Low-Rank Few-Shot Node Classification by Node-Level Graph Diffusion
Authors: Yancheng Wang
Chengshuai Zhao
Dongfang Sun
huan liu
Yingzhen Yang
Venue: ICLR 2026 Poster (2026)
Keywords: Few-Shot Node Classification
Low-Rank Few-Shot Graph Diffusion Model
Low-Rank Learning
Primary area: learning on graphs and other geometries & topologies
TL;DR: We propose a novel node-level graph diffusion method with low-rank feature learning for few-shot node classification, termed Low-Rank Few-Shot Graph Diffusion Model  (LR-FGDM), with strong theoretical guarantee and extensive empirical results.

Abstract: In this paper, we propose a novel node-level graph diffusion method with low-rank feature learning for few-shot node classification (FSNC), termed Low-Rank Few-Shot Graph Diffusion Model or LR-FGDM. LR-FGDM first employs a novel Few-Shot Graph Diffusion Model (FGDM) as a node-level graph generative method to generate an augmented graph with an enlarged support set, then performs low-rank transductive classification to obtain the few-shot node classification results. Our graph diffusion model, FGDM, comprises two components, the Hierarchical Graph Autoencoder (HGAE) with an efficient hierarchical edge reconstruction method and a new prototypical regularization, and the Latent Diffusion Model (LDM). The low-rank regularization is robust to the noise inherently introduced by the diffusion model and empirically inspired by the Low Frequency Property. We also provide a strong theoretical guarantee justifying the low-rank regularization for the transductive classification in few-shot learning. To further enhance the performance of LR-FGDM, we introduce LRA-LR-FGDM with a novel efficient LR-Attention layer, or the LRA layer, which applies self-attention to the output of the LR-FGDM encoder. The LRA layer further reduces the kernel complexity of LR-FGDM and contributes to a tighter generalization bound, leading to improved performance. Extensive experimental results evidence the effectiveness of LR-FGDM for few-shot node classification, which outperforms the current state-of-the-art. The code of the LR-FGDM is available at \url{https://github.com/Statistical-Deep-Learning/LR-FGDM}.

Published as a conference paper at ICLR 2026
LOW-RANK FEW-SHOT NODE CLASSIFICATION BY MODE-LEVER GRAPH DIFFUSION
Yancheng Wang. Chengshuai Zhao, Dongfang Sun, Huan Liu & Yingzhen Yang School of Computing and Augmented Intelligence
Arizona State University Tempe, AZ 85281, USA
{yancheng wang, czhao93, huanliu, iu, yingzhen. yang) edu
ABSTRACT
In this paper, we propose a novel node-level graph diffusion method with low-rank feature learning for few-shot node classification (FSNC), termed Low-Rank Few- Shot Graph Diffusion Model LR-FGDM. LR-FGDM first employs a novel Few- Shot Graph Diffusion Model (FGDM) a node-level graph generative method to generate an augmented graph with an enlarged support set, then performs low- rank transductive classification obtain the few-shot node classification results. Our graph diffusion model, FGDM, comprises two components, the Hierarchical Graph Autoencoder (HGAE) with an efficient hierarchical edge reconstruction method and new prototypical regularization, and the Latent Diffusion Model (LDM). The low-rank regularization is robust the noise inherently introduced by the diffusion model and empirically inspired by the Low Frequency Property. We also provide strong theoretical guarantee justifying the low-rank regularization for the transductive classification in few-shot learning. To further enhance the performance of LR-FGDM, we introduce L.R.-.R-FGIM with a novel efficient LR-Attention layer, the LRA layer, which applies self-attention to the output of the LR-FGDM encoder. The LRA layer further reduces the kernel complexity LR-FGDM and contributes to tighter generalization bound, leading improved performance. Extensive experimental results evidence the effectiveness of LR- FGDM for few-shot node classification, which outperforms the current state-of- the-art. The code of the LR-FGDM is available at https: / / thub. com/ Stati st cal Deep- Lea rn www.nninnnn/////
INTRODUCTION
Graph Neural Networks (GNNs) (Kipf & Welling, 2016b; Hamilton et al., 2017) are widely used for semi-supervised node classification (Veličković et al., 2018), but their effectiveness relics on ample labeled data. This challenge motivates few-shot node classification (FSNC), where only few la- beled nodes per class are available. Most FSNC methods (Zhou et al., 2019; Ding et al., 2020; Wang et al., 2022; Huang & Zitnik, 2020; Qian al., 2021; Lan et al., 2020: Liu et al., 2021b) follow a meta-learning framework (Finn et al., 2017; Snell et al., 2017) to generalize across tasks. More recent approaches (Tan al., 2022: Liu et al., 2024) leverage self-supervised Graph Contrastive Learning (GCL) (Mo et al., 2022; Jin et al., 2021; Wang & Yang, 2022; Ding et al., 2023), and achieve superior performance despite using only unlabeled data. However, all existing methods re- main constrained by the limited support set size. Although techniques like mix-up (Liu et al., 2025b) and random perturbation (Wu et al., 2022) offer marginal gains, the potential of generative models to synthesize support nodes remains underexplored. Building on the success of diffusion models in vi- sion, recent works have extended them to synthetic graph generation. Recent works (Niu et al., 2020; Jo et al., 2022; Haefeli et al., 2022; Vignac et al., 2023; Limnios et al., 2023) adapt diffusion models to generate realistic structures that align well with real-world networks. However, these approaches focus graph-level generation and do not support structured node- edge-level synthesis. Node- level graph augmentation typically relies on GANs (Jia et al., 2023; Wu et al., 2023; Wang et al., 2018 8; Liang et al., 2020; Yang et al., 2019) generate minority class nodes in imbalanced graphs,

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despite known issues of training instability and poor distributional matching (Dhariwal & Nichol, 2021).
In this work, we propose novel node-level graph diffusion method with low-rank feature learning for FSNC, termed Low-Rank Few-Shot Graph Diffusion Model or LR-FGDM ARFFGDM employs novel Few-Shot Graph Diffusion Model (FGDM) to generate augmented graph with an enlarged support set. set. The FGDM in LR-FGDM consists of two components, including the Hierarchical Graph Autoencoder (HGAE) with an efficient hierarchical edge reconstruction method and the Latent Dif- fusion Model (LDM). The HGAE learns compact latent node features for LDM by incorporating a prototypical regularization to encourage semantic structure the latent space. The hierarchical edge reconstruction method enables efficient reconstruction of the edges connecting to node from the latent space a hierarchical manner to avoid the quadratic complexity in edge reconstruction of the regular GAE (Kipf& & Welling, 2016a). Given a FSNC task, the FGDM generates the synthetic graph structure, consisting of the synthetic support nodes and the edges connecting to the original graph. The synthetic graph structure is then incorporated into the original graph, forming augmented graph with an enlarged support set consisting of the original and synthetic support nodes.
Although prior methods enlarge the support set via random perturbations (Gao et al., 2023b) or mix-up (Liu et al., 2025b), they fail to generate faithful graph structures, often assigning edges to synthetic nodes by reusing neighbors of real nodes. In contrast, our FGDM jointly encodes nodes and edges into a semantically regularized latent space for training the LDM,
90
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Accuncy
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COGA
82
(LR-
capturing the true joint distribution of features and structure. Let Vya and Vsup denote the set of synthetic support nodes and the original support set. As shown in Figure 1, while adding synthetic support nodes with FGDM improves COLA's perfor- mance when V/ynl S 3) Vuup, further increasing the number of synthetic nodes leads to sharp performance drop. This is due to inherent noise in the diffusion generation process (Ho et al., 2020; Fu et al., 2024; Azizi et al., 2023; He et al., 2022). To this end, we propose a low-rank learning method inspired by the widely-studied Low Frequency Property (LFP) (Rahaman et al., 2019; Arora et al., 2019; Cao et al., 2021: Choraria et al., 2022; Wang et al., 2024; 2025b), which suggests that the pro- jection of the ground truth class labels mostly concentrates on the top eigenvectors of the kernel gram matrix of the model, to be detailed in Section 3.4. Motivated by LFP, the truncated nuclear norm (TNN) is added as a low-rank regularization term to the training loss of the few-shot classi- fier on the augmented graph. The effectiveness of low-rank learning also theoretically justified by Theorem A. in Section A, which shows that reducing kernel complexity yields a tighter upper bound on the test loss. Inspired by this result, we further propose LRA-LR- with a low-rank attention (LRA) layer that applies self-attention to the LR-FGDM encoder output, further reducing kernel complexity and tightening the generalization bound compared to LR-FGDM. It is observed from Figure 1 that the COLA trained with the low-rank regularization performs significantly bet- ter than the regular COLA when the synthetic nodes added in the augmented graph are more than 3 Vaupl-
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|Ppel Dxlll
Figure 1: 5-way 5-shot node classification accuracies on Cora- Full trained with different numbers of synthetic support nodes added. COLA (LR-FGDM) is trained on the augmented graph by FGDM with the low-rank regularization, while COLA (FGDM) is trained without the regularization.
Existing graph few-shot learning methods (Liu et al., 2024; Wang et al., 2023a; Ma et al., 2025; Zhao et al., 2025) show that training graph encoders without label supervision yields better generalization to novel classes. However, the diffusion-based generator DoG (Wang et al., 2025b) relies on class labels as conditioning signals during both training and generation, which is problematic as test- time labels are disjoint from the training labels. Although semi-supervise 2002: Bair, 2013) can be applied on the node attributes to obtain pseudo labels for conditioning, as illustrated in Figure 3 (a). it often leads to semantic drift and unreliable conditioning due to entangled base and novel class semantics. prototypes jointly learned with the prototypical regularization, as illustrated in Figure 3 (b), rather than pseudo labels, to avoid this issue. As another significant difference from DoG (Wang et al., 2025b), a new prototypical regularization is introduced to HGAE to improve cluster separability in the latent space, making the
K-means (Basu et al.,
contrast, our LR- RRFGGMM conditions the diffusion model on cluster
based conditioning semantically aligned and robust.
Contributions. The contributions of this paper are presented as follows.
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First, we propose the Low-Rank Few-Shot Graph Diffusion Model (LR-FGDM), novel generative framework for FSNC tasks by synthesizing labeled support nodes and the associated edges through node-level graph diffusion model, Few-Shot Graph Diffusion Model (FGDM). The FGDM fea- tures a new Hierarchical Graph Autoencoder (HGAE) that incorporates prototypical regularization to structure the latent space semantically, and our L.M uses the prototypes instead of class labels such as those in DoG (Wang et al., 2025b) as the conditioning features. To mitigate the inherent inefficiency of the quadratic complexity in edge reconstruction over the entire graph, FGDM also introduces hierarchical edge reconstruction method, which hierarchically reconstructs the edges connecting each synthetic support node. While prior methods (Liu et al., 2025b; Wu et al., 2022) have shown promising results in enlarging the support set for FSNC, Table 8 in Section G.2 shows that LR-FGDM substantially outperforms existing support set augmentation approaches. Moreover, we introduce the Frechet Node Distance (FND) and the Frechet Edge Distance (FED) to validate the faithfulness of the synthetic support nodes and the associated edges in Section G.7 of the appendix.
