h = 0}^R$  , weight vectors  $\{\mathbf{w}_k\in \mathbb{R}^M\}_{k = 1}^D$  
+3: Output: Node predictions  $\{y_i\}_{i=1}^N$ .  
+4: Filtering: Compute filtered features  $\tilde{\mathbf{X}} = \sum_{h=0}^{R} \alpha_h \tilde{L}_G^h \mathbf{X}$ .  
+5: RFF computation: Sample frequencies  $\{a_{k,m}\}_{m=1}^{M} \sim \mathcal{N}(0,1)$  and phases  $\{b_{k,m}\}_{m=1}^{M} \sim \mathrm{Uniform}(0,2\pi)$  for each  $k$ . RFF map:  $\Phi_{\mathbf{a}}(x) = \sqrt{\frac{2}{M}}\left[\cos(a_{k,1}x + b_{k,1}), \ldots, \cos(a_{k,M}x + b_{k,M})\right]^{\top}$  
+6: for each node  $i = 1, \dots, N$  do  
+7: for each feature  $k = 1, \dots, D$  do  
+8: Obtain feature embedding:  $f_{k}(x_{i,k}) = \Phi_{\mathbf{a}}^{\top}(\tilde{x}_{i,k})\mathbf{w}_{k}$  
+9: Prediction:  $y_{i} \gets \sum_{k=1}^{D} f_{k}(x_{i,k})$ .  
+10: return  $\{y_i\}_{i=1}^N$ .
+
+column through the graph filter parameterized by shared filter coefficients. The magnitude of the filter coefficients reflects the importance of the neighbourhood. In other words, they explain which hop information plays a key role in the prediction. We also emphasize that sharing filter parameters reduces the complexity without compromising the accuracy (more details in Section 6). In the following step, for the target node  $i$ , we extract  $D$ -features and feed them to RFF layer where it undergoes the cosine transformation with the sampling parameters  $\{a_m, b_m\}_{m=1}^M$  shared across the features. The output of the RFF layer is the  $M$ -dimension RFF vector  $(\Phi_{\mathbf{a}})$  for each feature. Each of these undergoes a linear transformation using a single layer NN with size as number of RFF features. The final predictions are obtained depending on the downstream tasks:
+
+For node classification: For node classification task, the predictions obtained following (8) are passed through activation function (for ex . , sigmoid or softmax).
+
+For graph classification: We obtain the representation for the complete graph by sum pooling as  $\mathbf{z}_{\mathcal{G}} = \sum_{i=1}^{N} \mathbf{z}_i$ . Then the prediction follows by passing representation through an activation function.
+
+Handling multi-class datasets: If the dataset contains  $C$ -classes, we project the weights of a single layer RFF to the number of classes thereby enabling class-wise interpretability (more details in Section 6). We present detailed step by step process in Algorithm 1.
+
+# 4 Theoretical Characterization and Learnable Parameters Computation
+
+In this section, we discuss some important properties of G-NAMRFF i.e., permutation equivariance and robustness to the perturbations, and then discuss parameter complexity.
+
+Theorem 4.1 (Permutation Equivalence). Let  $\mathcal{P} = \{\mathbf{P} \in \{0,1\}^{N \times N} : \mathbf{P}^{\top} \mathbf{P} = \mathbf{P} \mathbf{P}^{\top} = \mathbf{I}_N\}$  be the set of all  $N \times N$  permutation matrices. Then under the permutation of the graph Laplacian  $\mathbf{L}_{\mathcal{G}}$  and node-feature matrix  $\mathbf{X}$  by any  $\mathbf{P} \in \mathcal{P}$ , the outputs from  $G$ -NAMRFF also modify as  $\mathbf{y}_{perm} = \mathbf{P} \mathbf{y}$ , where  $\mathbf{y} \in \mathbb{R}^N$  is the predictions across all the nodes.
+
+The proof for Theorem 4.1 is relegated to Appendix A. Observe that Theorem 4.1 guarantees any simultaneous permutation of node indices in the Laplacian and feature matrix results in the same permutation of the learned embeddings. This symmetry is essential for ensuring that the model's predictions depend only on the graph's structure, not on an arbitrary ordering of its nodes.
+
+Theorem 4.2 (Robustness to perturbation of graph Laplacian). Let  $\widehat{\mathbf{L}}_{\mathcal{G}} = \mathbf{L}_{\mathcal{G}} + \Delta \mathbf{L}_{\mathcal{G}}$  be the Laplacian of the perturbed graph, with  $\| \Delta \mathbf{L}_{\mathcal{G}} \|_2 \leq \epsilon$ , and assume that the RFF map  $\Phi_a(.)$  is  $C_{\mathrm{RFF}}$ -Lipschitz continuous. Then each node prediction satisfies  $|\widehat{y}_i - y_i| \leq CK \in D\|\mathbf{X}\|_2$ , where  $C = C_{\mathrm{RFF}}\left(\max_k\|\mathbf{w}_k\|_2\right)$ ,  $K = \frac{1}{4}\|\alpha\|_1(R^2 - 1)\left(\frac{R - 1}{R + 1}\right)^R$  are constants.
+
+The proof for Theorem 4.2 is relegated to Appendix A. It emphasizes the error i.e., difference between the predictions from G-NAMRFF is bounded linearly with the norm of the error term  $\Delta \mathbf{L}_{\mathcal{G}}$ . Hence, the
+
+node predictions are robust to small changes in the graph structure. The proof follows by leveraging the perturbation bounds on FIR filters and then invoking the Lipschitz continuity of RFF features.
+
+# 4.1 Parameter Complexity
+
+Recall, GNAN employs a deep network  $f_{k}$  for each of  $D$  features plus an additional DNN for  $\rho$  to encode the graph structure. If each of these  $D + 1$  is an  $L$  layer feed forward neural network with  $H_{u}$  hidden units, the total number of parameters to be learnt are  $(D + 1)\times (H_u^2\times (L - 1) + (L + 2)\times H_u + 1)$ .
+
+For G-NAMRFF, the only learnable parameters are filter coefficients and weights of dimension  $M$ . Therefore, total number of learnable parameters are  $D \times M + R + 1$ , where  $R$  is the filter order. For numerical comparison, considering the values given in GNAN [5] with  $H_{u} = 64$ ,  $L = 5$  and  $D = 100$ , for the proposed method with  $M = 100$  and  $R = 5$ , the number of parameters are approximately  $168 \times$  fewer. This reduction in model size, translates to arguably smaller run times of G-NAMRFF (see discussion on run times in Appendix C.2).
+
+# 5 Related Works
+
+The existing works that are closely related to ours can be broadly classified into 3 categories as follows:
+
+Black-box GNN Architectures: Graph representation learning with GNNs has become ubiquitous across a wide range of applications. Broadly speaking, existing black-box GNNs can be categorized into two types: (a): Message-passing models, which iteratively aggregate and update node features via learned message functions [12, 35, 11]. (b): Spectral-convolutional architectures, which perform graph filtering in the Fourier domain to capture global structure [40, 6, 20, 42, 34]. Additionally, to better weigh the influence of individual neighbors while aggregating messages, attention-based extensions such as Graph Attention Networks (GAT) [32], and more recently, graph transformer variants [39, 33] have been proposed. [33, 39, 32]. Finally, several works employ Gaussian processes over graphs to capture predictive uncertainty directly [7, 10]. [7, 10]. Despite these advances, all these approaches model a black-box function, and their learned embeddings must be supplemented by separate interpretability techniques.
+
+Interpretable Additive Models: Additive modeling is the workhorse principle of interpretable architectures for tabular data since the inception of Generalized Additive Models (GAMs) [14] to recent methods such as GPAM [41] [15, 2, 24, 8]. However, these methods were developed primarily for tabular inputs, and naive application of them to graph-structured data fails to exploit the relational inductive biases inherent in the graph topology. To address this gap, [5] extends generalized additive modeling to graphs by learning node-wise contributions aggregated across neighborhoods. As discussed earlier, this approach suffers with high parameter complexity and employs aggregation schemes that do not fully capture the influence of local graph structure.
+
+Post-hoc GNN Explainers: Explanable models for graph data are mostly post-hoc methods that interpret pretrained GNNs by identifying critical substructures or features after model training. For instance, models like GNN-Explainer [37], PG-Explainer [22] uncover the important subgraphs and corresponding node features for explaining the feature predictions, whereas Graph-LIME [17] extends LIME [27] to GNN architectures to quantify each feature contribution to a node-level decision. Recently, GRAPHTRAIL[4] enhances GNN interpretability by translating learned representations into concise, human-readable Boolean formulas. A comprehensive study of post-hoc explainable models is detailed in [31, 38].
+
+# 6 Numerical Experiments and Discussion on Interpretability
+
+In this section, we evaluate G-NAMRFF performance and demonstrate its ability to learn interpretable representations on node and graph classification tasks.
+
+# 6.1 Interpretability on Node Classification Task
+
+To begin with, we analyze the univariate functions learnt on PubMed [29] dataset with node-classification task. PubMed is a multiclass citation-network dataset where nodes are research articles
+
+categorized into three diabetes classes: Type 1, Type 2 and gestational [5]. To reveal how individual features influence the model's predictions, we plot each learned function against its corresponding raw input values. In particular, as G-NAMRFF produces three outputs per feature (one score for each class), we plot three separate curves for each feature as shown in Fig. 3. At any given feature value, the curve with the highest output indicates the class that the feature most strongly supports.
+
+In Fig 3(a), we show the influence of feature "children" on model output. It can be observed that model learnt a stronger contribution toward Type 1 diabetes, indicating that the presence of this term in an article increases the likelihood of it being classified under this category. This observation aligns with clinical understanding, as Type 1 diabetes is autoimmune in nature with children being susceptible[30], leading to increased mention of pediatric-related terms in relevant literature. Similarly, in Fig. 3(c), we show the influence of BMI (Body Mass Index), where it is clear that model associates predominantly with Type 2 diabetes. This clearly aligns with the clinical findings that elevated BMI is a major risk factor for Type 2 diabetes [1]. In Fig. 3(b), we show the influence of "sex (gender)," where model shows relatively lower contributions to both Type 1 and Type 2 diabetes, which is consistent with the fact that sex plays a mere role for diagnosis of Type 1 and Type 2 diabetes. However, in the case of gestational diabetes, which by definition is diagnosed during pregnancy, the model attributes higher importance.
+
+Although these explanations merely match model prediction and interpretation with ground truth, for more challenging and high-stakes scenarios, a subject matter expert would better assess the correctness of such plots. Doing so, we can truly utilize the transparency of the model to increase the belief in such predictions and also use the feedback to improve the model further.
+
+![](images/e1b7a376caaa7e0d105f6650ca5949142de00dc2a32733ac52a61d175a9714b6.jpg)  
+(a)
+
+![](images/41f003a31d78033fee866d81ffb7cce54377ebcea2a48545a02103d6fb92289a.jpg)  
+(b)
+
+![](images/76a4a642895f8b9e70859fff6645b63367775ebf357444d81de1a847e19b837a.jpg)  
+(c)
+
+![](images/6bd387fa466043ebc99d4f6eb38555e840d574a0746f99a63aabe10a6fdaf1b7.jpg)  
+(d)  
+Figure 3: PubMed dataset: Univariate function outputs on different features.
+
+# 6.2 Empirical Results on Node Classification Task
+
+Having demonstrated G-NAMRFF's ability to learn interpretable representations, we empirically evaluate its performance on medium-scale and large scale datasets namely Cora, CiteSeer, PubMed [36], Cornell[23], ogbn-arxiv and ogbn-products [16]. In Table 1, we report mean classification accuracy along with the standard deviation over five independent runs with different random seeds. Hyperparameter details (e.g., number of RFFs  $(M)$ , scale  $(\Theta)$ , filter order  $(R)$ , and optimizer settings) are provided in the Appendix C.1.
+
+Across all six datasets, G-NAMRFF matches or exceeds the performance of state-of-the-art black-box GNNs, confirming that interpretability need not come at the cost of accuracy. We also observe that incorporating information from multiple hops (G-NAMRFF) outperforms the one-hop variant (G-NAMRFF  $(R = 1)$ ), highlighting the value of deeper neighborhood aggregation. Finally, among interpretable models, G-NAMRFF outperforms the state-of-the-art models in addition to scaling effectively. In particular, on ogbn-products, both NAM and GNAN encounter out-of-memory errors, whereas G-NAMRFF trains successfully. A comprehensive study on hyperparameters is presented in the Appendix C.3.
+
+# 6.3 Interpretability on Graph Classification Task
+
+In this section, we discuss the interpretability on the graph classification task with the widely studied Mutagenicity dataset [18, 9]. The dataset includes 4,337 molecules represented as graphs and each graph is associated with binary labels. In contrast to node classification, here we present both global level and local level interpretations. Recall that we obtain the graph-level embeddings by sum pooling node embeddings. To begin with, we discuss the feature level importance.
+
+Table 1: Performance comparison on node classification task.  