Second, we introduce a low-rank regularization method in LR-FGDM for the transductive node classifier trained on the augmented graph, which is empirically motivated by the Low Frequency Property (LFP) in deep learning (Rahaman et al., 2019; Arora et al., 2019; Cao et al., 2021; Choraria et al., 2022: Wang et al., 2024; 2025b) and theoretically justified by a novel generalization bound for the transductive few-shot node classifier in Theorem A. The low-rank regularization promotes lower kernel complexity (KC), thus leading to a lower generalization bound for the test loss of the transductive classifier. Inspired by Theorem A. we further introduce LRA-LR-FGDM with a novel efficient LR-Attention layer, or the LRA layer, which applies self-attention to the output of the LR-FGDM encoder. LRA further reduces the KC of LR-FGDM, leading tighter generaliza- tion bound for the test loss than LR-FGDM Table 9 in Section G.3 demonstrates that LR LR-FGDM achieves substantially lower kernel complexity (KC) and tighter upper bound for the test loss com- pared to the baseline without low-rank regularization, and LRA-LR-FGDM further reduces KC, leading to an even tighter upper bound for the test loss than LR-FGDM. Furthermore, as shown in Table 1, Table 6, and Table 10, LR- AR-FGDM significantly outperforms ofate-of-the-art FSNC meth- ods across multiple graph benchmarks, and ARA-LR-FGDDE further improves the performance of LR-FGDM.
RELATED WORKS
2.1 FEW-SHOT NODE CLASSIFICATION (FSNC)
While GNNs for node classification are commonly trained in semi-supervised fashion (Kipf & Welling, 2016b), many efforts (Sun et al., 2020; Hamilton et al., 2017: Veličković et al., 2018) aim to reduce label reliance; however, they struggle with unseen classes at inference, motivating the study of FSNC. Most previous FSNC methods (Zhou et al., 2019; Finn et al., 2017; Yao et al., 2020; Snell et al., 2017: Liu et al., 2024; Wang al., 2022; Wu et al., 2024; Zhang al., 2025a) adopt meta-learning framework by training the FSNC model through a series of meta tasks. More recently, several works have incorporated contrastive learning into meta-learning enhance task-specific rep- resentation learning. (Liu et al., 2021a) and CPLAE (Gao et al., 2021) apply supervised contrastive losses within meta-tasks using augmented views. while PsCo (Jang et al., 2023) and MetaCon- trastive (Ni et al., 2021) perform unsupervised contrastive meta-learning. COLA (Liu et al., 2024) contrasts support and query prototypes to promote class-level consistency, and COSMIC (Wang et al., 2023a) leverages vulti-view contrastive regularization.
2.2 GRAPH DIFFUSION MODELS AND GENERATIVE DATA AUGMENTATION ON GRAPH
Score-based diffusion models (Song et al., 2021b) have achieved state-of-the-as art performance in diverse generative tasks (Ho et al., 2020; Song & Ermon, 2019; Gao et al., 2023a; Rombach et al., 2022; Baranchuk et al., 2022; Song et al., 2021 c;;;; Song & Ermon, 2020; Wang et al., 2025a). Graph diffusion models have emerged for synthetic graph generation (Niu et al., 2020; Haefeli et al., 2022; Jo et al., 2022; Zhou et al., 2024), with early works (Jo et al., 2022; Haefeli et al., 2022; Vignac et al., 2023) designing discrete diffusion processes over adjacency matrices. SaGess (Limnios et al., 2023) performs conditional generation of graphs inspired by LDM. However, these models primarily target graph-level generation, limiting their utility in node-level tasks such as FSNC. To enhance the performance of GNNs, node-level data augmentation has been applied to structure (Zhao et al.,
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2021b; Rong et al., 2020; Feng et al., 2022; Lai et al., 2024), features (You et al., 2020; Kong et al., 2022; Azad & Fang, 2024), and labels (Han et al., 2022: Wang et al., 2021; Verma et al., 2021: Zhao et al., 2024c). In FSNC, recent methods enhance the support set by perturbing node features and leveraging pseudo-laheled queries (Wu et al., 2022), or by using LLMM-hased prompting to generate synthetic support nodes for text-attributed graphs (Zhang al., 2025b). Generative data augmentation has been used to enhance GNN performance by generating synthetic nodes and edges to address class imbalance and enrich minority class features and connectivity (Zhao al., 202 lb; Zhou et al., 2024; Qu et al., 2021; Zhao et al., 2021a; a; Hsu et al., 2024; Gao et al., 2023b; Hsu et al,, 2023). However, these approaches often rely on GANs (Jia al., 2023; Wu et al., 2023; Wang et al., 2018; Liang et al., 2020; Yang et al., 2019), which suffer from training instability and poor alignment with real data distributions (Dhariwal & Nichol, 2021). To the best of our knowledge, FGDM is among the first to synthesize synthetic graph structures via diffusion models for FSNC.
3 FORMULATION
We aim to boost the performance of existing FSNC methods by augmenting the support set a few-shot task, thereby alleviating the data scarcity in each novel class.
The Pipeline of Integrating LR-FGDM with an Existing Few-Shot Learning Method. The proposed LR-FGDM serves as plug-in module to enhance existing FSNC methods, such as COS- MIC (Wang et al., 2023a) and COLA (Liu et al., 2024), by augmenting the support set. LR-FGDM consists of three steps for FSNC: (1) training FGDM, which includes learning a Hierarchical Graph Autoencoder (HGAE) on the original graph and a Latent Diffusion Model (LDM) on its latent space; (2) generating augmented graph by injecting synthetic support nodes and their edges into the original graph; and (3) applying an existing FSNC method to learn node embeddings, followed by training a low-rank transductive classifier the augmented support Figure 2 (a) illustrates the entire training pipeline with LR-FGDM. Figure 2 (b) illustrates the training of the LR-FGDM. and Figure (b) in the appendix illustrates the generation of the synthetic support nodes by LR-FGDM. LR-FGDM improves few-shot classification by expanding the support set, leveraging the benefits of stronger supervision as in prior augmentation studies (Wu al., 2022; Liu et al., 2025b).
Our LR-FGDM contains two com- ponents. which are the generation of an augmented graph with synthetic support data by FGDM, and few- shot learning with low-rank trans- ductive classification on the aug- mented graph. Our FGDM features a novel Hierarchical Graph Autoen- coder (HGAE) with efficient hier- archical edge reconstruction method and new prototypical regulariza- tion, detailed in Section 3.2. Then, the generation of an augmented graph with the synthetic support nodes and edges explained in Section 3.3. The low-rank transductive linear classifier for few-shot classification with theoretical guarantee is de- tailed Section 3.4.
Training the HGAE with Prototypical Regularization
Training of the LDM
Generation of the SGS
Training of the Low-Rank Transductive Classifler with Rank Regularization
(a) Training Pipeline LR-
Node Amribures
Prototypical Regulunization
Protolype
Reconstrucied Node Attributes
Edges
HGAE
LDM
LDM
HGAE
Reconstracted
Encoder
Forward
Reverse
Decoder
Edges
(b)i.
(.R.-GDMM
Figure 2: Figure (a) illustrates the entire training pipeline of the FSNC method with LR-FGDM, and Figure (b) illus- trates the training of LR-FGDM for synthetic graph struc- ture generation.
3.1 PRELIMINARIES
Few-Shot Node Classification (FSNC). FSNC assumes disjoint label sets across splits, denoted as Cuain- Cval. and Crest (Liu et al., 2024; Luo et al., 2024; Wang et al., 2023a; Zhao et al., 2024a). An n-way task requires the model to classify nodes into n distinct classes randomly sampled from Crest with only & labeled instances per class provided in the support set. Each task consists of labeled support set of n X l nodes and an unlabeled query set. The support set guides the model to learn a transductive classifier to predict the labels of nodes in the query set.
Attributed Graph and Notations. An attributed graph with N' nodes is denoted by g = (V,E, X). Here, V {v1, (21,,2,.... .., "'y) represents the nodes, and & C V x V represents the edges. Node
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attributes are given by X E RN''D, where each row X; RD corresponds to the attributes of node ey. The adjacency matrix A E {0, 1 x of graph G defines connections. Each row A, represents the connections of node v. The neighborhood [ii)= = (j Ai, = 1] includes node v; itself and all nodes connected v;- The notation [NV] denotes all natural numbers from 1 to N' inclusive.
[A]; stands for the i-th row of matrix A. I1-IIp denotes the p-norm of a vector or a matrix.
3.2 FEW-SHOT GRAPH DIFFUSION MODEL (FGDM)
Hierarchical Graph Autoencoder (HGAE) with Prototypical Regularization. To encode node 44 we first generate a latent feature of the node attribute X; as f(X,), where f(() is a Multi-Layer Perceptron (MLP) layer. To incorporate the information from the edges connected to us we add positional embeddings (Ma et al., 2021; You et al., 2019) to the node attributes of v,'s neighbors. For each neighbor (i),,, we modify the node attributes X Xj + pos(j), where pos(-) is a function converting the position index into an embedding vector (Vaswani et al., 2017). We apply two Graph Attention Network (GAT) (Veličković et al., 2018) layers aggregate the information in (X, j E (i)) into single latent feature Z, Next, we concatenate Z; with f(X,) to obtain the latent feature of node ot by Z, (l where f' is another MLP layer encoding the concatenated features to the latent space of LDM with lower dimension D'. After encoding a node oy in the graph to a latent feature Z, the decoder of the HGAE reconstructs the node attribute X, by three consecutive MLP layers and its associated edges A, by the hierarchical edge reconstruction method to be introduced later.
Prototypical Regularization. Existing FSNC methods (Snell et al., 2017; Laenen & Bertinetto, 2021; Ding et al., 2020; Lin et al., 2022) show that prototypical learning improves node embeddings by promoting intra-class compactness and inter-class separability. To align latent features of seman- tically similar nodes, we add a prototypical regularization to the HGAE loss, encouraging nodes within the same cluster to approach shared prototypes. The cluster assignments are obtained via sumi-supervised K-means (Basu et al., 2002; Bair, 2013) utilizing labeled nodes to guide cluster- ing while incorporating unlabeled nodes for better generalization to unseen classes. Let Pe E D represent the prototype of cluster e E [K], where K is the number of prototypes. The prototypical
regularization loss is defined as Lproro = Σm= Zi - P=(a l| where P=(() |T+col Ejevecs Zj.
(i) is the cluster index of node v, and V=(1) is the set of nodes in the cluster क (i)).
To address the quadratic complexity inherent in conventional GAEs (Zhai et al., 2018; Kipf & Welling, 2016a), we introduce a prototype-guided hierarchical edge reconstruction framework de- signed to promote efficient edge decoding. The edge reconstruction is conducted hierarchically based on clusters induced by learned prototype representations. We define an inter-cluster neighbor map C E {0, 1 xK where Cak = 1 indicates that node n2 connects to at least one node within the cluster represented by prototype k. Additionally, an intra-clusted neighbor map M E {0, 1)x's N' xxx constructed, where M, sigmifies that node 2 is connected to the s-th node within cluster ks with S denoting the maximum number of nodes in any cluster. In contrast to the Bi-Level Neigh- borhood Decoder (BLND) employed in DoG (Wang et al., 2025b) using balanced K --menns applied to node attributes, our method leverages prototype clusters learned jointly with the encoder, because nodes in the same prototype cluster have similar latent features, thus tend to connect with each other. The structure of the network used for the hierarchical edge reconstruction in the HGAE with prototypical regularization is illustrated in Figure 5 in Section F of the appendix.
Training the HGAE with Prototypical Regularization. For each node v, the hierarchical edge reconstruction method first reconstructs its inter-cluster neighbor map C, with one MLP layer. Af- ter that, the predicted cluster indices C(i) = {K E [K]]Cc = 1) are separately fed to an embed- ding layer to generate a set of class-conditional features ( = { g(k) E I Ik E C(i) using the class-conditional embedding method in Classifier-Free Guidance (Ho & Salimans, 2022), where g contains one text embedding layer followed by an MLP layer. Next, each of the class-conditional features g(k) E Z(i) is concatenated with the latent feature of the other branch for decoding the intra-cluster neighbor map by M. = g' o (k)), where g' is another MLP layer. The HGAE is trained by minimizing the sum of the node reconstruction loss, the hierarchical edge reconstruction
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loss, and the prototypical loss ppooto as follows,
| M ml
LUGAE =
X
+
(1)
Node Reconstruction Loss
Hierarchical Edge Reconstruction Loss
where I1-112 denotes the Euclidean norm. We perform detailed complexity analysis of the hierarchi- cal edge reconstruction method in Section C of the appendix. Table in Section 4.4 of the appendix demonstrates the improved efficiency of the proposed hierarchical edge reconstruction method com- pared to the decoder in regular GAF.