+
+<table><tr><td>Model</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td><td>Cornell</td><td>ogbn-arxiv</td><td>ogbn-products</td></tr><tr><td>GCN [20]</td><td>81.23 ± 1.1</td><td>71.20 ± 1.7</td><td>78.50 ± 1.3</td><td>65.90 ± 0.5</td><td>71.74 ± 0.3</td><td>75.64 ± 0.3</td></tr><tr><td>GAT [32]</td><td>80.32 ± 2.3</td><td>70.26 ± 2.3</td><td>77.12 ± 2.4</td><td>72.50 ± 0.7</td><td>71.95 ± 0.6</td><td>79.45 ± 0.5</td></tr><tr><td>GraphSAGE [12]</td><td>79.94 ± 3.4</td><td>65.12 ± 1.9</td><td>78.25 ± 1.2</td><td>75.90 ± 5.0</td><td>71.49 ± 0.2</td><td>75.63 ± 0.3</td></tr><tr><td>Graph Transformer [39]</td><td>80.70 ± 0.5</td><td>76.00 ± 0.9</td><td>78.80 ± 1.4</td><td>70.50 ± 1.7</td><td>70.13 ± 0.5</td><td>74.74 ± 0.5</td></tr><tr><td>NAM [2]</td><td>51.35 ± 2.3</td><td>55.40 ± 1.9</td><td>58.16 ± 2.3</td><td>59.15 ± 2.6</td><td>56.12 ± 3.4</td><td>OOM</td></tr><tr><td>GPAM [41]</td><td>59.96 ± 3.2</td><td>60.30 ± 3.9</td><td>62.30 �� 3.7</td><td>60.12 ± 3.6</td><td>62.35 ± 4.2</td><td>60.13 ± 3.9</td></tr><tr><td>GNAN [5]</td><td>77.89 ± 5.1</td><td>65.23 ± 3.7</td><td>75.13 ± 2.4</td><td>71.76 ± 4.2</td><td>69.56 ± 0.9</td><td>OOM</td></tr><tr><td>G-NAMRFF (R=1)</td><td>75.32 ± 1.8</td><td>67.12 ± 1.1</td><td>75.12 ± 3.8</td><td>64.12 ± 2.9</td><td>66.94 ± 1.6</td><td>55.73 ± 1.8</td></tr><tr><td>G-NAMRFF</td><td>79.84 ± 1.7</td><td>69.45 ± 2.5</td><td>77.30 ± 1.4</td><td>73.54 ± 4.9</td><td>70.02 ± 3.9</td><td>72.13 ± 0.4</td></tr></table>
+
+![](images/cd7d0a27832b4ce22e9abd9b099d89e4206b33186e8c6fdb9c64498a16f7413f.jpg)  
+(a)
+
+![](images/f7471a2eeb0957b51d105274f4b21e8a694cf702bf5667676753d207798fbddc.jpg)  
+(b)
+
+![](images/562a4a5d33397eafea1aacef6e9230665b087e5f79952a39f43ba97229baed3a.jpg)  
+Figure 4: (a). Feature level explanation on Mutagenicity dataset. (b). Local structure level explanation on Mutagenicity dataset. (c). Learnt FIR filter coefficients.
+
+![](images/61846e277a8bdf2922d2f36d41ccbe7445b0c4779539992d6223d64a78407568.jpg)  
+(c)
+
+Focusing on mutagenic class, we obtain the feature level scores globally by taking the sum of each atom's contribution across the graphs. In Fig. 4(a), we present per-atom importance for the mutagenic class. Notably, Nitrogen, Oxygen, and Lithium atoms exhibit the highest positive contributions towards predicting mutagenicity. This interpretability not only helps us understand why the model makes a given prediction, but also directs domain experts to inspect and, if necessary, adjust the features or model. For example, if the model misclassifies a molecule, one can review the atom importance scores to assess which atomic features drove the error and representation learning can be modified accordingly.
+
+Although feature-level importance gives insight into individual atom contributions, understanding which substructures are responsible for mutagenicity is crucial for molecular datasets. Using our model, we obtain local explanations by aggregating the feature-level attributions along the neighborhood to highlight the subgraph responsible for a prediction.
+
+In Fig. 4(b), we visualize these extracted substructures for two representative molecules classified as mutagenic. In the first molecule, the nitro  $(NO_2)$  and amino  $(NH_2)$  groups emerge as the dominant substructures each exhibiting strong positive attribution toward the mutagenic label. In the second, the aromatic carbon ring itself drives the mutagenicity score. These findings align closely with prior toxicology studies [9, 37], which identify  $NH_2$ ,  $NO_2$ , and polyaromatic rings as key mutagenic motifs. Importantly, our model reveals that hydrogen and oxygen atoms directly bonded to nitrogen also carry a significant contribution towards a mutagenic class, whereas isolated hydrogens on carbon rings do not play that significant role. This observation supports the global atom-importance ranking in Fig. 4(a), where nitrogen outranks hydrogen. The study presented again emphasizes the G-NAMRFF ability in global-level and local-level attributions. Recall that each FIR filter coefficient  $\alpha_h$  quantifies how much the model weights information from its  $h$ -hop neighborhood. To demonstrate this on the Mutagenicity dataset, in Fig. 4(c), we plot the filter coefficients for a fixed filter order. It can be observed that there is a clear decay in magnitude as  $h$  increases, confirming that our model naturally prioritizes close neighbors and progressively down-weights more distant nodes. As explained earlier, these learned coefficients thus offer direct, interpretable insight into the receptive field of each atom.
+
+# 6.4 Empirical Evaluation on Graph Classification Task
+
+In this section, we report G-NAMRFF performance on several standard graph-classification benchmarks: Proteins, NCI1, Mutagenicity, MUTAG, and PTC [16]. We compare the proposed model against both black-box GNNs and interpretable models. In Table 2, we present the mean graph classifica
+
+Table 2: Performance comparison on graph classification task.  
+
+<table><tr><td>Model</td><td>Proteins</td><td>Mutag</td><td>Mutagenicity</td><td>NCI1</td><td>PTC</td></tr><tr><td>GCN [20]</td><td>70.97±4.6</td><td>68.07±6.3</td><td>75.69±0.9</td><td>66.35±1.3</td><td>56.98±5.8</td></tr><tr><td>GAT [32]</td><td>69.92±4.0</td><td>67.20±3.4</td><td>69.40±1.2</td><td>66.12±2.1</td><td>55.60±11.1</td></tr><tr><td>GraphSAGE [12]</td><td>67.35±2.3</td><td>64.12±2.4</td><td>69.25±3.9</td><td>65.56±3.9</td><td>57.12±4.9</td></tr><tr><td>Graph Transformer [39]</td><td>69.76±3.2</td><td>66.30±5.3</td><td>73.10±0.9</td><td>68.24±3.4</td><td>55.90±3.5</td></tr><tr><td>NAM [2]</td><td>62.45±4.2</td><td>63.12±9.1</td><td>67.35±2.5</td><td>57.15±1.2</td><td>54.97±7.5</td></tr><tr><td>GPAM [41]</td><td>65.68±4.1</td><td>64.30±8.4</td><td>65.46±2.2</td><td>53.80±2.7</td><td>52.65±8.1</td></tr><tr><td>GNAN [5]</td><td>59.64±2.4</td><td>67.35±3.9</td><td>66.64±4.7</td><td>50.87±1.4</td><td>55.07±5.2</td></tr><tr><td>G-NAMRFF (R=1)</td><td>67.83±4.4</td><td>71.20±6.7</td><td>68.98±3.4</td><td>63.65±2.8</td><td>55.65±5.6</td></tr><tr><td>G-NAMRFF</td><td>69.94±3.7</td><td>79.81±5.3</td><td>71.70±2.0</td><td>66.10±1.7</td><td>61.91±3.4</td></tr></table>
+
+tion accuracies from 10-fold cross-validation, each repeated over three random seeds. The results highlights the strength of proposed model in obtaining the interpretable graph embeddings without compromising the accuracy. Detailed hyperparameter settings and ablation studies are provided in Appendix C.1 and C.3.
+
+# 7 Limitations and Future Directions
+
+While G-NAMRFF models the prediction output as a sum of independent feature-wise contributions, this formulation overlooks potential correlations among features. A straightforward extension that explicitly models both individual and pairwise feature interactions would substantially increase the parameter count, compromising the model efficiency. Hence designing a parameter-efficient formulation that can capture feature dependencies and higher-order correlations without sacrificing interpretability remains an important direction for future research. Further, recall that the smoothness prior imposed on the function limits its effectiveness on heterophilic graphs, where connected nodes often have dissimilar labels. Therefore, extending the current approach to better handle heterophilic data presents an interesting and promising direction for future research.
+
+# 8 Conclusions
+
+In this work, we introduced G-NAMRFF, a self-interpretable, lightweight additive model for graphs. We proposed a unified interpretable architecture that integrates both graph topology and node features into univariate functions via Random Fourier Features. Further, we provided theoretical guarantees showing that G-NAMRFF is robust to graph perturbations and permutational equivariant. Our analysis on parameter complexity revealed that G-NAMRFF requires  $168 \times$  fewer parameters than GNAN. Finally, through experiments on node and graph classification benchmarks, we demonstrated that although being light-weight and interpretable, G-NAMRFF does not compromise predictive performance.
+
+# References
+
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+
+# NeurIPS Paper Checklist
+
+The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit.
+
+Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist:
+
+- You should answer [Yes], [No], or [NA].  
+- [NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available.  
+- Please provide a short (1–2 sentence) justification right after your answer (even for NA).
+
+The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper.
+
+The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While "[Yes]" is generally preferable to "[No]", it is perfectly acceptable to answer "[No]" provided a proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering "[No]" or "[NA]" is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found.
+
+IMPORTANT, please:
+
+- Delete this instruction block, but keep the section heading "NeurIPS Paper Checklist",  
+- Keep the checklist subsection headings, questions/answers and guidelines below.  
+- Do not modify the questions and only use the provided macros for your answers.
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: The abstract and introduction clearly state our two key contributions: (1) a lightweight model with far fewer parameters than the state of the art interpretable GNN model, and (2) unified self-interpretable structure via graph-aware Random Fourier Features and learnable FIR filters. These two contributions are strongly supported by the emphasis on parameter-complexity analysis.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+# Answer: [Yes]
+
+Justification: The paper acknowledges that the proposed model obtains predictions by assuming feature-wise independence, thereby overlooking potential correlations among features. This design choice is intentional, as the primary objective is to develop a lightweight and interpretable model without compromising predictive performance. Extending the current framework to efficiently capture feature correlations while maintaining interpretability is identified as an important direction for future work.
+
+# Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+# Answer: [Yes]
+
+Justification: The outline of proofs and assumptions are discussed in the main paper whereas, complete proof for the Theorems are provided in the Appendix.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+# Answer: [Yes]
+
+Justification: The detailed hyperparameter analysis and step by step process followed to generate the results are detailed in the Appendix.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.
+
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+
+# 5. Open access to data and code
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+# 6. Experimental setting/details
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+
+# 7. Experiment statistical significance
+
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+
+# 8. Experiments compute resources
+
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+
+# 9. Code of ethics
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+
+# 10. Broader impacts
+
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+
+# 11. Safeguards
+
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+
+# 12. Licenses for existing assets
+
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+# 13. New assets
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+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
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+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
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+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
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+Justification: LLMs are only used for formatting the tables.
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+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# Appendix
+
+We organize the content in the Appendix as follows: In section A, we present proofs that supports permutation equivariance and robustness properties of G-NAMRFF. In section B, we include an additional discussion on interpretability with node and graph classification tasks. In section C, we present an ablation studies with respect to the hyperparameters.
+
+# A Proofs for the Theoretical Characterization of G-NAMRFF
+
+In this section, we present the proofs for the permutation equivariance and robustness properties of G-NAMRFF.
+
+Theorem. 4.1. (Permutation Equivalence) Let  $\mathcal{P} = \{\mathbf{P} \in \{0,1\}^{N \times N} : \mathbf{P}^{\top} \mathbf{P} = \mathbf{P} \mathbf{P}^{\top} = \mathbf{I}_N\}$  be the set of all  $N \times N$  permutation matrices. Then under the permutation of the graph Laplacian  $\mathbf{L}_{\mathcal{G}}$  and node-feature matrix  $\mathbf{X}$  by any  $\mathbf{P} \in \mathcal{P}$ , the output from  $G$ -NAMRFF also modifies as  $\mathbf{y}_{perm} = \mathbf{P}\mathbf{y}$ , where  $\mathbf{y} \in \mathbb{R}^N$  is the predictions across all the nodes.
+
+Proof. Let  $\mathbf{P} \in \mathcal{P}$  be the permutation matrix acting on the graph  $\mathcal{G}$ . Under this permutation, the graph Laplacian and feature matrix transforms as  $\tilde{\mathbf{L}}_{\mathcal{G},\mathrm{perm}} = \mathbf{P}\tilde{\mathbf{L}}_{\mathcal{G}}\mathbf{P}^T$  and  $\mathbf{X}_{\mathrm{perm}} = \mathbf{P}\mathbf{X}$ . Let us call the prediction of the node  $i$  under the permutation as  $y_{i,\mathrm{perm}}$ , whereas  $\mathbf{y}_{\mathrm{perm}}$  is obtained by stacking  $y_{i,\mathrm{perm}}$ .