Training the LDM. Once the HGAE with the hierarchical edge reconstruction method is trained,
obtain a of latent representations Z = {Z, E IR'" Ug E V) encoding both node attributes and edges. Traditional class-conditional diffusion models typically condition on class labels, including DoG (Wang et al., 2025b), as illustrated in Figure (a). However, in FSNC, the classes in the support and query sets are novel and disjoint from those used during training. As the diffusion model trained prior test-time adaptation, it cannot directly condition on these unseen class labels. To overcome this limitation, we leverage the prototypical regularization introduced in the HGAE, which encourages the latent representations Z; to cluster around their respective prototype representations. These prototypes, computed as the mean latent representation of each cluster, serve as semantically meaningful and continuous conditioning signals. Instead of relying on discrete class labels, the LDM is conditioned directly on the corresponding prototype representation for each training node, enabling prototype-basee based conditional generation. As illustrated in Figure 3 (b), each latent feature is paired with its assigned prototype the conditioning input under the Classifier Free Guidance (CFG) framework (Ho & Salimans, 2022). This design allows the LDM to learn to generate latent features aligned with the semantic structure of the data without requiring access to class labels. The training algorithm of FGDM is presented in Algorithm in Section H of the appendix.
3.3 GENERATION OF AUGMENTED GRAPH WITH SYNTHETIC SUPPORT DATA BY FGDM
Generation of Synthetic Graph Structures with FGDM. Once the FGDM trained, we aim to generate synthetic graph structures consisting of synthetic support nodes and edges connecting to the original graph. We first obtain the cluster label of each of the support nodes obtained from the prototypical regularization, which is used to get the prototype representation for the conditional generation of the synthetic graph structure, as illustrated in Figure 4 (b) in the appendix. Let Vusp S V be the original support nodes and Vayn be the synthetic support nodes. Let the node attributes of Vay be X.yn and the affinity matrix encoding edges between the synthetic nodes and real nodes be Asya- Then the synthetic graph structure is denoted as (Vsyn:
Asya). Let M be the number of nodes in the synthetic graph structure. The adjacency matrix of the augmented graph is Aaug =
0) [A Asya A syn
and the node attributes of the augmented graph is X
[ X.yn]= E (M))+ The augmented graph, which is the combination of the original graph and the synthetic graph structure, is then denoted by 9 = (V U Vayn: Xmug: Augg). Let Ve Vsup U Van denote the augmented support set. For a n-way k-shot FSNC task, we generate the synthetic graph structures, consisting of M q X n x k synthetic support nodes and their edges connecting to the original graph, where q denotes the number of synthetic nodes generated per real support node, The value of y for different tasks on different datasets selected by cross-validation as detailed in Section G.5. The augmented support set V/2 then consists of (g + 1)nk support nodes with (q + 1)Ak nodes in each of the n novel classes. The augmented graph Goug is then encoded using existing few-shot graph encoders, such COSMIC (Wang et al., 2023a) and COLA (Liu et al., 2024), yielding the representation for all the nodes in the augmented graph, which is denoted as He E (N(N+M)xx The generation of the augmented graph is described in Algorithm 2 in Section H of the appendix.
3.4 LOW-RANK TRANSDUCTIVE LINEAR CLASSIFIER FOR FEW- FEW-SHOT LEARNING
Due to the inherent stochasticity of diffusion models (Ho et al., 2020; Rombach et al., 2022), the synthetic graph structures generated by LR-FGDM may introduce noise, leading to semantic mis- matches between synthetic support nodes and their labels (Azizi et al., 2023; He et al., 2022). To address this, we follow prior FSNC methods (Wang et al., 2023a; Liu et al., 2024) by training a transductive node classifier on embeddings from few-shot graph encoder. Motivated by the Low

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Frequency Property (LFP) (Rahaman et al., 2019; Arora et al., 2019; Cao et al., 2021; Choraria et al., 2022: Wang et al., 2024; 2025b), which suggests that class labels concentrate on top eigenvectors of the model's kemee gram matrix, introduce novel low-rank regularization for the classifier with theoretical guarantees.
Notation Definition. Let u E RN' be a vector, we use [ to denote a vector formed by elements of with indices in A for A C [NT. If u is a matrix, then [u]_ denotes submatrix formed by rows of u with row indices in A. 11-I|; denotes the Frobenius norm, and Il-1lp denotes the p-norm. Let Vrs = Ve U Vur denote all the nodes from the n novel classes in an n-way k-shot task, where Ve and Vu are the labeled support set and the unlabeled query set, respectively, with [Vll m. Let N denote the number of nodes in VES. L and U denote the indices of the nodes in Ve and Vu- Let Vis = {vi 12g ..... 2 ). where oa is the i-th node in Vrs- Let Y: E IR be the ground-truth one-hot class label vector for of in Vrs, and define Yrs == (2; -- y yN] R be the ground- truth label matrix defined on the novel classes for all the nodes in Vrs- Let Hrs R be the representations of all the nodes in Vis- We define F(W)- = Hps W as the linear output of the transductive few-shot classifier with W E Rdx= being the weight matrix. Let K be the gram matrix N of the node representations HES, which is calculated by K = Hrs E NNN. Let {x, 1
with Rn 2 72.... 2 Amin(N.a. 2 Amin(N = = be the eigenvalues of K.
Low-Rank Transductive Few-Shot Node Classification. In order to encourage the features HAs or the gram matrix K to be low-rank, we explicitly add the truncated nuclear norm ||K]| = EN t-1 -i-ro+11 X to the loss function of the transductive few-shot node classifier. The starting rank ro < min(N, N, d) is the rank of the features Hes we aim to keep in the node representation, that is, if [K]], lr = 0, then rank(K) K) = ro- Therefore, the overall loss function is
min L(W)= W
Σ 8: VEVE
KL (y:; [softmax (HІs W)l,) - - - Ing?
where KL is the KL divergence. T >> 0 is the weighting parameter for the truncated nuclear norm [K/. We use regular gradient descent to optimize (2) with learning rate 1) (0, x-) W is initialized by ( 0, and at the t-th iteration of gradient descent for t 2 1, W is updated by W(') W w (W) w-wu-n 1)- The optimal rank ro different datasets is decided by cross-validation as detailed in Section 4.1 of the appendix. Here | can be efficiently computed by the Nystrom method detailed in Section I of the appendix.
Motivation of the Low-Rank Regularization. We study how the information of the ground-truth class label defined on the novel classes is distributed on different eigenvectors of the feature gram matrix K by performing Section G.9 of the appendix that the projection of the ground truth labels for the novel classes mostly concentrates on the top eigenvectors of K, known as the Low Frequency Property (LFP) widely studied in other areas of machine learning (Rahaman et al., 2019; Arora et al., 2019; Cao et al., 2021; Choraria et al., 2022; Wang et al., 2024; 2025b). We remark that the low-rank regularization ensures that mostly only the low-rank part of the node representations HFs is used for the FSNC, so that our transductive node classifier trained by (2) is free of the noise in the high-rank part of the Hps, thus being robust to the noise in the synthetic graph structures introduced by LR-FGDM. The low-rank learning is also theoretically justified by Theorem A. in Section A of the appendix, showing that the low-rank learning reduces the kernel complexity and renders a tighter bound for the test loss.
projenrrocttionn in Section G.9 of the appendix. It is observed in Figure in
3.5 LRA-LR- IMPROVING LR-FGDM BY Low RANK ATTENTION
To further improve the performance of LR-MGDM, we introduce LRA-LR-FGDY. with a novel I.R-Attenton layer, or the I.AA layer, which applies self-attention to the output of the LR-FGDM encoder by F = BHFs, where HBS E xd is the low-rank node representations produced by the LR-FGDM encoder. is the attention output and B E N our new attention matrix in the LRA layer. We recall that the kernel gram matrix of the node features Hys is K = Hrs The
attention weight matrix B is set B K/Xi
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Efficiency of LRA with the Low-Rank Attention Weight Matrix B. The attention matrix in LRA is computed by B K K/A1, which implies rank(B) I min {N, d]. In practice, the embedding dimension d is significantly smaller than the number of nodes N. In all our experiments, we set d = 256, following standard settings in existing FSNC methods (Wang et al., 2023b; Liu et al., 2024). As a result, is inherently low-rank, and the LRA operation BHrs = HrsHI, Hos X can be computed in ((Nd)) time by first computing the d x d matrix H Hrs and then multiplying with Hps, without explicitly constructing the dense N x N attention matrix.
The gram matrix Kr of the node representations F E R is then KF FF K Let (A.) J. be the eigenvalues of KF with X 2 X2 2 ...AN 2 O, then we have X, X /27 for every i E [N]. Noting that A; = πu /2 < X; due to the fact that λι 2 X for all i E [N], therefore, the LRA layer reduces the kernel complexity of the kernel gram matrix K, because the KC of KF is always not greater than that of K. We then train a transductive classifier on top of F similar to Section 3.4 by minimizing the loss function
min L(W)= W
Σ i: oeVE
(y; [softmax
+
(3)
Such linear classifier trained with the LRA layer through the optimization of (3) is termed LRA- LR-FGDM. It then follows from the above discussion and the upper bound for the test loss (5) in Theorem A.1 that ARA-LR-FGDM has lower KC, so that the test loss Uhea (t) of LLA--LR-FDDM can be even lower than that of LR-FGDM, suggesting a better prediction accuracy of LRA-LR- FGDM than LR-FGDM. This is empirically justified in Table 9 where L.A-LR-FGDM exhibits lower KC and lower upper bound for the test loss than that of LR-FGDM.
4 EXPERIMENTS
We evaluate the performance of the LR-FGDM for shot augmentation combined with the low-rank regularization for FSNC. In Section 4.1, we present the implementation details of the proposed LR- FGDM. In Section 4.2, we present the results for different FSNC settings. An ablation study on the effectiveness of the prototypical regularization in the HGAE and low-rank regularization on the few-shot classifier is performed in Section 4.3. In Section 4.4, we perform the efficiency analysis of LR-FGDM. Additional experiment results are presented in Section G the appendix. In Sec- tion G.I. we present the results for FSNC on three additional graph datasets. We also compare the LR-FGDM against existing state-of-the- shot augmentation methods in Section G.2. In Sec- tion G.3, we study the effectiveness of LR-FGDM in reducing the kernel complexity of the kernel gram matrix and the upper bound for the test loss of the transductive linear classifier in L.R-GDDM. heterophilic graph dataset, the Roman- Empire dataset (Platonov et al., 2023). In Section G.5, we present the details about cross-validation used to select the number of synthetic support nodes. In Section G.6, we perform the sensitivity anal- ysis of the hyperparameters т, K, and q. We have proposed the Frechet Node Distance (FND) and the Frechet Edge Distance (FED) to validate the faithfulness of the synthetic support nodes and the associated edges with comparison to existing shot augmentation methods in Section G.7. The statistical significance of the improvements achieved by L.R-FGDM in Section 4.2 and Section 4.3 validated by the student (-test detailed in Section G.8. In Section G.9, we analyze how the ground- truth label signal distributes across the eigenvectors of the feature Gram matrix. In Section 10 we study an automatic eigengap-based strategy for selecting To. In Section G. 1. we provide t-SNE visualization comparing the embeddings of synthetic nodes generated by LR-F with those of real nodes in the novel classes. In Section G. conduct ablation study to evaluate the impact of the LDM. In Section G.13, we report the training time of LR-FGDM and LRA-LR-FGDM on the augmented graphs. In Section 1.,, evaluate whether augmented graphs generated by LR- FGDM preserve key graph structural properties. In our experiments, we apply LR-FGDM on top of existing FSNC methods, COSMIC (Wang et al., 2023a) and COLA (Liu et al., 2024), which are the most recent slateoff-hhe-at FSNC methods with the best performance.