+
+Under the permutations, filter output modifies as
+
+$$
+\begin{array}{l} \tilde {\mathbf {X}} _ {\text {p e r m}} = \sum_ {h = 0} ^ {R} \alpha_ {h} \tilde {\mathbf {L}} _ {\mathcal {G}, \text {p e r m}} ^ {h} \mathbf {X} _ {\text {p e r m}} = \sum_ {h = 0} ^ {R} \alpha_ {h} \mathbf {P} \tilde {\mathbf {L}} _ {\mathcal {G}} ^ {h} \mathbf {P} ^ {T} \mathbf {P} \mathbf {X} \\ \stackrel {(a)} {=} \sum_ {h = 0} ^ {R} \alpha_ {h} \mathbf {P} \tilde {\mathbf {L}} _ {\mathcal {G}} ^ {h} \mathbf {X}, \tag {9} \\ \stackrel {(b)} {=} \mathbf {P} \tilde {\mathbf {X}}, \\ \end{array}
+$$
+
+where  $(9)(a)$ , follows from the property of permutation matrix i.e.,  $\mathbf{P}^T\mathbf{P} = \mathbf{I}$ . Observe that from  $(9)(b)$  under the permutation, filtered outputs also gets permuted. Leveraging this, we now evaluate the prediction of G-NAMRFF under permutation as
+
+$$
+\begin{array}{l} \mathbf {y} _ {\text {p e r m}} = \left[ y _ {1, \text {p e r m}}; \dots ; y _ {N, \text {p e r m}} \right], \\ = \sum_ {k = 1} ^ {D} \underbrace {\left[ \Phi_ {\mathbf {a}} ^ {T} \left(\tilde {x} _ {1 , k , \text {p e r m}}\right) ; \dots ; \Phi_ {\mathbf {a}} ^ {T} \left(\tilde {x} _ {N , k , \text {p e r m}}\right) \right]} _ {\boldsymbol {\Psi} _ {k, \text {p e r m}}} \mathbf {w} _ {k}, \tag {10} \\ \stackrel {(a)} {=} \mathbf {P} \sum_ {k = 1} ^ {D} \left[ \boldsymbol {\Phi} _ {\mathbf {a}} ^ {T} (\tilde {x} _ {1, k}); \dots ; \boldsymbol {\Phi} _ {\mathbf {a}} ^ {T} (\tilde {x} _ {N, k}) \right] \mathbf {w} _ {k}, \\ = \mathrm {P y}, \\ \end{array}
+$$
+
+where (10)(a), follows from (9)(b) where it can be observed that for a fixed feature  $k$  i.e.,  $\tilde{\mathbf{x}}_{k,\mathrm{perm}}\in \mathbb{R}^{N\times 1} = \mathbf{P}\tilde{\mathbf{x}}_k$ , therefore mappings get reordered with  $\Psi_{k,\mathrm{perm}}\in \mathbb{R}^{N\times M} = \mathbf{P}\Psi_k$ . Also it can be observed that  $\mathbf{z}_{k,\mathrm{perm}} = [f_k(x_{1,k}),f_k(x_{2,k}),\dots ,f_k(x_{N,k})]\in \mathbb{R}^{N\times 1} = \mathbf{P}\mathbf{z}_k$  from the permutational equivariance of  $\Psi_{k}$ .
+
+Theorem. 4.2. (Robustness to perturbation of graph Laplacian) Let  $\widehat{\mathbf{L}}_{\mathcal{G}} = \mathbf{L}_{\mathcal{G}} + \Delta \mathbf{L}_{\mathcal{G}}$  be the Laplacian of the perturbed graph, with  $\| \Delta \mathbf{L}_{\mathcal{G}} \|_2 \leq \epsilon$ , and assume that the RFF map  $\Phi_a(.)$  is  $C_{\mathrm{RFF}}$ -Lipschitz continuous. Then each node prediction satisfies  $|\widehat{y}_i - y_i| \leq CK \in D\|\mathbf{X}\|_2$ , where  $C = C_{\mathrm{RFF}}\left(\max_k\|\mathbf{w}_k\|_2\right)$ ,  $K = \frac{1}{4}\|\pmb{\alpha}\|_1(R^2 - 1)\left(\frac{R - 1}{R + 1}\right)^R$  are constants.
+
+Proof. Before proceeding to the proof, we state the following Lemma that bounds the error between the FIR filters built with  $\mathbf{L}_{\mathcal{G}}$  and  $\hat{\mathbf{L}}_{\mathcal{G}}$ . With slight abuse of notation from here on we represent normalized Laplacian with  $\mathbf{L}_{\mathcal{G}}$  instead of  $\tilde{\mathbf{L}}_{\mathcal{G}}$ .
+
+Lemma A.1. [19] Consider a polynomial filter  $\mathbf{H}(\mathbf{L}_{\mathcal{G}}) = \sum_{h=0}^{R} \alpha_h \mathbf{L}_{\mathcal{G}}^h$ , and perturbed Laplacian as  $\hat{\mathbf{L}}_{\mathcal{G}} = \mathbf{L}_{\mathcal{G}} + \Delta \mathbf{L}_{\mathcal{G}}$  with  $\| \Delta \mathbf{L}_{\mathcal{G}} \|_2 \leq \epsilon$  then
+
+$$
+\left\| \mathbf {H} \left(\mathbf {L} _ {\mathcal {G}}\right) - \mathbf {H} \left(\hat {\mathbf {L}} _ {\mathcal {G}}\right) \right\| _ {2} \leq \frac {1}{4} \| \boldsymbol {\alpha} \| _ {1} \left(R ^ {2} - 1\right) \left(\frac {R - 1}{R + 1}\right) ^ {R} \epsilon , \tag {11}
+$$
+
+where  $\| \pmb {\alpha}\| _1 = [\alpha_0,\dots ,\alpha_R]\in \mathbb{R}^{R + 1}$
+
+Lemma A.1 shows that the error between the outputs of two filters fed with true and perturbed Laplacian is bounded and scales linearly with the error norm. We leverage the above Lemma to bound the difference between the prediction obtained from G-NAMRFF under perturbations.
+
+Assume that the prediction of node  $i$  under the graph perturbation as  $\hat{y}_i$ . To begin with we evaluate the difference between the filter outputs. Considering the filter outputs with and without perturbation of graph as  $\tilde{\mathbf{X}}$  and  $\tilde{\mathbf{X}}_{\mathrm{per}}$ . We have
+
+$$
+\bar {\mathbf {X}} = \mathbf {H} (\mathbf {L} _ {\mathcal {G}}) \mathbf {X},
+$$
+
+$$
+\tilde {\mathbf {X}} _ {\text {p e r}} = \mathbf {H} (\hat {\mathbf {L}} _ {\mathcal {G}}) \mathbf {X}. \tag {12}
+$$
+
+Then the difference between the filtered outputs are bounded as
+
+$$
+\| \tilde {\mathbf {X}} _ {\text {p e r}} - \tilde {\mathbf {X}} \| _ {2} = \| \mathbf {H} (\hat {\mathbf {L}} _ {\mathcal {G}}) \mathbf {X} - \mathbf {H} (\mathbf {L} _ {\mathcal {G}}) \mathbf {X} \| _ {2}
+$$
+
+$$
+\stackrel {(a)} {\leq} \left\| \mathbf {H} \left(\hat {\mathbf {L}} _ {\mathcal {G}}\right) - \mathbf {H} \left(\mathbf {L} _ {\mathcal {G}}\right) \right\| _ {2} \| \mathbf {X} \| _ {2} \tag {13}
+$$
+
+$$
+\stackrel {(b)} {\leq} \frac {1}{4} \| \boldsymbol {\alpha} \| _ {1} (R ^ {2} - 1) \left(\frac {R - 1}{R + 1}\right) ^ {R} \epsilon \| \mathbf {X} \| _ {2},
+$$
+
+where (13)(a) follows from a norm inequality and (13)(b) follows from (11). From (13)(b) it is clear that the difference of filter outputs is bounded linearly with the energy in the error term. Leveraging this we now evaluate the difference in the prediction output. We index the  $i, k$  element of  $\tilde{\mathbf{X}}_{\mathrm{per}}$  with  $\tilde{x}_{i,k,\mathrm{per}}$ . Then,
+
+$$
+\begin{array}{l} | \hat {y} _ {i} - y _ {i} | = | \sum_ {k = 1} ^ {D} \boldsymbol {\Phi} _ {a} ^ {T} (\tilde {x} _ {i, k, \text {p e r}}) \mathbf {w} _ {k} - \sum_ {k = 1} ^ {D} \boldsymbol {\Phi} _ {a} ^ {T} (\tilde {x} _ {i, k}) \mathbf {w} _ {k} | \\ \leq \sum_ {k = 1} ^ {D} \| \boldsymbol {\Phi} _ {a} ^ {T} (\tilde {x} _ {i, k, \text {p e r}}) - \boldsymbol {\Phi} _ {a} ^ {T} (\tilde {x} _ {i, k}) \| _ {2} \| \mathbf {w} _ {k} \| _ {2} \\ \stackrel {(a)} {\leq} C _ {\mathrm {R F F}} \sum_ {k = 1} ^ {D} | \tilde {x} _ {i, k, \text {p e r}} - \tilde {x} _ {i, k} | \| \mathbf {w} _ {k} \| _ {2} \tag {14} \\ \stackrel {(b)} {\leq} \sum_ {k = 1} ^ {D} C _ {\mathrm {R F F}} \max  _ {k} \left\{\| \mathbf {w} _ {k} \| _ {2} \right\} \underbrace {\frac {1}{4} \| \boldsymbol {\alpha} \| _ {1} \left(R ^ {2} - 1\right) \left(\frac {R - 1}{R + 1}\right) ^ {R}} _ {K} \epsilon \| \mathbf {X} \| _ {2} \\ \stackrel {(c)} {\leq} C K \epsilon D \| \mathbf {X} \| _ {2}, \\ \end{array}
+$$
+
+where (14)(a) follows from Lipschitz continuity of  $\Phi_{\mathbf{a}}$ , (14)(b), follows from (13)(b). Whereas, (14)(c) follows by considering the  $C = C_{\mathrm{RFF}}\max_k\| \mathbf{w}_k\| _2$ . From (14)(c), it is clear that difference in the node predictions scales linearly with the norm of the error term  $\Delta \mathbf{L}_{\mathcal{G}}$ .
+
+# B Additional Discussion on Interpretability
+
+# B.1 Interpretability on Node Classification Task
+
+In this section, we present additional examples of the learned univariate functions on the PubMed dataset for the node classification task. Specifically, in Fig. 5, we illustrate the univariate function
+
+outputs for the features i.e., keywords as insulin, fat, liver, and diet. Recall that at any given feature value, the curve with the highest output indicates the class that the feature most strongly supports.
+
+In Fig. 5a, we examine the influence of the presence of diet keyword in the document on the model's predictions. It can be observed that the model associates higher output with type 2 diabetes, suggesting that diet is a strong predictor for this class. This observation aligns well with existing medical literature, which indicates that type 2 diabetes is heavily influenced by dietary factors, whereas type 1 diabetes, being an autoimmune condition is less affected by diet [21]. Consistent with this, the model assigns less importance to type 1 diabetes for this feature. In Fig. 5b, we plot the influence of fat keyword in the document where it is clear that it contributes less significantly to the prediction of type 1 diabetes, further supporting the model's ability to capture medically relevant patterns. Similarly, in Fig. 5c and Fig. 5d we plot the impact of these features on model prediction where it is clear that presence of this word aligns prediction towards type 2 and gestational diabetes. Although for illustration purposes we have presented few examples one can follow a similar procedure to obtain the contribution of different features on different datasets.
+
+![](images/f6036a8f29dea4d4fdb28dafb9f5ee1c6ef844e71afe09450bbd7723c72e6ebf.jpg)  
+(a)
+
+![](images/96b16a16e829d85b79038cbb7fbfe62224e0a0051fa0e6932d2692671d798433.jpg)  
+(b)
+
+![](images/59a4de10250d5c19ddfd56f950ec1e351976464960d9337fd20dda3f288078b6.jpg)  
+(c)  
+Figure 5: PubMed dataset: Univariate function outputs on features.
+
+![](images/85b09cfcd6c63bb49070190c9fa190b63ca03d78a935608afb0305491ff31bbe.jpg)  
+(d)
+
+# B.2 Interpretability on Graph Classification Task
+
+In this section, we include more discussion on interpretability on the mutagenicity dataset. Here the main target is to give local level (structural) explanations and also compare them with the GNN-Explainer [37] which is a post-hoc explainer.
+
+With the focus on mutagenic class, we give structure level explanations for the model prediction. In Figs. 6a, 6c and 6e we present three example molecules that are predicted as mutagenic and highlight predicted substructures responsible for the mutagenic prediction. It is clear that the  $\mathrm{NH}_2$ ,  $\mathrm{NO}_2$  and aromatic rings act as key contributors to mutagenic class. These explanations align with the existing studies which advocate the presence of these substructures highly influences the mutagenicity [9]. For a fair comparison we present the substructures revealed from GNN-Explainer in Fig. 6b, 6d and 6f. It can be observed that compared to GNN-Explainer, our proposed model identifies  $\mathrm{NH}_2$ ,  $\mathrm{NO}_2$  and aromatic rings more precisely.
+
+![](images/a71af2ef17fd937d8c952f4f5aea8c21b0a14afb459db257944a4da3a4d648e5.jpg)
+
+![](images/6ad8eb2f23415a2f15f693f8f106fb6f9c3480fe1e8d57f306dfd28cb90db348.jpg)  
+(b)
+
+![](images/7387df8113dbbf5a1eba63ee0889740edfb6379fd0cf55f8ebc52067b47c6ad0.jpg)  
+(a)
+
+![](images/fc7a91cd91dc92403eed0aad0778c40a5228095e73894618ed9cec2b1e126919.jpg)  
+(d)
+
+![](images/1c19ef6b2e6fd5df8a8165c06d4e5e0903d02e8be737f851728804dbd7d5a55a.jpg)  
+(c)  
+(e)  
+Figure 6: Mutagenicity dataset: Local level explanations.
+
+![](images/55abe18cc8dfa9fd4c28d836a08af6b580ca3d2db7a798bde8d5e41c6f0cc43e.jpg)  
+(f)
+
+# C Hyperparameter Details and Ablation Studies
+
+In this section, we first present the hyperparameter details used in our node and graph classification tasks. Then we analyze how each of these hyperparameters affects G-NAMRFF performance across datasets.
+
+# C.1 Hyperparameter Details
+
+We have conducted all the experiments using an NVIDIA A30 GPU. In particular, we set the tuning range as follows: Number of RFFs  $(M)$  -  $\{20, 40, 50, 100, 200\}$ , filter order  $(R)$  - [1, 7], kernel width  $(\Theta)$  - [1.0, 4.0], learning rate (Lr) -  $[1e - 4, 2e - 1]$  and weight decay (Wd) -  $[1e - 5, 1e - 2]$ . The hyperparameter configurations that yielded the best validation performance are detailed in Table 3. For node classification tasks, we follow the standard dataset splits as specified in [16, 20, 23]. In the case of graph classification tasks, where no standard splits are available, we employ 10-fold cross-validation. All models are trained for 1000 epochs using the AdamW optimizer with randomly initialized parameters. The specific learning rates and weight decay values used across different experiments are also reported in Table 3. For node classification, we report the mean classification accuracy computed over five independent random seeds. For graph classification, we report the mean classification accuracy across 10 folds, with each fold averaged over three random seeds.
+
+![](images/82bc877ed4342bc278b15144d1a0eb5b5b264a27967c51911d015bc46cf631f7.jpg)  
+Figure 7: Run time comparison.
+
+# C.2 Run time analysis
+
+In this section, we compare the runtime performance of G-NAMRFF, against GNAN, which is the only existing glass-box GNN architecture. In Fig. 7, we present the per-epoch runtime comparison on both node and graph classification tasks. Notably, it can be observed that even when GNAN is configured
+
+<table><tr><td>Dataset</td><td>Number of RFFs (M)</td><td>Filter Order (R)</td><td>Kernel width (Θ)</td><td>Learning rate (Lr)</td><td>Weight Decay (Wd)</td></tr><tr><td>Cora</td><td>100</td><td>7</td><td>2.2</td><td>1.8e-2</td><td>2.4e-4</td></tr><tr><td>Citeseer</td><td>100</td><td>5</td><td>2.5</td><td>5.1e-2</td><td>3.1e-4</td></tr><tr><td>Pubmed</td><td>100</td><td>5</td><td>3.8</td><td>2.7e-3</td><td>4e-4</td></tr><tr><td>Cornell</td><td>100</td><td>6</td><td>2.9</td><td>7.3e-2</td><td>3.7e-4</td></tr><tr><td>ogbn-arxiv</td><td>100</td><td>4</td><td>3.4</td><td>5.2e-2</td><td>1.3e-4</td></tr><tr><td>ogbn-products</td><td>40</td><td>5</td><td>2.3</td><td>2.9e-2</td><td>5e-4</td></tr><tr><td>Proteins</td><td>50</td><td>5</td><td>1.9</td><td>3.2e-3</td><td>5e-3</td></tr><tr><td>Mutag</td><td>200</td><td>7</td><td>1.2</td><td>3.7e-2</td><td>5e-3</td></tr><tr><td>Mutagenicity</td><td>200</td><td>6</td><td>1.4</td><td>1.1e-2</td><td>5e-4</td></tr><tr><td>NCI1</td><td>50</td><td>7</td><td>3.7</td><td>1.6e-2</td><td>5e-4</td></tr><tr><td>PTC</td><td>50</td><td>5</td><td>1.7</td><td>2.9e-3</td><td>2.5e-5</td></tr></table>
+
+Table 3: Hyperparameter details.
+
+![](images/fdcc62a8b95f4e58d8b0552e588961a7d9c84bbd1a0ba7fbe6ad550ce7f313ca.jpg)  
+(a) Mutagenicity  
+Figure 8: Classification accuracy against filter order.
+
+![](images/443c19d7ceddb2cff5691cf723d5940780db0968a477dc56125a7f5e14a60b13.jpg)  
+(b) Pubmed
+
+with a relatively small hidden dimension of 32, it still requires 10 to 100 times more runtime per epoch compared to G-NAMRFF.
+
+Furthermore, it is worth emphasizing that on small-scale datasets such as Citeseer and Pubmed, increasing the hidden dimension to 64 can already result in out-of-memory (OOM) errors when training GNAN. On large-scale datasets like ogbn-products, training GNAN becomes entirely infeasible due to excessive memory requirements.
+
+# C.3 Ablation studies
+
+In this section, we analyze the impact of hyperparameters on the model performance. Recall the key hyperparameters in G-NAMRFF includes the filter order  $(R)$  and number of RFF  $(M)$ .
+
+Filter Order: In Fig. 8, we illustrate the effect of the filter order on classification accuracy for two tasks: node classification on the Pubmed dataset and graph classification on the Mutagenicity dataset. Recall that a filter of order  $R$  aggregates information from  $R$ -hop neighborhoods in the graph. It can be observed that as the filter order increases, the model is able to incorporate information from larger receptive fields, leading to an initial improvement in classification accuracy. However, beyond a certain point, further increasing the filter order causes the accuracy to saturate or decline, likely due to oversmoothing. Notably, we observe that using higher-order filters ( $R > 1$ ) significantly improves performance compared to  $R = 1$ , highlighting the benefit of aggregating information from larger graph neighborhoods.
+
+To further highlight the relative importance of different node neighborhoods, we present the magnitudes of the learned filter coefficients for a fixed filter order in Fig. 9, using the Mutagenicity and Cornell datasets. It can be observed that the model assigns higher weights to closer neighborhoods, emphasizing the greater importance of immediate node interactions in the classification task.
+
+![](images/a86adc7718a3902cee82129cad36cfa2d768f9a53bdbfac012d18c0a5289fac5.jpg)  
+(a) Mutag
+
+![](images/6f317c11ed875be7d17615b4c9891165691eae3c20743155399536fdc8e4dad1.jpg)  
+(b) Cornell  
+Figure 9: Filter coefficients magnitude against filter order.
+
+# Random Fourier Features:
+
+We analyze the effect of the number of RFFs on classification accuracy. Figures. 10a, 10b, and 10c show the performance variation on the Cornell, PubMed, and NCI1 datasets, respectively. As the number of random Fourier features  $(M)$  increases, the classification accuracy initially improves due to better kernel approximation. However, beyond a certain point, performance drops as the model may begin to overfit with larger  $M$ . Across all datasets, we observe that good performance can be achieved with relatively small values of  $M$ , which translates to a lower number of learnable parameters. Recall that the total number of parameters in G-NAMRFF is  $D \times M + R + 1$ . As observed  $M$  being small the model remains lightweight while being effective.
+
+![](images/734dac63aabff4c9d2622b05d17271f503d678cc29bc296b89a6006564f566eb.jpg)  
+(a) Cornell
+
+![](images/02ee6c992b28544c6e5a0f6901ec61c5d7ab579d5779578f51490e161f784a65.jpg)  
+(b) Pubmed
+
+![](images/9a9f2743d0e99cd4fb3610bfd83ab92e433a663870f7b65457892056591c69bf.jpg)  
+(c) NCI1  
+Figure 10: Classification accuracy vs number of RFFs.

2025/Interpretable and Parameter Efficient Graph Neural Additive Models with Random Fourier Features/images.zip

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2025/Interpreting Arithmetic Reasoning in Large Language Models using Game-Theoretic Interactions/1ac993ea-9d1c-4e92-9f4d-2a3d48247f19_origin.pdf

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2025/Interpreting Arithmetic Reasoning in Large Language Models using Game-Theoretic Interactions/full.md

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+# Interpreting Arithmetic Reasoning in Large Language Models using Game-Theoretic Interactions
+
+Leilei Wen $^{1}$  Liwei Zheng $^{1}$  Hongda Li $^{1}$  Lijun Sun $^{1}$  Zhihua Wei $^{1}$  Wen Shen $^{1*}$
+
+Tongji University, Shanghai, China
+
+{2331934, 2432010, 2410907, sunlijun, zhihua_wei, wenshen}@tongji.edu.cn
+
+# Abstract
+
+In recent years, large language models (LLMs) have made significant advancements in arithmetic reasoning. However, the internal mechanism of how LLMs solve arithmetic problems remains unclear. In this paper, we propose explaining arithmetic reasoning in LLMs using game-theoretic interactions. Specifically, we disentangle the output score of the LLM into numerous interactions between the input words. We quantify different types of interactions encoded by LLMs during forward propagation to explore the internal mechanism of LLMs for solving arithmetic problems. We find that (1) the internal mechanism of LLMs for solving simple one-operator arithmetic problems is their capability to encode operand-Operator interactions and high-order interactions from input samples. Additionally, we find that LLMs with weak one-operator arithmetic capabilities focus more on background interactions. (2) The internal mechanism of LLMs for solving relatively complex two-operator arithmetic problems is their capability to encode operator interactions and operand interactions from input samples. (3) We explain the task-specific nature of the LoRA method from the perspective of interactions.
+
+# 1 Introduction
+
+In recent years, the arithmetic reasoning capabilities of large language models (LLMs) have improved significantly, but the internal mechanism is still unclear. Some studies identified neurons that have great effects on arithmetic reasoning [Yu and Ananiadou, 2024, Rai and Yao, 2024]. Other studies evaluated the impact of modifying words in input arithmetic questions on neuron activations and network outputs [Stolfo et al., 2023, Zhang et al., 2024]. However, previous studies have not mathematically guaranteed that the explanations faithfully reflect the arithmetic reasoning logic of LLMs. Recently, a series of studies have used game-theoretic interactions between input variables to explain the representation power of traditional DNNs [Ren et al., 2024, Deng et al., 2024], e.g., the interaction between "green" and "hand" forms the concept of "beginner." The interaction has been proven to be faithful explanations by a series of theoretical guarantees [Ren et al., 2023a].
+
+Inspired by these studies, we use interactions to explain the arithmetic reasoning capability of LLMs. As Figure 1 shows, given an input arithmetic question  $x$ , e.g., "What is 2 plus 7? Answer is," the interaction  $S$  (e.g.,  $S = \{2, \text{plus}, 7\}$ ) represents an AND relationship between the words in  $S$ , which is equivalently encoded by the LLM. Each interaction  $S$  contributes a numerical effect  $I_S$  to push the output score towards generating the answer "9." Thus, we can construct a logic model  $\phi(x)$  based on interactions. The faithfulness of using the logical model to explain an LLM is reflect as follows.
+
+- As Figure 1(a) shows, given the same input sentence  $\pmb{x}$ , the output score of the LLM is equal to the output of the logical model, that is,  $v(\pmb{x}) = \phi(\pmb{x}) = \sum_{S \subseteq N} I_S$ , where  $N$  is the set containing all the
+
+![](images/cb07ce66f3201dd991bd1be8c166970b6832339bde1911ede6b2a6efdcad159e.jpg)  
+Figure 1: Illustration of using interactions to explain the internal mechanism of LLMs for arithmetic reasoning. (a) Given an arithmetic question  $\pmb{x}$  consisting of  $n$  words, let  $N = \{x_1, x_2, \dots, x_n\}$ , where each  $x_i$  represents a single word in the input question. We use a logic model based on interactions, i.e.,  $\phi(\pmb{x}) = \sum_{S \subseteq N} I_S$  to explain the detailed inference patterns encoded by LLMs. Note that superscripts a and b are not input to the LLM, but are used to distinguish two words of "is" in two positions. (b) Based on interactions, we investigate how an LLM encodes different types of interactions during forward propagation when solving arithmetic problems.
+
+words in  $\pmb{x}$ , i.e.,  $N = \{x_{1}, x_{2}, \ldots, x_{n}\}$ . This means that the output score of the LLM on any input sentence can be represented as the sum of the effects of all interactions.
+
+- More crucially, no matter how we  $\mathbf{mask}^3$  input words in  $\pmb{x}$ , the output score of an LLM on the masked sentence  $\pmb{x}^{\mathrm{masked}}$  is equal to the output of the logical model, that is,  $v(\pmb{x}^{\mathrm{masked}}) = \phi(\pmb{x}^{\mathrm{masked}}) = \sum_{S \subseteq N} \mathbb{1}(\pmb{x}^{\mathrm{masked}} \text{ triggers } S) \cdot I_S$ . This means that the output score of an LLM on any masked sentence can be represented as the sum of the effects of all triggered interactions. For example, if we mask the input word "7" in Figure 1(a) and obtain the masked sentence "What is 2 plus [pad_token]? Answer is," then all interactions that contain "7" (e.g.,  $S = \{2, \text{plus}, 7\}$ ) will not be triggered, meaning that  $\mathbb{1}(\pmb{x}^{\mathrm{masked}} \text{ triggers } S) \cdot I_S = 0$ . As a result, these non-triggered interactions have no effects on the network output. Please see Theorem 1 for details.
+
+In this way, interactions can be considered as inference patterns encoded by an LLM. Therefore, we quantify how LLMs encode different interactions during forward propagation. We find that different LLMs have preferences for encoding certain interactions. For example, as Figure 2 shows, when computing features closer to the output layer, the Llemma-7B [Azerbayev et al., 2023] model tends to encode interactions that contain both operands and operators, while the OPT-1.3B [Zhang et al., 2022b] model tends to encode interactions that only contain background words.
+
+Inspired by the above observations, we further define four types of interactions encoded by the LLM for arithmetic reasoning, including (1) operand interactions, (2) operator interactions, (3) operand-operator interactions, and (4) background interactions, as shown in Figure 1. Furthermore, we propose a new metric to quantify the focus of LLMs on different types of interactions and discover the following insights.