In Section G.4, we study the effectiveness of LR-FGDM
4.1 IMPLEMENTATION DETAILS
We conduct experiments for FSNC on CoraFull (Bojchevski & Günnemann, 2018), ogbn-arxiv (Hu et al., 2020), Coauthor-CS (Shchur al., 2018), DBLP (Tang et al., 2008), Roman-Empir (Platonov
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Published as a conference paper at ICLR 2026
et al., 2023), Amazon- Mmazon-Computers, Amazon- (Shchur et al., 2018), and Citeseer (Sen et al., 2008), with details in Section D.1 of the appendix. The training settings of LR-FGDM and the hyper-parameter tuning are described in Section D.2 of the appendix.
Table 1: The overall FSNC results of all methods under different settings. The best result is in bold, and the second-best result is underlined. The statistical significance of the results is deferred to Table 17 of the appendix.
Dataet
CocaFoll
ogllnsaraie
Took
2-way I-shot
2-way S-shot
S-ttay
I-shot
S-ery
S-dhon
2-way I-shot
2-way
S-shot
S-uny
I-chot
S-way Swcher
ProneNin (Snell et 2017)
2.42
72.31
2.55
$2.45
1.6
51.54 1,88
2.50
75.83
2.00
53.31 1:21
Mena GNN (2hos et al. 2019)
75.25 3.55
84,20
2,80
55.03
70,30 4 2.02
02.52
3.41
70.15 2.65
27.14
1.54
31.32 1: 1.71
GPN (Ling atal.. 2020)
74.29
3.47
85.28
2.53
52:75
2,32
72.82 1.88
64.00
5.71
79.78 11 3.50
37.81
2.34
50.30
1 2.13
G Meta Illungg& Zimik 2020)
78.23
9.41
20.40
2.04
03:44
75.84
1.70
60.00
5.32
75.36
2.89
31.45
1.70
47.16
1.73
TENT (Wing et 2022)
77:15
26:00
2.01
55-41
70 10:10-
173
70.30
2.85
81.35
2:77
18:25
1:73
01.38
1.72
KD-RSNC (Wuetal. 20024)
83.02
2.08
90.08
2.12
71.55
85 06.89-1
5
70.90
3.15
84.67
2.39
52:74
64.91
1.20
NomPPop /Phang
20225a)
83.01
2.61
03.87
2.39
85.47
70.33
5.10
84:36-
52.37
2:11
64:28
1.72
STAR (Line al. 2025a)
1.08
96.96
1.48
75.85
57.51 1.55
70.45
2.03
86.1
1.75
00.98
1:25
DaG (Wing ct 202550
1.08
96.35
1.82
75.13
56.47 1.18
77.33
86.80 2:21
58.42 1.47
66:20
1.85
COSMIC (Wang et 20133a)
2.70
96.51
T499 2.40
56.31
2.17
70.71 =
85.10
50.28 2.30
65.42
(Liu
1.92
06.17
T5.47 2.12
57.83
77.32
86.42
55.24 2.04
67.02
COSMIC (.FFGDM
09.21
2.58
96.74
16.3 2.15
58.81
1.53
77.08 2.75
K5-85 2.00
07:30
COSMIC (.RA--B--FGIMD
99.78
97.29-
1.88
77.41
89.47
1.79
78-33 2.04
87.80
1.73
1.49
COLA (LR-FGDM)
1.74 57.51
07.38-
1.67
80.05
1.72
70.00
68.34
57
1.57
COLA L.FGMM
97.97
1.65
1.80
70.02
1:08
1.84
1.71
70.22
Dannect
Commbee-C
DELF
Tiok
Zoway
I-dhot
Toway
Sahoe
S-aoy
L-chet
Seny
S-dton
Toway
L-shon
2uway S-ahel
S-wny
1-chot
5-way
S-hhe
PrnneNer /Snell et at. 2017]
10.02
2.70
11.00
2211
02.15
1,02
10.25
1:30
C1.37 2:50
12.11:22 2:73
11.01 1::1::10 1:55
21.20 1 1.88
Meta CNN (2hon et al. 2019)
$3.00
2.00
90.11
2.17
52.86 1 2.14
68.
1.40
$2.00 0 3:20
85.15
3.29
07.24
2.72
72.15 1: 2:40
GPN (1mng at... 2020)
$4.51
2:73
90.30
0 1.90
03 a 2.07
S1.79
1:18
79.35
5.46
85.85 1: 2.61
50.30 08.38 1: 2.40
75.46 1.87
G-Mets (flangg 8 7imik. 2020)
54.19
2.50
91.00
1.01
59.05 8: 2.16
74.18 01 1:29
80.2
00 = 5.20
65.55 2:56
13.322 0 2.70
75.82 + 2:11
TENT (Wong et al. 2022)
57.
2.58
91.75-
1.60
03:70
1.88
70.00
1.19
64.00
2.75
90.05
2:51
61.56 2.23
79.51 + 2.0c
KD-FSNCCWW (Wu 2024)
80.75
2.30
95.21
2.01
07.05
1.06
61:02
1.17
91.81
2.41
91:37
1.70
74.85
+ 2.15
63.75 + 1.91
NomPPop (Zhun st. 31255a
60.51
2.41
95.22 = 1.97
07.16
1.66
54.61
1:14
91.52
2.45
94.05
1.72
75.59
2.38
64.32
1.89
STAR (Lin et al. 2025a)
01.28
1.15
96.11 = 1.86
09:25
1.25
57.00
21 1.38
98.30
0000110 1.47
07.52
1.65
77.14 4: 1.35
67.30 1.05
DoG (Wang et al., 20250)
01.10
1.84
96.88 = 1.58
68.96
1,80
57.85 2: 1.41
98.55
1.36
96005
1.22
T8.87 2: 1.37
87.30
1.25
COSMIC (Wung et al., 20123a)
00.29 2.30
96.32 = 1.98
69.21
1 1.63
55.47
1::1
02.35
2.52
94.82
1.00
76.02
2.23
85.31
1.92
CDL A (1is al., 2014)
01.53+2.00
96.78 - 1.84
70.16
1.57
98-16
2.07
05.02
+ 1.05
78. 2.05
67.23
1.87
COSMIC CI CERGD
02.05.2220
96.71 1.07
10.11
2 1.36
54.72
1.08
51.76
= 2:39
96336
1.65
18.06 2.00
61.40 1.1
COSMIC (LRAALLR-FGDMM
02.91 + 1.58
97.24 1.59
70.90
1.96
68.21
1:12
96.83
2.30
97.44
1.41
79.11 0: 1.92
57.06 + 1.58
COLA d.R-PGDM)
53.81 2 1.85
97.91 = 1.56
72:08
1.43
80.83
2: 1:21
90.85
= 2.08
07.98
1.47
50.16 1.58
89.51
= 1.65
COLA (..RA-IR-FGGM)
94.33
1.71
98.41 + 1.49
73346
+ 1.29
90.39
1.03
96.
+ 1.92
98.52 + 1.39
80:65
1.75
96.07 + 1.57
4.2 RESULTS
We compare the performance of the proposed LR-FGDM with state-of-the-art FSNC methods, in- cluding ProtoNet (Snell et al., 20,7), Meta-GNN (Zhou et al., 2019), GPN (Ding et al., 2020), G-Meta (Huang & Zitnik, 2020). TENT (Wang al., 2022), KD-FSNC (Wu al., 2024). Norm- Prop (Zhang et al., 2025a). COSMIC (Wang et al., 2023a). COLA (Liu et al., 2024), and STAR (Liu et al,,, 2025a). We also compare LR-FGDM with the diffusion--bsse synthetic graph structure gener- ation method DoG (Wang et al., 2025b). Since DoG requires label-conditionng during training and generation, which is unavailable for unseen classes in few-shot settings, we employ semi-supervised K-means (Basu et al., 2002; Bair, 2013) to obtain pseudo labels as conditioning signals for DoG. The number of clusters and the number of synthetic nodes are all decided by cross-validation. We integrate LR-FGDM and LRA-LR-FGDM into COSMIC and COLA. resulting in four variants, denoted as COSMIC (LR-FGDM), COLA (LR-FGDM), COSMIC (LRA-LR-FGDM) and COLA (LRA-----GGDM). The experiments are conducted for 2-way and 5-way classification tasks, each with 1-shot and 5-shot settings following (Liu et al., 2024; Wang et al., 2023a). The mean accu- racy and standard deviation across 20 independent runs for each setting are reported. Is observed in Table 1 that LR-FGDM consistently improves the performance of COSMIC and COLA on all the datasets. For example, COLA (LR-FGDM) outperforms COLA by 2.29% on Coauthor-CS for the 5-way 5-shot FSNC. LRA-LR-FGDM models show further improved performance over our LR-FGDM models, which significantly outperform the COSMIC and COLA. The results on the het- erophilic graph dataset, Roman- Roman--mmire, and three additional graph datasets, Amazon-Computerr Amazon-Photo, and Citeseer, are deferred to Table 10 and Table the appendix.
4.3 ABLATION STUDY
To thoroughly study the effectiveness of the prototypical regularization in the HGAE and low-rank regularization on the classifier, we conduct an ablation study CoraFull, ogbn-arxiv, Coauthor- CS, and DBLP under the 5-way 5-shot setting for FSNC. We evaluate three variants of the COLA (LR-FGDDM, which are the COLA (LR-FGDM) without the prototypical regularization, COLA (LR-FGDM) without the low-rank regularization, and COLA (LR-FGDM) without both the low- rank regularization and the prototypical regularization. It is observed from Table 2 that both the

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Published as a conference paper at ICLR 2026
low-rank regularization training the few-shot node classifier and the prototypical regularization on training the HGAE play important roles in improving the performance of the baseline method.
Table Ablation study on the low-rank regularization and the prototypical regularization. The study performed under the 5-way 5-shot setting for FSNC. The statistical significance of the results is deferred to Table 18 the appendix.
Method
CoraFull
ogbn-arxiv
Coauthor-CS
DBLP
COLA (Liu al., 2024)
87.83
67.52
87.54
87.23
COLA (LR-FGDM) w/ both low-rank and protorypical regularization
88.12
67.91
87.93
87.55
COLA (LR-FGDM) w/o low-rank regularization
88. 74
68 38.60
88.72
88.28
COLA (LR-FGDM) w/o prototypical regularization
88.79
68.45
89.02
88.64
COLA (LR-FGDM)
89.66
69.63
89.83
89.51
COLA L.RA-LR-FGDM.
90.32
70.22
90.39
90.07
4.4 TRAINING EFFICIENCY ANALYSIS
To study the efficiency of the FGDM, we compare the training time between our HGAE with hi- erarchical edge reconstruction and the regular GAE without hierarchical edge reconstruction. In addition, we also compare the time for the generation our FGDM and FGDM without hierar- chical edge reconstruction in its HGAE. All evaluations are conducted using single Nvidia A100 GPU. It is observed from the results in Table that the hierarchical edge reconstruction method significantly reduces the computation cost of the training and synthetic graph structure generation. For instance, the training of GAE without hierarchical edge reconstruction takes over five times the training time of our GAE with hierarchical edge reconstruction on ogbn-arxiv. In addition, the hier- archical edge reconstruction method also significantly reduces the time for synthetic graph structure generation. For instance, the data generation without the hierarchical edge reconstruction method takes over four times the data generation time of our FGDM on ogbn-arxiv.
Table 3: Time for the training of GAE and LDM in FGDM and data generation with FGDM on different datasets. The training time for the FSNC models on the augmented graphs generated by FGDM shown in Table 22 of the appendix.