+
+- Insight 1: The internal mechanism of LLMs for solving simple one-operator arithmetic problems is their capability to encode $^2$  operand-Operator interactions and high-order interactions from input samples. For simple one-operator arithmetic problems, we observe that LLMs with strong performance tend to increase their focus on operand-Operator interactions and high-order interactions, while reducing their focus on operator interactions and background interactions. A high-order interaction contains a large number of input words, while a low-order interaction contains a small number of input words. In contrast, LLMs with weak performance tend to focus on background interactions and extremely low-order interactions.  
+- Insight 2: The internal mechanism of LLMs for solving relatively complex two-operator arithmetic problems is their capability to encode $^{2}$  operator interactions and operand interactions from input samples. As an LLM learns to solve two-operator arithmetic problems, we observe that it consistently focuses more on operator interactions and gradually increases its focus on operand interactions during the final stages of training.
+
+- Insight 3: We explain the task-specific nature of the LoRA method from the perspective of interactions. As an LLM with strong one-operator arithmetic capabilities is fine-tuned using LoRA to solve relatively complex two-operator arithmetic problems, its capability to solve simpler one-operator problems declines. This phenomena is consistent with the task-specific nature of LoRA [Hu et al., 2022]. From the perspective of interactions, we observe that the LLM reduces its focus on operand-operator interactions and high-order interactions, which helps explain the underlying mechanism behind the task-specific nature of LoRA in arithmetic reasoning tasks.
+
+# 2 Related Work
+
+Explaining arithmetic reasoning in LLMs. Some studies identified and analyzed key neurons to explain arithmetic reasoning in LLMs. Yu and Ananiadou [2024] identified an internal logic chain. Rai and Yao [2024] investigated neuron activations and identified reasoning-related neurons. Some studies identified key components by changing the inputs of the LLM to perturb activations and then measuring the changes in the output logits. Hanna et al. [2023] explained the capability of the GPT2-small model on the "greater than" task by identifying a specific circuit. Zhang et al. [2024] further selectively fine-tuned the key components to boost the LLM's arithmetic performance. Stolfo et al. [2023] used a causal mediation analysis framework to provide a mechanistic explanation of how LLMs solve arithmetic problems. Wu et al. [2023] proposed a method to scale causal analysis of language models to billions of parameters. However, previous studies failed to provide strong mathematical support for their explanations. In contrast, in this paper, we use interactions to explain how LLMs solve arithmetic tasks, which has been proven to be a faithful explanation.
+
+Using game-theoretic interactions to explain DNNs. Ren et al. [2023a] proposed using interactions to explain DNNs and provided a series of theoretical guarantees to ensure the faithfulness of this explanation. A series of studies further explored using interactions to explain the representational power of DNNs, including adversarial robustness [Wang et al., 2021], adversarial transferability [Wang et al., 2020], and the overfitting problem [Ren et al., 2023b, Zhou et al., 2023]. Additionally, Deng et al. [2021] demonstrated that DNNs struggle to encode mid-order interactions. Ren et al. [2024] discovered that DNNs encode interactions of different orders in two distinct phases. Some other studies used interactions to analyze common mechanisms across various deep-learning methods. Deng et al. [2024] showed that the core mechanism of fourteen attribution methods can be explained as a redistribution of interactions. Zhang et al. [2022a] found that the twelve previous methods for improving transferability all reduce interactions between local adversarial perturbations. However, due to terabytes of data and billions of parameters in LLMs, as well as the inherent complexity of arithmetic reasoning tasks, whether interactions can be used to explain the arithmetic reasoning of LLMs while ensuring the faithfulness of the explanation remains to be verified.
+
+# 3 Using interactions to explain inference patterns encoded by LLMs
+
+# 3.1 Preliminaries: disentangling the network output using interactions
+
+Given a DNN and an input sample  $\pmb{x}$  with  $n$  variables indexed by  $N = \{\pmb{x}_1, \pmb{x}_2, \dots, \pmb{x}_n\}$ , we can obtain  $2^n$  masked<sup>3</sup> sentences  $\{\pmb{x}_T \mid T \subseteq N\}$  by masking<sup>3</sup> each of  $n$  variables. Specifically,  $\pmb{x}_T$  denotes the input sentence when we keep variables in  $T$  unchanged and mask<sup>3</sup> variables in  $N \setminus T$ .  $\pmb{x}_0$  denotes the input sentence when all variables in  $\pmb{x}$  are masked<sup>3</sup>. Ren et al. [2021b] have proven that there exists a surrogate logical model  $\phi(\cdot)$  that can well predict/fit the scalar output of the DNN for all  $2^n$  different masked sentences, as Theorem 1 shows.
+
+$$
+\forall T \subseteq N, \phi (\boldsymbol {x} _ {T}) = \phi (\boldsymbol {x} _ {\emptyset}) + \sum_ {S \subseteq N, S \neq \emptyset} \mathbb {1} (S | \boldsymbol {x} _ {T}) \cdot I _ {S}, \tag {1}
+$$
+
+where the trigger function  $\mathbb{1}(S|\boldsymbol{x}_T)$  represents an AND relationship among the input variables in a set  $S \subseteq N$ . That is, if all variables in  $S$  are present in  $T$ , then  $\mathbb{1}(S|\boldsymbol{x}_T) = 1$ . Otherwise, if any variable in  $S$  is masked<sup>3</sup>, then  $\mathbb{1}(S|\boldsymbol{x}_T) = 0$ . Each interaction  $S$  has a scalar weight  $I_S$ .
+
+Theorem 1. (Universal matching property, as proven by Ren et al. [2023a]). When the scalar weight  $I_S$  in the logical model  $\phi(\cdot)$  is defined as follows,
+
+$$
+\forall S \subseteq N, S \neq \emptyset , I _ {S} = \sum_ {S ^ {\prime} \subseteq S} (- 1) ^ {| S | - | S ^ {\prime} |} \cdot v \left(\boldsymbol {x} _ {S ^ {\prime}}\right), \tag {2}
+$$
+
+then we have  $\forall T\subseteq N$  , the output score of the DNN  $v\left(\pmb {x}_T\right) = \phi \left(\pmb {x}_T\right)$
+
+The universal matching property guarantees the theoretical faithfulness of using interactions to explain the DNN. That is, the interaction truly reflects the inference logic of the DNN. Besides, the interaction has been proven to serve as the basis for many game-theoretic metrics, further ensuring its theoretical faithfulness. Please see Appendix C for more details about game-theoretic metrics.
+
+# 3.2 Explaining the LLM using interactions
+
+Although interactions have been widely used to explain traditional DNNs, the following two challenges remain in using interactions to explain the forward propagation process of the LLM.
+
+(1) How to quantify interactions encoded by an LLM in intermediate layers. Given an LLM with  $L$  layers $^4$ , we set the output score  $v^{(l)}(\boldsymbol{x}_T)$  at layer  $l$  as follows.
+
+$$
+\forall T \subseteq N, v ^ {(l)} \left(\boldsymbol {x} _ {T}\right) = \cos \left(f ^ {(l)} \left(\boldsymbol {x} _ {T}\right), f ^ {(l)} \left(\boldsymbol {x} _ {N}\right)\right) = \frac {\left(f ^ {(l)} \left(\boldsymbol {x} _ {T}\right)\right) ^ {\top} f ^ {(l)} \left(\boldsymbol {x} _ {N}\right)}{\| f ^ {(l)} \left(\boldsymbol {x} _ {T}\right) \| _ {2} \| f ^ {(l)} \left(\boldsymbol {x} _ {N}\right) \| _ {2}}, \tag {3}
+$$
+
+where  $f^{(l)} \in \mathbb{R}^d$  denotes the embedding of the last input token. The score  $v^{(l)}(\pmb{x}_T)$  measures the cosine similarity between  $f^{(l)}(\pmb{x}_T)$  and  $f^{(l)}(\pmb{x}_N)$ . If the value of  $v^{(l)}(\pmb{x}_T)$  is large, then it indicates that the feature  $f^{(l)}(\pmb{x}_T)$  of the masked sentence is similar with the feature  $f^{(l)}(\pmb{x}_N)$  of the original sentence. This means that input variables in  $T$  have great impact on the feature of the  $l$ -th layer. Thus, in Equation (2), the interaction computed based on the score can be considered as an inference pattern encoded by the features at the  $l$ -th layer. Note that we use  $f^{(l)}(\pmb{x}_T) - f^{(l)}(\pmb{x}_{\emptyset})$  to replace  $f^{(l)}(\pmb{x}_T)$ , where  $f^{(l)}(\pmb{x}_{\emptyset})$  denotes the embedding when we mask all input variables in  $\pmb{x}$ . This shifting operation is used to ensure that  $f^{(l)}(\pmb{x}_{\emptyset}) = \mathbf{0}$ .
+
+(2) Why we use words instead of tokens as input variables. It is because the tokenizers of different LLMs adopt different strategies for encoding numbers. For example, the Llama-2-7B [Touvron et al., 2023] model divides the word "15" into two tokens "1" and "5," while the OPT-1.3B [Zhang et al., 2022b] model obtains a single token "15." To ensure a fair comparison of interactions encoded by different LLMs, we treat all tokens within a word as a single input variable. When generating the masked<sup>3</sup> sentence  $\boldsymbol{x}_T$ , we use a specific padding token (see Appendix B for details) to mask all tokens that belong to the words in  $N \setminus T$ .
+
+Based on the above two settings, we quantify interactions in LLMs when solving arithmetic problems. Figure 2 shows the top 10 interactions (i.e., those interactions with largest absolute values) encoded $^2$  by the Llemma-7B model [Azerbayev et al., 2023] and the OPT-1.3B model [Zhang et al., 2022b] when solving a one-operator problem "How much is 4 times 2? Answer is." The Llemma-7B model predicts the correct answer "8," while the OPT-1.3B model predicts the incorrect answer "4." We observe that the Llemma-7B model mainly focuses on interactions that contain only background words in an early (5th) layer $^4$ , and mainly focuses on interactions containing both operands and operators in a late (30th) layer. In comparison, the OPT-1.3B model mainly focuses on interactions that contain only background words in a late (24th) layer, which might be the reason why the OPT-1.3B model answers incorrectly.
+
+To better observe the changing trends of different interactions encoded by an LLM during forward propagation, Figure 2 reports the curves of interactions encoded by LLMs when computing features. To ensure a fair comparison of interactions at different layers, we compute the normalized strength of interactions at different layers, that is,  $\frac{|I_S^{(l)}|}{Z^{(l)}}$ , where  $Z^{(l)} = \mathbb{E}_{S\subseteq N}|I_S^{(l)}|$ . As Figure 2 shows, the LLemma-7B model enhances its focus on interactions containing both operands and operators (e.g.,  $S = \{is^{\mathrm{a}},4,times,2\}$ ) during forward propagation. In comparison, the OPT-1.3B model does not show a preference for any interactions in the middle layers and shifts its focus to interactions that contain only background words (e.g.,  $S = \{is^{\mathrm{b}}\}$ ) in the very few layers close to the output layer. Therefore, interactions provide us with a new perspective to explain the internal mechanism of LLMs for arithmetic reasoning.
+
+Given a simple one-operator arithmetic problem as input: "How much is  $a$  4 times 2? Answer is"
+
+# The Llemma-7B model predicts: 8 (Correct)
+
+![](images/945647fec59f8d2554a59f31d810fd4c266a4bfb67002f6480acc1b7e121a7b9.jpg)  
+(a) Interactions encoded by the Llemma-7B model  
+Figure 2: Visualization of top 10 interactions encoded by LLMs when computing features in different layers. Results show that the Llemma-7B model mainly focuses on interactions containing both operands (in blue) and operators (in orange) in a late layer (i.e., the 30th layer). In comparison, the OPT-1.3B model mainly focuses on interactions that contain only background words in a late layer (i.e., the 24th layer), which may be the reason why the OPT-1.3B model answers incorrectly. Note that superscripts a and b are not input to the LLM, but are used to distinguish two words of "is."
+
+# The OPT-1.3B model predicts: 4 (Incorrect)
+
+![](images/1457da3511e56203a0ea38c02892c17f4327f950166753195e0029ae3e9331c3.jpg)  
+(b) Interactions encoded by the OPT-1.3B model
+
+# 3.3 Defining and quantifying different types of interactions
+
+Inspired by the above observations obtained in Section 3.2, we classify the variables in an input arithmetic query  $x$  into the following three categories: operand variables  $x^{\mathrm{opd}}$ , operator variables  $x^{\mathrm{opr}}$  and background variables  $x^{\mathrm{bg}}$ . Thus, all  $2^{n}$  interactions can be classified as the following four types to describe the inference patterns encoded by LLMs for arithmetic reasoning.
+
+Operand interactions. If an interaction  $S$  contains at least one operand variable but no operator variables, we regard  $S$  as an operand interaction. Let  $\Omega^{\mathrm{opd}}$  denote the set of all operand interactions.
+
+$$
+\Omega^ {\mathrm {o p d}} = \{S \mid \exists \boldsymbol {x} ^ {\mathrm {o p d}} \in S \wedge \notin \boldsymbol {x} ^ {\mathrm {o p r}} \in S \}. \tag {4}
+$$
+
+Operator interactions. If an interaction  $S$  contains at least one operator variable but no operand variables, we regard  $S$  as an operator interaction. Let  $\Omega^{\mathrm{opr}}$  denote the set of all operator interactions.
+
+$$
+\Omega^ {\mathrm {o p r}} = \{S \mid \exists \boldsymbol {x} ^ {\mathrm {o p r}} \in S \wedge \not \exists \boldsymbol {x} ^ {\mathrm {o p d}} \in S \}. \tag {5}
+$$
+
+Operand-operator interactions. If an interaction  $S$  contains at least one operand variable and at least one operator variable, we regard  $S$  as an operand-operator interaction. Let  $\Omega^{\mathrm{opd -opr}}$  denote the set of all operand-operator interactions.
+
+$$
+\Omega^ {\mathrm {o p d - o p r}} = \{S \mid \exists \boldsymbol {x} ^ {\mathrm {o p d}} \in S \wedge \exists \boldsymbol {x} ^ {\mathrm {o p r}} \in S \}. \tag {6}
+$$
+
+Background interactions. If an interaction  $S$  contains neither operand nor operator variables, we regard  $S$  as a background interaction. Let  $\Omega^{\mathrm{bg}}$  denote the set of all background interactions.
+
+$$
+\Omega^ {\mathrm {b g}} = \{S \mid \notin \boldsymbol {x} ^ {\mathrm {o p d}} \in S \wedge \notin \boldsymbol {x} ^ {\mathrm {o p r}} \in S \}. \tag {7}
+$$
+
+Note that the union of the four sets above is the complete set of  $2^n$  interactions. According to Theorem 1, the sum of numerical effects of these four types of interactions can accurately fit the output scores of the LLM, thereby ensuring the faithfulness of the analysis. We design the following metric to quantify how an LLM focuses on a specific type of interaction  $\Omega^{type} \in \{\Omega^{\mathrm{opd}}, \Omega^{\mathrm{opr}}, \Omega^{\mathrm{opd-opr}}, \Omega^{\mathrm{bg}}\}$ .
+
+Definition 1. (Focality on a specific type of interaction). Let  $R^{(l)}(\Omega^{type})$  denote the focality on a specific type of interaction  $\Omega^{type}$  at layer  $l$ . It is computed as follows,
+
+$$
+R ^ {(l)} \left(\Omega^ {\text {v t y p e}}\right) = \frac {\mathbb {E} _ {S \in \Omega^ {\text {v t y p e}}} \left| I _ {S} ^ {(l)} \right|}{Z ^ {(l)}}, \tag {8}
+$$
+
+where  $Z^{(l)} = \mathbb{E}_{S\subseteq N}|I_S^{(l)}|$  is a normalization term used to ensure a fair comparison of the interaction effects across different layers.
+
+In Equation 8, a higher  $R^{(l)}(\Omega^{type})$  value suggests that the LLM focuses more on this type of interaction  $\Omega^{type}$  when computing features at layer  $l$ . If  $R^{(l)}(\Omega^{type}) = 1$ , that means the strength of this type of interaction  $\Omega^{type}$  encoded by the LLM is equal to the average strength of all interactions encoded by the LLM, which reveals that the LLM does not show a preference for this type of interaction. If  $R^{(l)}(\Omega^{type}) > 1$ , that means the strength of this type of interaction  $\Omega^{type}$  exceeds the average, indicating that the LLM exhibits a preference for this type of interaction.
+
+We also quantify interactions of different orders to measure the representation complexity of an LLM. The order  $m$  of an interaction  $S$  is defined as the number of variables in  $S$ , that is,  $m = |S|$ . We design the following metric to quantify how an LLM focuses on  $m$ -order interactions.
+
+Definition 2. (Focality on interactions of a specific order). Let  $\kappa_m^{(l)}$  denote the focality on  $m$ -order interactions at layer  $l$ . It is computed as follows,
+
+$$
+\forall m \in \{1, 2, \dots , n \}, \kappa_ {m} ^ {(l)} = \frac {\mathbb {E} _ {| S | = m} \left| I _ {S} ^ {(l)} \right|}{Z ^ {(l)}}. \tag {9}
+$$
+
+If  $\kappa_{m}^{(l)}$  has a larger value when  $m$  is higher, it indicates that the LLM encodes interactions of greater complexity when computing features at layer  $l$ . If  $\kappa_{m}^{(l)}$  has a larger value when  $m$  is lower, it indicates that the LLM encodes interactions of lower complexity when computing features at layer  $l$ .
+
+# 4 Comparative studies
+
+In this section, we conduct comparative studies to analyze the internal mechanism of LLMs for arithmetic reasoning (see Section 4.1). We also fine-tune an LLM to improve its capability to solve arithmetic problems and explore how the LLM encodes different types of interactions during the training process (see Section 4.2).
+
+LLMs. We use interactions to analyze seven LLMs for arithmetic reasoning, including the OPT-1.3B [Zhang et al., 2022b] model, the GPT-J-6B [Wang and Komatsuzaki, 2021] model, the Llama-2-7B [Touvron et al., 2023] model, the Llemma-7B [Azerbayev et al., 2023] model, the MathCoder-L-7B [Wang et al., 2023] model, the MathCoder-CL-7B [Wang et al., 2023] model, and the CodeLlama-13B [Roziere et al., 2023] model. Appendix B shows how to mask words for these LLMs.
+
+Data. We follow Karpas et al. [2022], Razeghi et al. [2022], Stolfo et al. [2023] to conduct experiments on a set of arithmetic problems hand-crafted by humans, including 6 templates for one-operator two-operand queries and 29 templates for two-operator three-operand queries. For example, "The sum of  $n_1$  and  $n_2$  is" and "What is the ratio between  $n_1$  minus  $n_2$  and  $n_3$ ? The answer is." Each template for one-operator queries includes all four arithmetic operators, i.e.,  $\{+, -, \times, \div\}$ , and each template for two-operator queries corresponds to a unique com
+
+Table 1: Overall accuracy  $(\%)$  of different LLMs on arithmetic queries.  
+
+<table><tr><td>Model</td><td>1-opr</td><td>2-opr</td></tr><tr><td>OPT-1.3B</td><td>3.2</td><td>1.7</td></tr><tr><td>GPT-J-6B</td><td>14.7</td><td>5.8</td></tr><tr><td>Llama-2-7B</td><td>65.1</td><td>10.1</td></tr><tr><td>Llemma-7B</td><td>75.1</td><td>15.3</td></tr><tr><td>MathCoder-L-7B</td><td>74.0</td><td>8.2</td></tr><tr><td>MathCoder-CL-7B</td><td>62.6</td><td>9.3</td></tr><tr><td>CodeLlama-13B</td><td>71.1</td><td>15.0</td></tr><tr><td>OPT-1.3B Fine-tuned</td><td>83.6</td><td>69.7</td></tr></table>
+
+bination of two operators. Please see Appendix F for details of templates. For each template of one-operator queries, we generate 20 prompts by randomly sampling operands  $(n_1,n_2)$ , and for each template of two-operator queries, we generate 5 prompts following the same procedure. Table 1 shows the overall accuracy of different LLMs on one-operator queries and two-operator queries. Please see Appendix G for details of the accuracy tests. We observe that the Llama-2-7B model, the Llemma-7B model, the MathCoder-L-7B model, the MathCoder-CL-7B model, and the CodeLlama-13B model perform well on one-operator queries, while the OPT-1.3B model and the GPT-J-6B model perform relatively poorly. However, for two-operator queries, all seven LLMs perform poorly.
+
+![](images/9a83bb5dacb4f136ac62dd558146abebe9185c5bdc10b4eaa79dfbd9cc38fc04.jpg)  
+Figure 3: Focality across different types of interactions  $R^{(l)}$  encoded by LLMs during forward propagation. Each curve in the figure is averaged over various one-operator arithmetic queries.
+
+![](images/9b9f46fc6c360e3a02d456f0023855e40d42339d5bec7d9089ba01a4a96c9ff9.jpg)  
+Figure 4: Focality across interactions of different orders  $\kappa^{(l)}$  encoded by LLMs during forward propagation. Each curve in the figure is averaged over various one-operator arithmetic queries.
+
+# 4.1 Exploring the internal mechanism of LLMs for solving arithmetic problems
+
+In this subsection, we analyze how LLMs focus on different types of interactions during forward propagation and obtain the following insights.
+
+# Insight 1: The internal mechanism of LLMs for solving simple one-operator arithmetic problems is their capability to encode operand-Operator interactions and high-order interactions.
+
+Figure 3 reports the focality  $R^{(l)}$  on different types of interactions encoded by LLMs during forward propagation. The results are averaged over all one-operator queries. We observe that in LLMs with strong one-operator arithmetic capabilities (the Llama-2-7B model, the Llemma-7B model, the MathCoder-L-7B model, the MathCoder-CL-7B model, and the CodeLlama-13B model), the focality on operand-operator interactions  $R^{(l)}(\Omega^{\mathrm{opd -opr}})$  tends to increase and exceeds 1.0 in the later layers, while the focality on operator interactions  $R^{(l)}(\Omega^{\mathrm{opr}})$  and the focality on background interactions  $R^{(l)}(\Omega^{\mathrm{bg}})$  tend to fall below 1.0. We notice that in the MathCoder-L-7B model, the focality on operator interactions and the focality on background interactions increase and exceed 1.0 in the very few layers close to the output layer. However, the encoding of operand-operator interactions in the later layers remains sufficient to support its arithmetic reasoning. This suggests that solving one-operator arithmetic problems needs LLMs to increase their focus on operand-operator interactions, and reduce their focus on operator interactions and background interactions in the later layers. Note that in the Llama-2-7B model, the Llemma-7B model, and the MathCoder-L-7B model, the focality on operand interactions  $R^{(l)}(\Omega^{\mathrm{opd}})$  also tends to increase and exceeds 1.0 in the later layers, which may contribute to their strong performance on one-operator arithmetic tasks.
+
+In comparison, in LLMs with weak one-operator arithmetic capabilities (the OPT-1.3B model and the GPT-J-6B model), the focality on different types of interactions is similar and remains around 1.0 in the middle layers, while the focality on background interactions  $R^{(l)}(\Omega^{\mathrm{bg}})$  suddenly increases in the very few layers close to the output layer. This suggests that these LLMs do not exhibit a clear preference for any specific type of interaction in the middle layers, but subsequently shift their focus to background interactions in the very few layers close to the output layer, which weakens their performance on one-operator arithmetic tasks.
+
+Figure 4 reports the focality  $\kappa^{(l)}$  on interactions of different orders encoded by LLMs during forward propagation. The results are averaged over queries from a single one-operator template. We observe that in LLMs with strong one-operator arithmetic capabilities (the Llama-2-7B model, the LLemma-7B model, the MathCoder-L-7B model, the MathCoder-CL-7B model, and the CodeLlama-13B model), the focality on high-order interactions tends to increase and exceeds 1.0 in the later layers. This suggests that solving one-operator arithmetic problems needs LLMs to increase their focus on high-order interactions in the later layers. In comparison, in LLMs with weak one-operator arithmetic capabilities (the OPT-1.3B model and the GPT-J-6B model), the focality on high-order interactions in the middle layers is around 1.0, while in the very few layers close to the output layer, the focality on
+
+![](images/14703be2aa51ae72345ae38a237a8c376ddf52345285417eaadb0b8847371fa2.jpg)  
+Figure 5: Visualizing the dynamic changes in different types and orders of interactions encoded by the OPT-1.3B model during the training process on one-operator queries. Each curve is averaged over various one-operator queries. Results show that as accuracy improves, the model gradually increases its focus on operand-operator interactions (top) and high-order interactions (bottom).
+
+the extremely low-order interactions suddenly increase. This suggests these LLMs exhibit insufficient encoding of higher-order interactions in the middle layers and tend to focus on extremely low-order interactions, which weakens their performance on one-operator arithmetic tasks.
+
+# 4.2 Dynamic encoding of different types of interactions during the training process
+
+We further explore how an LLM learns to solve arithmetic problems. That is, we investigate how an LLM encodes different types of interactions when trained on arithmetic problem data. To this end, we use the LoRA method [Hu et al., 2022] to fine-tune the OPT-1.3B model on arithmetic queries in the following three different ways. $^{8}$  (1) We fine-tune the OPT-1.3B model on one-operator queries, improving its accuracy on one-operator queries from  $3.2\%$  to  $83.6\%$ . This version is termed the OPT-1.3B-One model. (2) We fine-tune the OPT-1.3B model on two-operator queries, improving its accuracy on two-operator queries from  $1.7\%$  to  $69.7\%$ . (3) Building upon the OPT-1.3B-One model, we further train it on two-operator queries. We analyze the dynamic encoding of different types of interactions during the above three training processes and obtain the following insights.
+
+Figure 5 reports the dynamic changes in different types of interactions encoded by the OPT-1.3B model during training on one-operator queries. The results of  $R^{(l)}$  are averaged over all one-operator queries, and the results of  $\kappa^{(l)}$  are averaged over queries from a single one-operator template. We observe that as the accuracy of the OPT-1.3B model on one-operator queries improves, the focality on operand-operator interactions  $R^{(l)}(\Omega^{\mathrm{opd -opr}})$  and the focality on high-order interactions gradually increase, while the focality on operator interactions  $R^{(l)}(\Omega^{\mathrm{opr}})$  and the focality on background interactions  $R^{(l)}(\Omega^{\mathrm{bg}})$  gradually decrease. The results validate our Insight 1.
+
+We also observe that the OPT-1.3B model consistently enhances its focus on background interactions in the very few layers close to the output layer during the training process. This may be due to the inherent preference of the OPT-1.3B model, which tends to enhance its focus on background interactions in the very few layers close to the output layer, as Figure 3 shows.