Training Time (seconds)
Generation Time (s/sample)
Datssets
HGAE
GAE wio lierarchical Edge Reconstruction
FGDM
FGDM wio
Edge Reconstruction
CorFull
41
129
154
0.067
0.073
Coouthor-CS
52
145
179
0.074
0.088
ogbn-arxiv
301
1690
315
0.130
0.426
DBLP
0.049
CONCLUSION
In this paper, propose novel node-level graph diffusion method with low-rank feature learn- ing for FSNC, termed Low-Rank Few-Shot Graph Diffusion Model or LR-FGDM. LR-FGDM ad- dresses the limitation of data scareity in few-shot settings by augmenting the support through a novel node-level graph diffusion model and enforcing low-rank regularization on the training of the few-shot node classifier. FGDM integrates Hierarchical Graph Autoencoder (HGAE) with a hierarchical edge reconstruction method and a Latent Diffusion Model (LDM). The low-rank regularization motivated by the Low Frequency Property (LFP) and theoretically justified by theorem to show lower generalization eттоr. Inspired by the novel theorem, we further introduce LRA-LR-FGDM with novel LR-Attention layer, or the LRA layer, which further reduces the KC of FR-MGMM leading to tighter generalization bound for the test loss than LR- LR-FGDM. Extensive experiments on multiple graph benchmark datasets show that LR-FGDM significantly improves the performance of few-shot node classifiers, demonstrating superior generalization capabilities com- pared to state-of-the-art methods.
ACKNOWLEDGMENTS
Y. Wang and Y. Yang are supported by the 2023 Mayo Clinic and Arizona State University Alliance for Health Care Collaborative Research Seed Grant Program under Award No. AWD00038846 and by the NIH grant under Award No. 10T20D037955 U.S. National Science Foundation (NSF) under grant I1S-2229461.
C. Zhao and H. Liu are supported by the
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A THEORETICAL JUSTIFICATION FOR THE LOW-RANK REGULARIZATION
We have the following theoretical result, Theorem A. on the Mean Squared Error (MSE) loss of the unlabeled query nodes measured by the gap between [F(W, t)l and [Yssly when using the low-rank feature Hps with ro E [N], which is the generalization eтror bound for the linear transductive classifier using F(W) to existing works such as (Kothapalli et al., 2023) that use the MSE to analyze the optimization and the generalization of GNNs, we employ the MSE loss provide the generalization error of the node classifier in the following theorem. It is remarked that the MSE loss is necessary for the generalization analysis of transductive learning using transductive local Rademacher complexity (Tolstikhin al., 2014; Yang. 2025a;b). We use ( denote the i-th column of matrix A, and suppose that the eigendecomposition of K is K = UEUT. T, where U is an orthogonal matrix and Σ
H W predict the labels of the query nodes. Similar
a diagonal matrix with its diagonal elements being X 1"
R5 is an eigenvalue of with U()
i=1
as the corresponding eigenvector for all E [N].
Theorem A.1. Suppose that Ve is sampled uniformly without replacement from Ves, and Vi
VFs Ve is the query set containing the test nodes where U/ is the set of the indices of nodes in Vu
where Vell m and [Vul u. Suppose that for each E [n]] the j-th column of Y YFs. Y (1) satisfies
N
FS
12
for some positive number μ. Then after the t-th iteration of gradient descent on the training loss
L(W)= == 1/(2m). Enjeve lyy, - [Hys [H,sW],11, 112 for all 2 1, for every x > 0 and every d (0, 1), with probability at least - exp(- -r) - 8, we have
W1,1l
1 (tea = |F(W, 1) - YrSle
L(K, YES, KC(K) + N Nu,,,6
CO
(5)
where
min {u, m}
Nu,m,& 14g2(4 min {u, m) /8)*
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co is positive number depending n, ja, and with TO | Ni Kis
More-
over, L(K, t) = [Im. </.,c) (Ysle l F' Ke,e = [Hss (His] /m, and KC is
the kernel complexity of the kernel gram matrix K Hs /N defined KC(K) min ro ] + | ( vm
This theorem is proved in Section B of the appendix. Detailed explanation about Theorem A.I is deferred to Section B.1 of the appendix.
B THEORETICAL RESULTS
We present the proof of Theorem A.1 in this section.
Proof of Theorem A.1. can be verified that at the t-th iteration of gradient descent for 2
have
W w(-1) m
Hrsle HFs w - YrS) -YFs
(6)
It follows from (6) that
[esl W(t Hesle - Tk HPs W((-1) YFS c
where ke,c c [1s]e T /m E Rmxm With F(W, 1) His W()), it follows from (7) that [F(W YrS) = (Im c)) [(W, 1- Y
then follows from the above recursion that
F(W,t) Yıslo= = (Im RKc,c)² YYslle
(8)
It can be verified that
F(W, (TOP)), Vi [N], [n].
We apply (Yang, 2025b, Eq.
(B.70)) obtain the following bound for the test loss
[[F(Wtt) YrShu ll::
1 [F(, Yeslelle + co min r(u, m, Q) + co oSQsn
Yrsheelf m
(9)
Nu.m,
with
X
Σ
r(u, m, (=: Q
(==++++++) u m
+
q g=
u
778
where co is positive number depending n, μl, and To with Ti maxie[N] Kii-
It follows from (8) and (9) that for every E [N], we have
1 m
T)Ke,)) [Yssle 2 (Im
[F(, 1) Y 1l <
Zg
(
1 + u m
g
g=ro+1
Co
+ corD
+
N Nu,m,ō
II kc.e) [Ysllc
+ coro
(1 (++
+ co |K
V=++
+
Cox
Nu,m,8
(10)
where ① (8) and EN Eg R, |K],, (5) then follows directly from (10).
n
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B.1 FURTHER EXPLANATION OF THEOREM A.
Define F(W. His W the output of the classifier after the f-th iteration of gradient descent for / 2 1. It is noted that (t) is the test loss of the unlabeled query nodes measured by the distance between the classifier output F F(W,U () and Yss. There are terms on the upper bound for the test loss in (5), L(K, YES. t) and KC(K), which are explained as follows. L(K, Yss, Fs.st) t) corresponds to the training loss of the node classifier with the ground truth label for the novel classes. KC(K) is the kernel complexity (KC), which measures the complexity of the kernel gram matrix from the node representation HIs- We remark that the TNN |||| appears on the RHS of the upper bound (5), theoretically justifying why we learn the low-rank features K for FSNC by adding the TNN || to the training loss. Moreover, when the low frequency property holds, L(K, Y t)
would be very small with enough iteration number t. K = Hps Hps is approximately a low-rank matrix of rank ro since Hes approximately rank-To matrix with its TNN optimized through the optimization of the encoder of the HGAE. A smaller ||,,, is obtained by optimizing the training loss in Equation (2), which in turn ensures a smaller kernel complexity (KC) defined in Theorem A. contributing to a smaller generalization bound for transductive node classification.
C COMPLEXITY ANALYSIS OF THE HIERARCHICAL EDGE RECONSTRUCTION METHOD
In our work, we have proposed an efficient hierarchical edge reconstruction method to reconstruct the edges connected to a node in the graph. To show its efficiency, we analyze the inference time complexity and the parameter size of the HGAE with the hierarchical edge reconstruction method. For comparison, we also analyze the inference time complexity and the parameter size of GAE, where the hierarchical edge reconstruction method is replaced by a regular edge decoder that directly reconstructs the adjacency matrix A (Kipf & Welling, 2016a). For ease of comparison, we denote the number of parameters and inference cost of all the MLP and GAT layers except the hierarchical edge reconstruction process as SMIP and CMIP, respectively. For node ey in the graph, let d; E Csk be the number of clusters predicted to be connected U,. Let D' be the dimension of the input feature for the hierarchical edge reconstruction. The inference time complexity of HGAE with hierarchical edge reconstruction is O(K + d, + CMLP), where O(KD') the additional complexity for computing the inter-cluster neighbor map and encoding the cluster indices. D'dD'M the computation cost for computing the intra-cluster neighbor map. In contrast, the inference time complexity of regular GAE with regular edge decoder is O(D' D'M + CMLP)- We note that d, is upper bounded by the degree of the node uy In most graph datasets, the average degree of nodes is usually very small. For instance,
CoraFull, where the average node degree is 6.41, we have d; & 6.41. As result, D'(K + M) < M. For example, setting = 200 and M = 100 on Pubmed, we find that the inference time complexity of HGAE with hierarchical edge reconstruction is OS41DD' + CMLP), which is much more efficient than the regular edge decoder whose inference time complexity is (20000 + CMLP). In general, the inference time complexity of HGAE with hierarchical edge reconstruction is much lower than that of GAE with a regular edge decoder.
Table 4: Statistics of the graph datasets.
Datasct
# Nodes
# Edges
# Features
# Classes
CoraFull
19,793
63,421
8,710
70
ogbn-arxiv
169.343
1,166.243
128
Coauthor-CS
18,333
81,894
6,805
15
DBLP
40,672
144,135
7,202
137
Roman-Empire
22,662
32,927
64
18
Citeseer
3,327
4,732
3,703
AmazonnCoopuute
13,752
245,861
767
10
Amazon-Photo
7,650
119,081
745
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D ADDITIONAL EXPERIMENT DETAILS
D.I DATASETS
To evaluate the performance of method FSNC, we conduct experiments on eight widely used real-world benchmark datasets, which are CoraFull (Bojchevski & Günnemann, 2018), ogbn- arxiv (Hu et al., 2020), Coauthor-CS (Shchur et al., 2018), DBLP (Tang al., 2008), Roman- Empire (Platonov et al., 2023), Amazon-Computery Amazon- Photo (Shchur et al., 2018), and Cite- seer (Sen al., 2008) with their statisties summarized in Table CoraFull an extended version of the Cora dataset, constructed from the entire citation network, where nodes represent papers and edges denote citation links; node classes correspond to paper topics. ogbn-arxiv is a directed ci- tation graph derived from the arXiv Computer Science category in the Microsoft Academic Graph (MAG) (Wang et al., 2020a), where nodes are arXiv papers and edges represent citation relations. Node labels are based on 40 CS subject areas. Coauthor-CS is co-authorship graph extracted from MAG during the KDD Cup 2016 challenge, where nodes denote authors and edges indicate co-authorship. Node features are derived from paper keywords, and node classes correspond the authors' most active research fields. DBLP is another citation network in which nodes represent papers and edges denote citation links. Node features based on paper abstracts, and labels corre- spond to publication venues. Roman-Empinn is synthetic dependency graph designed to simulate extreme heterophily where adjacent nodes often belong to different classes. Nodes represent words and edges reflect syntactic dependencies, with class labels assigned based on grammatical roles. Amazon-Computee and Amazon- Photo are two product co-purchase networks from the Amazon dataset, where nodes represent products and edges indicate frequently co-purchased items. Node features are extracted from product reviews, and class labels represent product categories. Citeseer a citation network where nodes are scientific publications and edges represent citation links. Node features are TF-IDF weighted word vectors, and classes correspond to research topics.
D.2 TRAINING SETTINGS OF HGAE AND LDM
The training of the HGAE is divided into two phases. In the first phase, we pre-train the HGAE by only minimizing the node reconstruction loss and the edge reconstruction loss in Equation (1) for 500 epochs. In the second phase, we minimize LUGAE with the prototypical loss for another 500 epochs. We use the Adam optimizer with learning rate of 0.001 for the training. The weight decay is set to x 50 We train the LDM in the LR-FGDM after finishing the training of the HGAE. We use the Adam optimizer with learning rate of 0.0002 to train the LDM for 1000 epochs.