+
+Insight 2: The internal mechanism of LLMs for solving relatively complex two-operator arithmetic problems is their capability to encode operator interactions and operand interactions. Figure 6 reports the dynamic changes in different interactions encoded by the OPT-1.3B model during training on two-operator queries. The results are averaged over all two-operator queries. We observe that the focality on operator interactions  $R^{(l)}(\Omega^{\mathrm{opr}})$  remains consistently high throughout the training process. As the accuracy of the OPT-1.3B model on two-operator queries improves, the focality on operand interactions  $R^{(l)}(\Omega^{\mathrm{opd}})$  gradually increases in the later layers during the final stage of training. This suggests that solving two-operator arithmetic problems requires an LLM to focus more on operator interactions and operand interactions. We notice that the OPT-1.3B model consistently focuses on background interactions, which may be due to its inherent preference. Notably, the focality on operand-operator interactions gradually increases in the very few layers close to the output layer, which may contribute to the model's strong performance on two-operator arithmetic tasks.
+
+![](images/c38d0001ced0a9747c92399a55853b387ed15ac32b05a1f1d842907956a029dd.jpg)  
+Figure 6: Visualizing the dynamic changes in different types of interactions encoded by the OPT-1.3B model during the training process on two-operator queries. Each curve is averaged over various two-operator queries. Results show that the OPT-1.3B model consistently focuses more on operator interactions and gradually increase its focus on operand interactions during the final stage of training.
+
+![](images/e342b58ff813dd2a14ed099491a3dfde702cc8b32ed4a580c667cd42c97cd994.jpg)  
+Figure 7: Visualizing the dynamic changes in different types and orders of interactions encoded by the OPT-1.3B-One model during the training process on two-operator queries. Each curve is averaged over various one-operator queries. Results show that as accuracy declines, the model reduces its focus on operand-operator interactions (top) and high-order interactions (bottom).
+
+Insight 3: We explain the task-specific nature of the LoRA method from the perspective of interactions. Figure 7 reports the dynamic changes in different types of interactions encoded by the OPT-1.3B-One model during training on two-operator queries. The results of  $R^{(l)}$  are averaged over all one-operator queries, and the results of  $\kappa^{(l)}$  are averaged over queries from a single one-operator template. We observe that the accuracy of the OPT-1.3B-One model on one-operator queries declines, which means that the OPT-1.3B-One model gradually forgets how to solve simpler arithmetic problems while learning more complex knowledge. Meanwhile, the accuracy on two-operator queries steadily improves during training (please see Appendix M for details). This phenomena is consistent with the task-specific nature of LoRA [Hu et al., 2022]. From the perspective of interactions, we observe that the focality on operand-operator interactions  $R^{(l)}(\Omega^{\mathrm{opd -opr}})$  and the focality on high-order interactions gradually decrease during the training process, with most values tending to stay around 1.0. The results help explain the underlying mechanism behind the task-specific nature of LoRA in arithmetic reasoning tasks and validate our Insight 1.
+
+# 5 Conclusion and discussions
+
+In this paper, we use interactions to provide a deep understanding of the internal mechanism of LLMs for arithmetic reasoning. Through comparison studies of different types and orders of interactions encoded by LLMs during forward propagation, we find that the internal mechanism of LLMs for solving simple one-operator arithmetic problems is their capability to encode operand-operand interactions and high-order interactions. We further fine-tune an LLM to explore how an LLM encodes different types and orders of interactions when learning to solve arithmetic problems. We find that the internal mechanism of LLMs for solving relatively complex two-operator arithmetic problems is their capability to encode operator interactions and operand interactions. We also explain the task-specific nature of the LoRA method from the perspective of interactions.
+
+Limitations. As the number of input variables increases significantly, the computational cost of interaction analysis indeed rises. To address this issue, we can analyze informative variables while treating uninformative variables as background or adopt some other methods (see Appendix L). On the other hand, we have only studied simple arithmetic problems and have not yet extended our research to more complex math word problems. In the future, we will work on this.
+
+# Acknowledgment
+
+This work is partially supported by the National Nature Science Foundation of China (No.62376199,62206170).
+
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+
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+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
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+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [No]
+
+Justification: While the code and dataset generated are not yet ready for anonymous open sourcing, we plan to open-source the code and the data with appropriate licensing.
+
+Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
+- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).  
+- The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.  
+- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.  
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: Experimental details are included in the main paper and in Appendix B, Appendix F, Appendix H, Appendix K.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [No]
+
+Justification: For all experiments, the do_sample parameter of the large language model was set to False to ensure deterministic outputs.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
+
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
+- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)  
+- The assumptions made should be given (e.g., Normally distributed errors).  
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We have added computational resources needed in the Appendix K.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: Our work is to study arithmetic reasoning in large language models, and it doesn't violate any code of Ethics.
+
+Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: We discuss scalability and limitations of our work in conclusion, Appendix ?? and Appendix L. Since our work is about the internal mechanism of arithmetic reasoning in LLMs, we do not believe that there are any direct negative implications.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: Our work is not focused towards releasing novel pretrained language models, image generators, or scraped datasets. For the data generated using existing arithmetic reasoning templates with randomly sampled one- and two-operand numbers, and the large language model fine-tuned on the data based on existing pretrained models, we believe there are no risks of misuse.
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: We make sure to cite all relevant and related work. The templates we use are publicly available.
+
+Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.
+
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Justification: We do not release any new assets in this work. For the data generated in our work, we include templates mentioned in Appendix F.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: We do not do any crowdsourcing experiments.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: We do not do any experiments with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
+
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: We only use LLMs for translation and grammar checking.
+
+Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# A Strategies of masking input variables
+
+In the research of attribution methods, it is common to use a specific token or embedding to mask input variables of a DNN [Lundberg, 2017, Ancona et al., 2019, Fong et al., 2019], and use changes in network outputs on the masked samples to estimate attributions of input variables. However, each method has certain limitations. The mean baseline value [Dabkowski and Gal, 2017], i.e., setting the baseline value for each input variable to its mean across all samples, introduces additional signals, e.g., grey dots in images. Similarly, the zero baseline value [Ancona et al., 2019, Sundararajan et al., 2017], i.e., setting baseline values for all input variables to zero, would also introduce additional signals to the input, such as black dots.
+
+# B Details about how to mask input words for different LLMs
+
+In this paper, we analyze seven LLMs for arithmetic reasoning, including the OPT-1.3B model, the GPT-J-6B model, the Llama-2-7B model, the CodeLlama-2-7B model, the MathCoder-L-7B model, the MathCoder-CL-7B model, and the Llemma-7B model. For the OPT-1.3B model, we use the “</s>” token with the token  $id = 2$  to mask the words in  $N \backslash T$ . For the GPT-J-6B model, we use the “<|endoftext|>” with the token  $id = 50256$  to mask the words in  $N \backslash T$ . For the other five models, including the Llama-2-7B model, the CodeLlama-2-7B model, the MathCoder-L-7B model, the MathCoder-CL-7B model, and the Llemma-7B model, with Llama as their base model, we use the “<unk>” with the token  $id = 0$  to mask the words in  $N \backslash T$ .
+
+# C Properties of interactions
+
+The Harsanyi interaction  $I_{S}$ , i.e., the interaction in this paper, can explain the elementary mechanism of existing game-theoretic metrics [Ren et al., 2021a], including the Shapley value [Shapley, 1953], the Shapley interaction index [Grabisch and Roubens, 1999], and the Shapley-Taylor interaction index [Sundararajan et al., 2020].
+
+(1) Connection to the Shapley value [Shapley, 1953]. Let  $\phi(i)$  denote the Shapley value of an input variable  $i$ , given the input sample  $\mathbf{x}$ . Then, the Shapley value  $\phi(i)$  can be explained as the result of uniformly assigning attributions of each Harsanyi interaction to each involving variable  $i$ , i.e.,  $\phi(i) = \sum_{S \subseteq N \setminus \{i\}} \frac{1}{|S| + 1} I_{S \cup \{i\}}$ . This also proves that the Shapley value is a fair assignment of attributions from the perspective of the Harsanyi interaction.  
+(2) Connection to the Shapley interaction index [Grabisch and Roubens, 1999]. Given a subset of variables  $T \subseteq N$  in an input sample  $\pmb{x}$ , the Shapley interaction index  $I_T^{\text{Shapley}}$  can be represented as  $I_T^{\text{Shapley}} = \sum_{S \subseteq N \setminus T} \frac{1}{|S| + 1} I_{S \cup T}$ . In other words, the index  $I_T^{\text{Shapley}}$  can be explained as uniformly allocating  $I_{S'}$  such that  $S' = S \cup T$  to the compositional variables of  $S'$ , if we treat the coalition of variables in  $T$  as a single variable.  
+(3) Connection to the Shapley Taylor interaction index [Sundararajan et al., 2020]. Given a subset of variables  $T \subseteq N$  in an input sample  $\mathbf{x}$ , the  $k$ -th order Shapley Taylor interaction index  $I_T^{\text{Shapley-Tay}}$  can be represented as a weighted sum of interaction effects, i.e.,  $I_T^{\text{Shapley-Tay}} = I_T$  if  $|T| < k$ ;  $I_T^{\text{Shapley-Tay}} = \sum_{S \subseteq N \setminus T} \binom{|S| + k}{k}^{-1} I_{S \cup T}$  if  $|T| = k$ ; and  $I_T^{\text{Shapley-Tay}} = 0$  if  $|T| > k$ .
+
+Given an input sample  $\mathbf{x}$ , the Harsanyi interaction  $I_S$  satisfies seven desirable axioms in game theory [Ren et al., 2021a], including the efficiency, linearity, dummy, symmetry, anonymity, recursive and interaction distribution axioms.
+
+(1) Efficiency axiom. The output score of a model can be decomposed into interaction effects of different patterns, i.e.,  $v(\pmb{x}) = \sum_{S \subseteq N} I_S$ .  
+(2) Linearity axiom. If we merge output scores of two models  $w$  and  $v$  as the output of model  $u$ , i.e.,  $\forall S \subseteq N, u(\boldsymbol{x}_S) = v(\boldsymbol{x}_S) + w(\boldsymbol{x}_S)$ , then their interaction effects  $I_S^{(v)}$  and  $I_S^{(w)}$  can also be merged as  $\forall S \subseteq N, I_S^{(u)} = I_S^{(v)} + I_S^{(w)}$ .  
+(3) Dummy axiom. If a variable  $i \in N$  is a dummy variable, i.e.,  $\forall S \subseteq N \setminus \{i\}$ ,  $v(\pmb{x}_{S \cup \{i\}}) = v(\pmb{x}_S) + v(\pmb{x}_{\{i\}})$ , then it has no interaction with other variables,  $\forall \emptyset \neq T \subseteq N \setminus \{i\}$ ,  $I_{T \cup \{i\}} = 0$ .
+
+(4) Symmetry axiom. If input variables  $i, j \in N$  cooperate with other variables in the same way,  $\forall S \subseteq N \setminus \{i, j\}$ ,  $v(\pmb{x}_{S \cup \{i\}}) = v(\pmb{x}_{S \cup \{j\}})$ , then they have the same interaction effects with other variables,  $\forall S \subseteq N \setminus \{i, j\}$ ,  $I_{S \cup \{i\}} = I_{S \cup \{j\}}$ .  
+(5) Anonymity axiom. For any permutations  $\pi$  on  $N$ , we have  $\forall S \subseteq N, I_S^{(v)} = I_{\pi S}^{(\pi v)}$ , where  $\pi S \triangleq \{\pi(i) \mid i \in S\}$ , and the new model  $\pi v$  is defined by  $(\pi v)(\boldsymbol{x}_{\pi S}) = v(\boldsymbol{x}_S)$ . This indicates that interaction effects are not changed by permutation.  
+(6) Recursive axiom. The interaction effects can be computed recursively. For  $i \in N$  and  $S \subseteq N \setminus \{i\}$ , the interaction effect of the pattern  $S \cup \{i\}$  is equal to the interaction effect of  $S$  with the presence of  $i$  minus the interaction effect of  $S$  with the absence of  $i$ , i.e.,  $\forall S \subseteq N \setminus \{i\}$ ,  $I_{S \cup \{i\}} = I_S^{(i \text{ is always present})} - I_S \cdot I_S^{(i \text{ is always present})}$  denotes the interaction effect when the variable  $i$  is always present as a constant context, i.e.,  $I_S^{(i \text{ is always present})} = \sum_{L \subseteq S} (-1)^{|S| - |L|} \cdot v(\boldsymbol{x}_{L \cup \{i\}})$ .  
+(7) Interaction distribution axiom. This axiom characterizes how interactions are distributed for "interaction functions" [Sundararajan et al., 2020]. An interaction function  $v_{T}$  parameterized by a subset of variables  $T$  is defined as follows:  $\forall S \subseteq N, v_{T}(\boldsymbol{x}_{S}) = c$ , if  $T \subseteq S$ ; otherwise,  $v_{T}(\boldsymbol{x}_{S}) = 0$ . The function  $v_{T}$  models pure interaction among the variables in  $T$  because only if all variables in  $T$  are present the output value will be increased by  $c$ . The interactions encoded in the function  $v_{T}$  satisfy  $I_{T} = c$ , and  $\forall S \neq T, I_{S} = 0$ .
+
+# D Details about network architectures and chosen layers for experiments in section 4
+
+OPT-1.3B model. The OPT-1.3B model is composed of the following parts: one word embedding layer (namely Embedding Layer), one position embedding layer (namely OPTLearnedPositionalEmbedding), 24 OPT decoder modules (namely OPTDecoderLayer), and one linear output layer (namely Linear Layer). The architecture can be summarized as Embedding Layer  $\rightarrow$  OPTLearnedPositionalEmbedding  $\rightarrow$  [OPTDecoderLayer]  $\times$  24  $\rightarrow$  Linear Layer. Each module of OPTDecoderLayer contains a self-attention mechanism layer (namely OPTAttention), an activation function (namelyReLU), a layer normalization operation (namely LayerNorm), two fully connected layers (namely Linear), and a final layer normalization operation (namely LayerNorm). In our experiments, we selected all 24 OPTDecoderLayer modules and conducted experiments based on the output features of the LayerNorm operation.