The rank parameter To and the weighting parameter T associated with the TNN loss are selected through crossvaalidation tailored to each dataset. We define the rank as ro [ min {N, d]1, where represents the rank ratio and d is the dimension of the learned node representations. The hyperparameter ? searched over the set (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9), while the TNN weight r is chosen from (0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5}. The number of the prototype clusters, K, is selected from {5, 10, 15, 20, 25).
E ILLUSTRATION OF THE SYNTHETIC SUPPORT NODE GENERATION BY LR-FGDM
Figure illustrates the encoding and decoding process of the node attributes and the associated edges by the LR-FGDM and the DoG (Wang et al., 2025b), with the difference between LR-FGDM and DoG marked in red.
Node Attrivntes
Clustering
Pscudo Label
Node Altrivutes
Recendracted
Nosee Attibunes
Prosiiypical Rogalarrrtion
Proietype
Node Atzihutes
Recossiiatee
Edges
DuG Easeder
LIDS
LIM
Desode:
Recemtracted lidges
Edge
Ercoder
RGAAE
Forwand
LDM
Beversi
LDM
HGAB Decodar
Resmsstrccll Edacs
DoG
FR-MGDM
Figure 3: Figure (a) and (b) illustrate the training of the DoG (Wang et al., 2025b) and the training of the LR-FGDM, respectively, with the difference between LR-FGDM and DoG marked in red.
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Figure illustrates the generation of the synthetic support nodes and the associated synthetic edges by the LR-FGDM and the DoG (Wang et al., 2025b), with the difference between LR-FGDM and DoG marked in red.
Support Node
Pscudo Label
Associated Synthetic Edges
Support Node
Nearest Prototype
Associated Syathetic Edges
Gauzaian
LDM
HIAAD
Systhetic
Gaussian
LDD
RGAA Decoder
Synthetic Support Node
Noise
Reverse
Decoder
Support Node
Nosse
Reverse
(a) DoG
(b) LR-FGDM
Figure 4: Figure (a) illustrates the generation of the synthetic support nodes and the associated synthetic edges by the DoG (Wang et al., 2025b). Figure (b) illustrates the generation of the synthetic support nodes and the associated synthetic edges by the LR-FGDM.
F DETAILS AND STUDIES ON THE HIERARCHICAL EDGE RECONSTRUCTION METHOD
Figure 5 illustrates the structure of the network used for the hierarchical edge reconstruction in the HGAE with prototypical regularization. In contrast the Bi-Level Neighborhood Decoder (BLND) employed in DoG (Wang et al., 2025b) using balanced K-means on node attributes, our method leverages prototype cluster assignment and prototype representations learned jointly with the encoder of the HGAE with prototypical regularization, because nodes in the same prototype cluster have similar latent features, thus tend to connect with each other.
Hierachical Edge Reconstruction Loss
Prototype Representation
Pz(1)
C
Cluster Assignment
m(1)
M
MUL
C
Intra-Cluster
Neighbor Map
Z
Embedding Layer
Latent Feature
Inter-Cluster
MLP
Neighbor Map
Concat
MIL
M.x
Concal
Ma
Figure 5: Illustration of the network architecture for the hierarchical edge reconstruction method in the HGAE with prototypical regularization.
To validate the effectiveness of the hierarchical edge reconstruction method compared to the BLND proposed in DoG (Wang et al., 2025b), we perform an ablation study by comparing the perfor- mance of the LR-FGDM with an ablation model where the hierarchical edge reconstruction module is replaced by the BLND in DoG (Wang et al., 2025b). The ablation model is denoted as LR- FGDM (BLND). The study is performed for the 5-way 5-shot FSNC task on CoraFull, ogbn-arxiv, Coauthor- CSS, and DBLP. It is observed in Table that LR-FGDM consistently outperforms LR- FGDM (BLND) across all datasets. For example, LR-FGDM achieves 1.16% improvement on ogbn-arxiv. highlighting the superiority of the proposed hierarchical edge reconstruction method in capturing meaningful structural patterns for few-shot learning. These results underscore the bene- fits of leveraging prototype-guide inter-cluster and intra-cluster connectivity over purely attribute- based neighborhood decoders like BLND.
Table 5: Performance comparison between the proposed hierarchical edge reconstruction method and the Bi-Level Neighborhood Decoder (BLND) DoG (Wang et al., 2025b) for the 5-way 5-shot FSNC task.
Data
CoraFull
ogbn-arxiv
Coauthor-CS
DBL.P
FR- (BLND)
88.52
68.47
88.96
88.37
LR-FGDM
89.66
69.63
89.83
89,51
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G ADDITIONAL EXPERIMENT RESULTS
G.1 FEW-SHOT NODE CLASSIFICATION ON CITESEER, AMAZON COMPUTERS, AND AMAZON PHOTOS
In this section, we further validate the effectiveness of LR-FGDM for FSNC on additional datasets, including Citeseer (Sen et al., 2008), Amazon Computers (Shchur al., 2018), and Amazon Pho- tos (Shchur et al., 2018). Due to the limited number of classes in these three datasets, we follow (Wu et al., 2024) and only perform FSNC under the 2-way 1-shot and the 2-way 5-shot settings. is observed in Table 6 that LR-FGDM consistently improves the performance of COSMIC and COLA on all three datasets and significantly outperforms all competing FSNC methods. To demonstrate the statistical significance of the improvements achieved by LR-FGDM over the baseline methods, we perform 1-tests on the few-shot classification accuracies obtained from 20 independent few-shot tasks for each setting and each dataset. It is observed in Table 7 that models enhanced by LR-FGDM consistently yield statistically significant improvements over the corresponding methods across all few-shot settings with p-values p < 0.05.
Table 6: The overall FSNC results of all methods under different settings for Amazon-Computee Amazon-Phot Photo, and Citeseer. The node classification accuracy and its standard deviation are in %. The best result under each setting in bold, and the second-best result is underlined.
Method
Amazzon C Compulers
Amazon Photo
Cieeeeer
I-shat
2-way
3-shot
I-shot
Eway S-shit
Z-way
T-shol
2-wny 5-shot
(Snell et al., 2017)
66.67 + 2.54
03.1
+ 1.80
66.74 = 2.08
12.04 = 1.94
67.39
1.65
79.02 + 2.33
Meta-GNN (7hou et al., 2019)
60.54
2.79
68.36 + 2.15
09.34 + 2.03
76.20
1.87
67.41
1.60
79.08
2.27
GPN (Ding el al., 2020)
63.85
2.31
71.02
1 2.07
72.35
1.92
17.88
1.74
69.12
1.68
80.02
2.14
G-Meta (Huang 7imik, 2020)
62.56 3.11
71.47 +- 2:97
70.18 + 2.10
77.45
1.81
65.53
1.58
78.01
1.80
TENT (Wang el al., 2022)
77.74 3.16
86.06 1 2.16
84.62 1 2.78
86.53
2.00
75.03
2.81
85.31
2.42
KD-FSNC (Wu et al., 2024)
86.92 1.74
95.30
4: 0.85
91.08 4 2.17
96.60 = 0.41
79.48
2.62
86.
1.32
NormPrap (Zhang al., 2025u)
85.10 2.08
94.35
1.30
90.42
2.21
96.15
0.53
78.41
2.37
85.60
1.61
COSMIC (Wang
2023a)
87.12 1.82
95.60 +: 1.01
91.54 4: 2.04
96.12
0.12
79.77
2.20
86.23
1.53
COLA (Liu et al., 2024)
87.52 1.78
95.89
1.02
91.74
1.04
96.38
0.33
80.13 2.11
87.02
1.30
COSMIC (LR-FGDM)
88.63 1.70
96.71 + 0.97
92.38 - 1.91
97.22
0.29
80.92
2.00
87.62 1.27
COSMIC FGGDMM
89.12
1.58
97.25 0.88
92.87 = 1.79
97.71 0.27
81.38
1.86
88 15
1.19
COLA (LR-FGDM)
89.14 + 1.66
9T.21 + 0.89
93.41 1.75
97.15
0.26
81.73 1.84
88.21 + 1.28
COLA (LRA-LR-FGDM)
89.63
1.55
0.82 97.74
93.92 - 1.63
97.96 + 0.25
82.25
1.71
88.71 1.16
Table 7: p-values from t-tests comparing COSMIC (LR-FGDM), COLA (LR-FGDM), COSMIC (LRA-LR-FGDM) and COLA (LRA-LR-FGDM) against their corresponding baseline methods, COSMIC and COLA, in Table 6.
Dataset
Amazon Computers
Amazon Photo
Citeseer
Task COSMIC (LR-FGDM)
2-way T-shot
2-way 5-shol
2-way T-shol
2-way 5-shol
2-way T-shot
2-way 5-shot
0.026
0.018
0.021
0.012
0.034
0.017
COLA (LR (LR-FGDM)
0.004
0.006
0.003
0.010
0.005
COSMIC (LRA-LR- LR-FGD
0.004
0.003
0.007
0.002
0.004
0.010
COLA (LRA-LR L.--FGDM
0.003
0.001
0.001
0.001
0.003
0.002
G.2 COMPARISON WITH EXISTING SHOT AUGMENTATION METHODS
In this section, we compare LR-FGDM with existing shot augmentation methods. IA-FSNC (Wu al., 2022) incorporates confidently predicted query nodes as additional labeled instances and in- troduces noise-based perturbations to the node features. We also compare LR-FGDM with the berf-perfonming GAN-based synthetic graph structure generation method, Semantic-awarg Node Synthesis (SNS) (Gao et al., 2023b), which is originally proposed to generate synthetic nodes in the minority class for imbalanced datasets with a Generative Adversarial Network (GAN) (Goodfellow et al., 2020). SNS is adapted to the FSNC scenario crating synthetic support nodes. We further compare LR-FGDM with another shot augmentation method SMILE (Liu et al., 2025b), which augments the support set using mix-up strategy. For a fair comparison, we apply IA-FSNC, SNS, and SMILE to augment the support set in COLA for 5-way 5-shot FSNC in the same manner as LR-FGDM. It is observed in Table 8 that LR-FGDM always outperforms all the competing shot augmentation methods across multiple graph datasets and few-shot settings.
baseline for shot augmentation by gen-
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Table Comparison of LR-FGDM with existing shot augmentation methods on 5-way 5-shot node classification. All methods are applied to augment the COLA (Liu et al., 2024). The statistical significance of the results is deferred to Table of the appendix.
Method
CoraFull
ogbn-arxiv
Coauthor-CS
DBLP
COLA (Liu al., 2024)
87.83
67.52
87.34
87.23
COLA (SMILE) (Liu al., 2025b)
88.21
68.01
88.02
88.07
COLA (IA-FSNC) (Wu al., 2022)
88.36
68.17
88.21
88.16
COLA (SNS) (Gao al., 2023b)
88.49
68.32
88.34
88.28
COLA (LR-FGDM)
89.66
69.63
89.83
89.51
COLA (LRA-LR-FGDMM)
90.32
70.22
90.39
90.07
G.3 STUDY ON THE KERNEL COMPLEXITY (KC) AND THE UPPER BOUND FOR THE TEST LOSS IN THEOREM A.1
In this section, we study the effectiveness of LR-FGDM in reducing the upper bound for the test loss in Equation (5) and the two terms in it, including the kernel complexity of the kernel gram ma- trix, KC(K), and the training loss of the node classifier with the ground shound-truth -ruth label, L(K, Yps t). The study performed for 5-way 5-shot node classification tasks Cora-Full and Coauthor-CS with two baseline models, COSMIC (Wang et al., 2023a) and COLA (Liu et al., 2024), as well as the corresponding models augmented by LR-FGDM, which are COSMIC (LR-FGDM) and COLA (LR-FGDM). It is observed from Table 9 that the upper bound for the test loss for the few-shot node classifiers trained by LR-FGDM is significantly lower than that of the baseline without low-rank regularization. Furthermore, LR-FGDM achieves substantially lower values in both KC(K) and LKK, YIS, f) in the upper bound, validating the theoretical motivation behind low-rank regulariza- tion and confirming the robustness and generalization benefits of LR-FGDM in FSNC.
L(K, Yrs, (), KC(K), and the upper bound for the test loss. The lowest values for each dataset are bold, and the second-lowest are underlined. The study is performed for 5-way 5-shot FSNC on CoraFull, ogbn-arxiv, Coauthor-CS, DBLP, and Koman-Empire and for 2- Amazon-Photo, and Citeseer.
Table 9: Comparisons
way shohot FSNC on Amazon-Compune
Datasets
Terms
COSMIC
COSMIC (LR-FGDM)
(LRA-LR-FGDM)
COSMIC
COLA (LR-FGDM)
(LRA-LR-FGDM)
COLA
L(K. Yos. C)
6.44
3.72
3.57
3.52
CoraFull
RC
0.35
0.20
0.19
0.40
0.18
0.17
Upper
L(K, Yia
4.54
4,02
3,89
4.69
3.95
3.82
ogba-arxiv
KC
0.47
0.24
0.23
0.50
0.21
0.20
Upper Bound
9.40
8.20
7.95
9.84
7.97
7.73
T.K. (
4,26
5,38
3.29
3.95
3:40
3.32
Coauthor-CS
KC
0.52
0,30
0.29
0.66
0.28
0,23
Upper Bound
7.99
6.25
6.09
7.63
6.16
6.03
L(K. Yos.
3.63
DBL.P
RC
0.48
0.26
0.25
0.53
0.23
0.21
Upper
L.K. (
5.12
3,88
3.74
4.95
3.79
3.62
Amazon- Conuuntee
KC
0.44
0.25
0.24
0.47
0.22
0.20
Upper Bound
9.12
7.10
6.93
8.80
6.85
6.60
L,K, Yss
4.41
3.52
3.45
4.28
3.48
3.38
Amazon-Photo
KC
0.20
0.18
Upper Bound
8.35
6.55
6.40
8.09
6.33
6.12
Yss.
Citeseer
RC
0.58
0.31
0.29
0.61
0.27
0.22
Upper
L.K. ()
8.35
6.62
6.48
8.10
6.55
6.31
Roman-Empire
KC
0.40
0.36
Upper Bound
13.92
11.08
10.90
13.55
10.82
10.55
G.4 EFFECTIVENESS LR-FGDM HETEROPIILIC GRAPHS
While most existing FSNC studies have primarily targeted homophilous graphs, where connected nodes tend to share similar labels, many real-world graphs exhibit heterophily, where neighboring nodes often belong to different classes. In such cases, standard GNNebased few-shot methods face fundamental limitations, as neighborhood aggregation mechanisms become less effective or even detrimental learning discriminative node representations. This poses even greater challenge
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under few-shot conditions, where only handful of labeled nodes per class are available to guide the model. To assess the generalization capability of our proposed LR-FGDM this challenging scenario, we conduct experiments
the Roman-Empire dataset (Platonov et al., 2023) following COLA (Liu et al., 2024), which is a syntactic word dependency graph characterized by extreme heterophily. In this graph, node labels reflect grammatical roles rather than local connectivity, mak- ing the graph structure highly non-homophilous. To demonstrate the statistical significance of the improvements achieved by LR-FGDM over the baseline methods, we perform f-tests on the few- shot classification accuracies obtained from 20 independent few-shot tasks for each setting. It is observed in Table 1I that models enhanced by LR-FGDM consistently yield statistically significant improvements over the corresponding methods across all few-shot settings with p-values p < 0.05.
Table 10: FSNC results on the Roman Aamaaeemppireee dataset, which features extreme heterophily. Accu- racy and standard deviation are reported in %. The best result for each setting is in bold, and the second-best underlined.
Method
2-way I-shol
2-way
S-shot
5-way
1-shot
5-way 5-shot
MAML (Finn et al., 2017)
12.83 = 2.31
50.12 1112:22 2.21
19.45 = 1.20
25.73
1:11
ProtoNet (Snell ul., 2017)
48.67 + 2.65
61.34
2:47
28.52
1.69
44.10
1.72
Meta-GNN (Zhou et al. 2019)
63.15 + 3.20
73.28 1: 2.85
11.80 = 2.14
60.33 + 2.06
GPN (Ding et 2020)
62.10 +
74.01 2.63
42.73
AMM-GNN (Wang et al., 2020b)
65.02 + 3.05
76
2.13
48.92 + 2.39
66.22
+ 1.81
G-Meta (Huang & Zitnik, 2020)
66.74 3.22
78.36 2.14
50.14 = 2.43
66.40
1 1.75
TENT (Wang
2022)
66.23 3.05
77.29 2.39
45.73 2.01
61.78
1.81
KD-FSNC (Wu et 2024)
69.15 2.59
80.16 2.11 2.11
58.92 = 2.34
73.28
+ 1.90
NormProp (Zhung al., 2025a)
69.02 2.67
80.13
67.84 2.30
72.46
COSMIC (Wang et al., 2023a)
71.84 +
$2.35 2.26
60.25
2.12
75.33 + 2.05
COLA (Liu et al., 2024)
70.96 + 2.44
$1.48 2.09
59.31 =: 2.37
74.02 + 1.88
COSMIC (LR-FGDMI
13.38 2.56
53.79 2.18
61.45 1LA552334
76.32 1.1
COSMIC (LRA--LR-FOAL --F--DDDM)
73.89 + 2.31
84.33 + 2.01
61.98 + 2.12
77.04 + 1.78
COLA (LR-FGDM)
75.62 + 2.35
85 2.02
63.18 2.11
78.23
1.76
COLA (LRA-LR-FGDDD)
76.17 2.18
85.93
1.89
63.69
1.97
78.79
1.63
Table 11: p-values from t-tests comparing COSMIC (LR-FGDM), COLA (LR-FGDM), COSMIC LGDA-, and COLA (LRA-LR-FGDM) against their corresponding baseline methods, COSMIC and COLA, on Roman-Fmpire.
w-w T-shot
2-way 5-shol
5-way T-shol
5-way 5-shol
(OSMICCRR-FDDM. COLA (LR- HGDM)
0.034
0.024
0.028
0.012
0.019
0.022
0.013
0.002
COSMIC (LRA-LR-FGDM.
0.007
0.006
0.008
0.005
COLA (LRA-LR R.F-GM
0.004
0.003
0.006
0.003
G.5 VOSSS-VALI ON THE NUMBER OF SYNTHETIC NODES
The number of synthetic nodes generated for each novel class given each support node, denoted as q, plays a crucial role in determining the effectiveness of LR-FGDM. While generating more synthetic nodes can potentially enrich the support set and provide stronger supervision signals, it may also introduce redundancy or noise if excessive synthetic samples are added. In our experiments, we select the value of q for different datasets using 5-fold cross-validation over the base training classes. The value of is selected from range of candidate values, including 2,, 3, 4, 5, 6, 7, 8, 9, 10).
Table 12: The selected number of synthetic nodes per support node (q) for each dataset and few-shot setting.
Dataset CoraFull
2-way 1-shot
2-way 5-shot
5-way 1-shot
5-way 5-shot
2
3
3
5
ogbn-arxiv
5
8
5
Coauthor-CS
DBLP
5
3
4
G.6 SENSITIVITY ANALYSIS ON THE HYPERPARAMETERS Fo, K, AND q
In this section, we first conduct a sensitivity analysis т, which is the weighting parameter for the TNN [K|], in Equation 2. The study is performed using COLA (LR-FGDM) on the CoraFull
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dataset for the 5-way 5-shot node classification task. We evaluate the performance of the COLA (LR-FGDM) with т varying in {0.1, 0.3, 0.5, 0.6, 0.7, 2.7,0.8,0.39 0.9}. As shown in Table 13, although the best performance is achieved at 0.6, the performance of COLA (LR-FGDM) remains stable across different values of r. Even the lowest performing setting, r = 0.1, results in only marginal 0.18% decrease in accuracy compared the best result. In addition, we perform an ablation study to examine the influence of the rank parameter TO = [ min N, d), where E (0, [] controls the effective rank used in the truncated nuclear norm. We evaluate COLA (LR-FGDM) with y varying from 0.05 to 0.5. As shown in Table 13, the accuracy is robust to different values of y, with the highest performance observed at y 0.2. We also conduct an ablation study on the hyperparameter K, which denotes the number of clusters used for prototype regularization in the HGAE. We vary K' from 5 to 50 with a step size of 5. As shown in Table 13, the performance remains stable across different values of K, with slight peak at K = 10. This suggests that the model is not sensitive to the choice of cluster number, K.
Table 13: Sensitivity analysis on the weighting parameter for the TNN, the rank ratio y used in To = mymin [nn]] and the number of clusters K for prototype computation in COLA (LR- FGDM) for the 5-way 5-shot node classification task on CoraFull.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Accuracy
89.48
89.55
89.59
89.57
89.63
89.66
89.62
89.65
89,58
7
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Accuracy
89.27
89.44
89,58
89.66
89.62
89.55
89.64
89.60
89,60
89,55
K
5
10
15
20
25
30
35
40
45
50
Accuracy
89.30
89.66
89.58
89.64
89.53
89.59
89.62
89.56
89.48
89.44
Furthermore, perform a sensitivity analysis to evaluate the effect of the number of synthetic nodes per support node, q. on FSNC performance under the 5-way 5-shot setting. For each dataset, we vary q from 1 to 10 and report the mean classification accuracy over 20 tasks. As shown in Table 14, increasing 9 generally leads to consistent improvements in performance up to a certain threshold, beyond which the gains tend saturate or marginally decline. This trend highlights the benefit of augmenting the support set with a moderate number of synthetic nodes, which helps enhance generalization by enriching the local representation space. Notably, while further increasing q beyond the optimal value does not continue to improve performance, the resulting degradation is marginal.
Table 14: Sensitivity analysis on the number of synthetic nodes per support node (q) for four datasets under the 5-way 5-hhot FSNC setting.
9
3
4
5
6
7
8
9
10
CoraFull
88.71
89.02
89.37
89.51
89.66
89.53
89.59
89. 89.44
89.41
89.40
oghn-arxiv
67.80
68.21
68.84
69.08
69.33
69.42
69.63
69.59
69.45
69.20
Coauthor-CS
88.33
88.82
89.08
89.31
89.83
89.62
89.54
89.48
89.30
89.12
DBLP
87.93
88.44
88.87
89.02
89.51
89.35
89.29
89.10
88.94
88.91
G.7 QUALITY EVALUATION OF THE AUGMENTED GRAPH
This paper introduces novel node-level graph diffusion model named FGDM, which synthesizes the graph structures. The synthetic graph structure, which consists of the synthetic support nodes and the associated edges, generated by the FGDM, is subsequently combined with the original graph to form an augmented graph. In Section 4.2, we have shown that the FSNC methods trained on the augmented graph achieve significantly better performance. In this section, we directly evaluate the data quality of the synthetic graph structures generated by the FGDM. In the visual domain, the Frechet Inception Distance (FID) is a widely used metric to evaluate the quality of the synthetic images generated by the generative models (Brock et al., 2019; Ho et al., 2020). The FID score measures the similarity between the distribution of the generated images and the distribution of the real images. To compute the FID score, the pre- pre-trained Inception v3 (Szegedy et al., 2016) is used to extract the features from both the real images and the generated images, which are then modeled as the multivariate Gaussian distributions. The FID score is then calculated using the Frechet Distance
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(FD) (Brock et al., 2019) between the two multivariate Gaussian distributions modeling the real and the generated images (Dowson & Landau, 1982). A lower FID score indicates that the generated images more similar to the real images, suggesting better quality.