+
+GPT-J-6B model. The GPT-J-6B model includes a single word embedding layer (namely Embedding Layer), followed sequentially by 28 Transformer blocks (namely GPTJBlock), culminating in an output layer normalization (namely LayerNorm) and a linear output layer (namely Linear Layer). This architecture can be succinctly described as: Embedding Layer  $\rightarrow$  [GPTJBlock]  $\times$  28  $\rightarrow$  LayerNorm  $\rightarrow$  Linear Layer. Delving into the details of each module of GPTJBlock, it comprises three integral components, including a layer normalization (namely LayerNorm), a self-attention mechanism (namely GPTJAttention), and a feed-forward network (namely GPTMLP). In our experiments, we selected all 28 GPTJBlock modules and conducted experiments based on the output features of the LayerNorm operation.
+
+Other Llama-based models. The other five models based on Llama include the Llama-2-7B model, the CodeLlama-2-7B model, the MathCoder-L-7B model, the MathCoder-CL-7B model, and the Llemma-7B model. These models share the same architecture, which is composed of various elements: a single word embedding layer (namely Embedding Layer), followed by a sequence of 32 LlamaDecoderLayer, an output layer normalization layer (namely LlamaRMSNorm), and culminating in a linear output layer (namely Linear Layer). This architecture can be summarized as Embedding Layer  $\rightarrow$  [LlamaDecoderLayer]  $\times 32 \rightarrow$  LlamaRMSNorm  $\rightarrow$  Linear Layer. Delving into each module of LlamaDecoderLayer, it integrates multiple components, including a self-attention mechanism layer (namely LlamaAttention), a feed-forward network layer (namely LlamaMLP) encompassing several linear layers and activation functions, an input layer normalization (namely Input LayerNorm), and a post-attention layer normalization (namely Post Attention LayerNorm), the latter serving to normalize the output from the self-attention mechanism layer. In our experiments, we selected all 32 modules of LlamaDecoderLayer and conducted experiments based on the output features of the LayerNorm operation.
+
+![](images/5a95cea7aa79410189a2e173ca88209f553414fd2cd9bcf598ee9b894cc0f5a5.jpg)  
+Figure 8: The visualization of  $v^{(l)}$  at different token positions across various layers of CodeLlama for the prompt "How much is 5 plus 1? Answer is" under different masked inputs. Results show that except for the final tokens of the complete sentence (i.e., token 10 and token 13), the  $v^{(l)}$  values of other tokens in the middle layers remain almost unchanged.
+
+Table 2: Question templates for one-operator arithmetic queries.  
+
+<table><tr><td>Type</td><td>Addition</td><td>Subtraction</td></tr><tr><td>1</td><td>How much is n1 plus n2? Answer is</td><td>How much is n1 minus n2? Answer is</td></tr><tr><td>2</td><td>What is n1 plus n2? Answer is</td><td>What is n1 minus n2? Answer is</td></tr><tr><td>3</td><td>How much is the sum of n1 and n2? Answer is</td><td>How much is the difference between n1 and n2? Answer is</td></tr><tr><td>4</td><td>What is the sum of n1 and n2? Answer is</td><td>What is the difference between n1 and n2? Answer is</td></tr><tr><td>5</td><td>The sum of n1 and n2 is</td><td>The difference between n1 and n2 is</td></tr><tr><td>6</td><td>Given two numbers n1 and n2, the sum of them is</td><td>Given two numbers n1 and n2, the difference between them is</td></tr><tr><td></td><td>Multiplication</td><td>Division</td></tr><tr><td>1</td><td>How much is n1 times n2? Answer is</td><td>How much is n1 over n2? Answer is</td></tr><tr><td>2</td><td>What is n1 times n2? Answer is</td><td>What is n1 over n2? Answer is</td></tr><tr><td>3</td><td>What is the result of n1 times n2? Answer is</td><td>What is the result of n1 over n2? Answer is</td></tr><tr><td>4</td><td>How much is the product of n1 and n2? Answer is</td><td>How much is the ratio between n1 and n2? Answer is</td></tr><tr><td>5</td><td>The product of n1 and n2 is</td><td>The ratio of n1 and n2 is</td></tr><tr><td>6</td><td>Given two numbers n1 and n2, the product of them is</td><td>Given two numbers n1 and n2, the ratio between them is</td></tr></table>
+
+# E The features of other tokens in the middle layers remain unchanged
+
+Through experiments, we found that, except for the final token of the complete sentence, the  $v^{(l)}$  values at other token positions in the middle layers remain almost unchanged. Figure 8 shows the  $v^{(l)}$  values at different token positions across various layers of CodeLlama for the prompt "How much is 5 plus 1? Answer is" under different masked inputs  $x_{T}$ . The prompt consists of 13 tokens {<s>, How, much, is, space, 5, plus, space, 1, ?, answer, is, space}. Except for token 10 (?) and token 13 (space), the  $v^{(l)}$  values of other tokens in the middle layers remain almost unchanged.
+
+# F Prompt Templates
+
+In Table 2 and 3, we report the question templates used as prompts for the model for one- and two-operator queries, respectively. For two-operator queries, we use one query template for each of the 29 possible two-operation combinations. To enable the model to output the answer directly, we appended "The answer is" at the end of each template.
+
+# G Performance of the LLMs
+
+To more accurately and fairly evaluate the LLMs' capabilities to solve one-operator and two-operator arithmetic problems, we evaluated the accuracy of all LLMs on the one-operator and two-operator test sets used for fine-tuning in Appendix H.
+
+For the OPT-1.3B and GPT-J-6B models, we treat each operand as a single token, while for other LLMs, each number is split into multiple tokens ("0", "1", "2", ..., "9") by the tokenizer.
+
+Table 3: Question templates for two-operator arithmetic queries.  
+
+<table><tr><td>Type</td><td>Formula</td><td>Format</td></tr><tr><td>1</td><td>f = ((A + B) * C)</td><td>Sum A and B and multiply by C</td></tr><tr><td>2</td><td>f = (A + B * C)</td><td>What is the sum of A and the product of B and C?</td></tr><tr><td>3</td><td>f = ((A - B) * C)</td><td>What is the product of A minus B and C?</td></tr><tr><td>4</td><td>f = (A / (B / C))</td><td>How much is A divided by the ratio between B and C?</td></tr><tr><td>5</td><td>f = (A - B * C)</td><td>What is the difference between A and the product of B and C?</td></tr><tr><td>6</td><td>f = (A * (B - C))</td><td>How much is A times the difference between B and C?</td></tr><tr><td>7</td><td>f = ((A + B) / C)</td><td>What is the ratio between A plus B and C?</td></tr><tr><td>8</td><td>f = (A - (B - C))</td><td>How much is A minus the difference between B and C?</td></tr><tr><td>9</td><td>f = ((A - B) / C)</td><td>What is the ratio between A minus B and C?</td></tr><tr><td>10</td><td>f = (A - B / C)</td><td>What is the difference between A and the ratio between B and C?</td></tr><tr><td>11</td><td>f = (A / (B + C))</td><td>How much is A divided bu the sum of B and C?</td></tr><tr><td>12</td><td>f = (A / (B - C))</td><td>How much is A divided by the difference between B and C?</td></tr><tr><td>13</td><td>f = (A + B / C)</td><td>what is the sum of A and the ratio between B and C?</td></tr><tr><td>14</td><td>f = (A * (B / C))</td><td>How much is A times the ratio between B and C?</td></tr><tr><td>15</td><td>f = (A * B + C)</td><td>How much is the sum of A times B and C?</td></tr><tr><td>16</td><td>f = (A * (B + C))</td><td>How much is A times the sum of B and C?</td></tr><tr><td>17</td><td>f = (A / B + C)</td><td>How much is the sum of A divided by B and C?</td></tr><tr><td>18</td><td>f = (A / B / C)</td><td>How much is A divided by B divided by C?</td></tr><tr><td>19</td><td>f = (A / B - C)</td><td>How much is the difference between A divided by B and C?</td></tr><tr><td>20</td><td>f = (A / B * C)</td><td>How much is A divided by B times C?</td></tr><tr><td>21</td><td>f = (A - (B + C))</td><td>How much is A minus the sum of B and C?</td></tr><tr><td>22</td><td>f = (A * B - C)</td><td>How much is the difference between A times B and C?</td></tr><tr><td>23</td><td>f = (A / (B * C))</td><td>How much is A divided by the product of B and C?</td></tr><tr><td>24</td><td>f = (A - B + C)</td><td>How much is A minus B plus C?</td></tr><tr><td>25</td><td>f = (A + B + C)</td><td>How much is A plus B plus C?</td></tr><tr><td>26</td><td>f = (A - B - C)</td><td>How much is A minus B minus C?</td></tr><tr><td>27</td><td>f = (A * B / C)</td><td>How much is A times B divided by C?</td></tr><tr><td>28</td><td>f = (A + B - C)</td><td>How much is A plus B minus C?</td></tr><tr><td>29</td><td>f = (A * B * C)</td><td>How much is A times B times C?</td></tr></table>
+
+# H Fine-tuning Details
+
+For the one-operator data, we sample 500 examples for each of the 6 natural language templates in Table 2 and their corresponding original mathematical expressions, resulting in a total of 3,500 samples. For the two-operator data, we sample 1,000 examples for each of the 29 natural language templates in Table 3, resulting in a total of 29,000 samples. We set the maximum operand value to 100 and exclude problems with final results exceeding 1000. For single-operator data, we use an 8/2 train-test split, while for two-operator data, we use a 9/1 train-test split.
+
+We fine-tune the OPT-1.3B model using the LoRA method [Hu et al., 2022] on one-operator and two-operator samples. For the one-operator templates, we train the model for 10 epochs with a batch size of 16. For the two-operator templates, we train the model for 20 epochs with a batch size of 32. The training uses a learning rate of 8e-4 with a linear decay scheduler. The LoRA configuration includes a rank of 8, a LoRA alpha of 32, and a dropout of 0.05.
+
+# I More experimental results
+
+Figure 9 shows that LLMs with good arithmetic capabilities gradually focus more on high-order interaction patterns. Figure 10 illustrates that the OPT-1.3B model gradually increases its focus on
+
+![](images/0ee566b781ef6e5bef52e986cbe8e2c5b38b1053ec19d158cb84b5ddc111048d.jpg)  
+Figure 9: Comparing the normalized average strength  $\kappa^{(l)}$  of interaction patterns of different orders encoded by LLMs during forward propagation. Each curve in the figure is averaged over various one-operator arithmetic queries, corresponding to template 4 in Table 2.
+
+![](images/58223ecc2a0a780ac6693b296b9c59b2d4bda44f4ff2c5f22a0d51176059682b.jpg)  
+Figure 10: Visualizing the dynamic process of different interaction patterns encoded by the OPT-1.3B model during the training process. Each curve is averaged over various one-operator queries, corresponding to (a) template 0 and (b) template 3 in Table 2.
+
+high-order interaction patterns during the learning process of simple arithmetic problems. Figure 11 demonstrates that the OPT-1.3B model gradually decreases its focus on high-order interactions while learning relatively complex arithmetic problems.
+
+# J Information about the use of AI assistants.
+
+In this paper, AI tools such as DeepSeek were used for translation and grammar checking.
+
+# K Computational budget
+
+We conducted our experiments on an NVIDIA GeForce RTX 4090 24GB GPU. For the Llama-2-7B model, the computation time per one-operator sample is around 30 seconds, while that for a two-operator sample is around 60 seconds.
+
+![](images/ea13e7927e359234799f83384740dd5d1909218fba6f73970de87f2bc1fe0053.jpg)  
+Figure 11: Visualizing the dynamic process of different interaction patterns encoded by the OPT-1.3B-One model during the training process. Each curve is averaged over various one-operator queries, corresponding to (a) template 0 and (b) template 3 in Table 2.
+
+# L Limitations
+
+When the number of input variables increases significantly, the computational cost of interaction analysis indeed rises. To deal with this issue, we can adopt the following approaches.
+
+Analyzing informative variables while treating uninformative variables as background to reduce computational complexity while preserving core semantic information. Previous research [Chen et al., 2024] has demonstrated that selecting informative input variables does not fundamentally compromise the faithfulness of interactions.
+
+Phrase-level aggregation, where related tokens are merged into meaningful phrases without compromising semantic integrity, can reduce the computational burden.
+
+Exploring approximation methods, such as variable selection based on attention weights and prioritizing tokens that have a greater impact on model predictions for interaction analysis, can help reduce computational overhead.
+
+# M Accuracy curve of the OPT-1.3B-One model on two-operator queries
+
+We visualize the accuracy of the OPT-1.3B-One model on two-operator arithmetic queries across different checkpoints. As shown in Figure 12, the model demonstrates a gradual improvement in performance during the training process. We also highlight five selected checkpoints (referred to as TimePoints 1 through 5) that are used in Figure 7.
+
+![](images/60a1e17110fe932f166e74345470148b7060179a05787427cddb240f0ac50229.jpg)  
+Figure 12: Accuracy of the OPT-1.3B-One model on two-operator queries evaluated at different checkpoints. Red markers indicate selected TimePoints 1-5 used in Figure 7.

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