Quality Evaluation of the Synthetic Nodes. Although the Inception model cannot be applied to the graph data, we can replace the Inception model in the computation of the FID score with the pre- trained GCN (Kipf& & Welling, 2017) for extracting node features to adapt the metric to evaluate the quality of synthetic nodes generated by the FGDM. To this end, we define the Frechet Node Distance (FND), which is the FD between the multivariate Gaussians modeling the node features extracted by pre-trained GCN. We randomly split the nodes from the novel classes in the original graph into two partitions of equal size, which are the base partition and the test partition. To mitigate the influence of the randomness, we compute the FND scores with 10 different random splits and report the mean and the standard deviation of the FND scores across different runs. The FND computed between the nodes in the test partition and the base partition of the original graph establishes the baseline of the expected FND score for high-quality support nodes. By computing the FND score between the features of the synthetic support nodes in the synthetic graph structures generated by the FGDM and the features of the nodes in the base partition of the original graph, we evaluate the quality of the synthetic support nodes. For simplicity, we refer to the FND score for the synthetic support nodes as the FND between their features and the features of the nodes in the base partition of the original graph. To show the effectiveness of the prototypical regularization in the training of the HGAE for the PGDM, we also compute the FND for the nodes in the synthetic graph structures generated by the PGDM without the prototypical regularization. To demonstrate the advantages of the FGDM over the vanilla diffusion model, the DDPM (Ho et al., 2020), we train baseline DDPM model on the input node attributes of the original graph and synthesize the same number of synthetic nodes as the FGDM. The synthetic edges are then generated by connecting each synthetic node to its K-nearest neighbors in the original graph using the K K-nearest neighbors (KNN) algorithm with K == [dave], where dave is the average degree of the original graph. The synthetic graph structures, including the synthetic nodes and edges generated by the baseline DDPM model, are combined with the original graph to form the augmented graph. Next, we compute the FND score for the nodes in the synthetic graph structures generated by the baseline DDPM model. In addition, we also compute the FND for the support nodes generated by three competing shot augmentation methods, including SMILE (Liu et al., 2025b), IA-FSNC (Wu et al., 2022), and SNS (Gao et al., 2023b). The ablation study is performed for the 5-way 5-shot setting of the FSNC. The lower FND scores indicate that the node features are more similar to the features of nodes in the base partition of the original graph. It is observed in Table 15 that the FND score of the nodes in the synthetic graph structures generated by the FGDM is closest to the FND score of the original graph, which demonstrates that the FGDM generates faithful synthetic nodes.
Table 15: Frechet Node Distance (FND) to the nodes in the base partition of the original graph. The
mean and standard deviation of the FND scores computed with 10 diffcrent random splits of the base partition and the test partition in the original graph are reported. The evaluation is performed for the 5-way 5-shot setting of the FSNC. The FND score for the original graph is computed between the nodes in the test partition and the nodes in the base partition of the original graph.
Data Baseline DDPM
CoraFull
oghn-arxiv
Coauthor-CS
DBLP
13.41
8,95
10.32_041
0.34
SMILE (Liu et al.. 2025b)
00.2
6.03-0.32
8.21
0.38
IA-FSNC (Wu et al., 2022)
9.92+40.41
5.87=0.29
7.88
6.12-0.37
SNS (Gao et al., 2023b)
9.73++0.40
5.69+0031
7.44+0.34
6.01=0.33
FGDM (w/ Prototypical Regularization)
9.24++0.88
5,23+0030
7.13+00.77
5.95=0.36
FGDM
8.10+0027
4.39+00.00
5.29+0.21
4.08-0.42
Original Graph
7.9510.32
4.3310.27
4.211
0.34
Quality Evaluation of the Synthetic Edges. Similar to the design of the FND score for evaluat- ing the quality of synthetic nodes, we replace the Inception model in the computation of FID with the pre-trained GNN (Zhu et al., 2021) for edge feature extraction to adapt the metric to evaluate the quality of the synthetic edges generated by the FGDM. To this end, we define Frechet Edge Distance (FED), which is the FD between the multivariate Gaussians modeling the edge features extracted by a pre-trained GNN. Similar to the evaluation of the FND, we randomly split the edges in the original graph into two partitions of equal size, which are the base partition and the test par-
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tition. To mitigate the influence of the randomness, compute the FED scores with 10 different random splits and report the mean and the standard deviation of the FED across different runs. The FED computed between the edges in the test partition and the edges in the base partition of the original graph establishes the baseline of the expected FED score for the high-quality edges. By computing the FED between the features of edges in the synthetic graph structures generated by the FGDM and the features of edges in the base partition of the original graph, we evaluate the quality of the edges in the synthetic graph structures. For simplicity, we refer to the FED score for the synthetic edges as the FED between their features and the features of edges in the base partition of the original graph. Similar to the evaluation of the synthetic nodes, we also compute the FED score for the edges in the synthetic graph structures generated by the FGDM without the prototypical regularization. We compute the FED score for the edges in the synthetic graph structures gener- ated by the baseline DDPM model. Since the edges in the synthetic graph structures are generated by the KNN algorithm, we evaluate the baseline DDPM models using different values of K from 1, X [d In addition, also compute the FED for
{[ /4,, [dave/2], dave the edges generated by three competing shot augmentation methods, including SMILE (Liu et al., 2025b), IA-FSNC (Wu et al., 2022), and SNS (Gao al., 2023b). The FED score for edges in the original graph is also computed. We use the same NBFNet (Zhu et al., 2021) pre-trained on the original graph to extract the edge features for computing the FED score. The ablation study is per- formed for the 5-way 5-shot setting of the FSNC. Lower FED scores indicate that the edge features are more similar the features of edges in the base partition of the original graph. It observed in Table 16 that the FED score of the edges in the synthetic graph structures generated by the FGDM is closest to the FED score of the original graph, which demonstrates that the FGDM generates more faithful synthetic edges compared to the competing methods.
Table 16: Frechet Edge Distance (FED) to the edges in the base partition of the original graph. The
mean and standard deviation of the FED scores computed with 10 different random splits of the base partition and the test partition in the original graph are reported. The evaluation is performed for the 5 5-way 5-shot setting of the FSNC. The FED score for the original graph is computed between the nodes in the test partition and the nodes in the base partition of the original graph.
Data
FolaFull
ogbn-arxiv
Coauthor-CS
DBLP
Baseline DDPM (K =
12.17:0.47
0.35
10.89
H..4
Baseline DDPM (K
[dwe/2]
0..3
0.53
9.43:0034
7.32+0.33
Baseline DDPM (K
dyve
10.48
8.0110.31
9.04:1..36
6.8811032
Baseline DDPM (K
22 x dave] D
10.51
7.9610.22
9.1710..7
6.94.10.3
Buseline DDPM (K
[4 dave
10.92-0.42
8.05
9.41-0.33
7.1040.36
SMILE (Liu et al., 2025b)
10.41=0.36
7.89+0.30
8.93+0.34
6.6740.31
IA-FSNC (Wu et al., 2022)
10.96=0.34
7,80=0.29
8.62+0033
6.39+00.0
SNS (Gao al., 2023b)
10.18=0.35
7.77+0.28
8.66+0.32
6.46++0.29
FGDM (w/o Prototypical Regularization)
10.13+0.38
7.70+0.27
8.79+0.35
6.33+00.00
FGDM
8.29-0030
5,33+00.2
6,56+00.6
5,39+0,27
Original Graph
8.10=0.27
5.1410.22
6.37+:025
5.19:0.26
Table 17: valuues from t-tests comparing COSMIC (LR-FGDM), COLA (LR-FGDM), COSMIC (LRA-LR-FGDM and COLA (LRA-LR-FGDM) against their corresponding baseline methods, COSMIC and COLA, in Table 1.
CoraPall
oghn-arsiv
Yask
2-way T-bbot
2-way S-bhot
5-way I-shol
5-way 3-shox
T-shot
2-way S-abot
I-hhot
S-shol
COSMIC FURFO
0,039
COLA (LR-FGMM
0.004
0.007
0.011
COSMIC (LRA-LR FGDM)
0.007
0.005
0.008
0.006
0.004
0.003
0.007
0.005
COLA (LRA-LR- FGDM)
0.004
0.003
0.007
0,005
0.006
0.004
0.008
0.002
Countbor CS
DBLF
Task
Joway F-bat
Jwway S-shot
S-way T-shol
S-way S-cbor
J-way T-sbot
J-way Swohot
Soway T-shut
S-way S-ahot
COSMIC FGDM
0,013
0.015
0024
0.025
0.019
0.011
0,027
0001
COLA (LR-PGDM)
D:007
0.003
0.009
0.004
0.006
0.002
0.005
0.003
COSMIC (LRA-LR FGDM)
0.006
0.004
0.008
0.007
0.005
0.004
0.009
0.006
COLA LR. FGDM)
0.006
G.8 IMPROVEMENT STATISTICAL SIGNIFICANCE ANALYSTS
To demonstrate the statistical significance of the improvements achieved by LR-FGDM over the
baseline methods, perform 1-tests the few-shot classification accuracies obtained from 20 in-
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Table 18: p-values from t-tests comparing COLA (LR-FGDM) and COLA (LRA-LR-FGDM) with the second-best ablation model in Table 2.
Dataset
CoraFull
ogbn-arxiv
Coauthor-CSS
DBLP
COLA (LR-FGDM)
0.012
0.009
0.043
0.008
COLA (LRD-LR-FGDMD
0.004
0.003
0.007
0.004
Table 19: p-values from t-tests comparing COLA (LR-FGDM) and COLA (LRA-LR-FGDM) with the second-best ablation model Table
Datasct
CoraFull
ogbn-arxiv
Coauthor-CS
DBLP
COLA FGDM)
0.025
0,032
U.026
0.018
COLA (LRA-LR-FGDM)
0.005
0.004
0.006
0.003
dependent few-shot tasks for each setting. For results in Table 1, we compare COSMIC (LR-FGDM) and COLA (LR-FGDM) against their corresponding baseline methods, COSMIC and COLA. It is observed in Table 17 that models enhanced by LR-FGDM consistently yield statistically signifi- cant improvements over the corresponding methods across all datasets and settings with p-values further validate the statistical significance of the improvements of COLA
p < 0.05. In addition, (LR-FGDM) over the ablation models in Table 2 and models augmented by other support set aug- mentation methods in Table 8. It is observed from Table 18 and Table 19 that COLA (LR-FGDM) significantly outperforms all competing variants, with all p-values below 0.05, further supporting the statistical significance of the improvements by LR-FGDM over existing methods.
G.9
AND CONCENTRATION RATIO ANALYSIS
We first compute the eigenvectors U of the feature gram matrix K. Let U R&xr be the top r-eigenvectors of K and U be the r-th eigenvector of K. Then, the eigen-projection value of the
ground-truth label YPS on U(r) is computed by pr Σ UO"
Y (e) forr [N].
PS
where ( the c-th column of Y We p = [pI. [Pt....,PNI E RR The eigen-projection pr reflects the amount of the signal in the label projected onto the r-th eigenvector of K, and the signal concentration ratio of rank reflects the proportion signal projected onto the top r eigenvectors of K. The signal concentration ratio for rank is computed by |/p(1 r)]l where p(1 r) contains the first r elements of p. Figure 6 illustrates the eigen-projection and concentration ratio of the min {N, D'), the signal concentration
ground truth label Cora-Full. By the rank To ratio the ground truth label is 0.792.
Conventration Ratio
1.0
Projecttion Value
0.5
0.8
0.4
0.6
0.3
0.4
0.2
0.1
0.2
0.0
0.0
10000
15000
20000
0
5000
10000
15000 20000
Rank
Rank
Figure 6: Eigen-projection (left) and concentration ratio (right) of the ground truth label on Cora- Full.
G.10 STUDY ON THE AUTOMATIC SELECTION OF THE RANK TO
We explore an automatic rank-selection strategy based on the eigenvalue decay of the feature ker- nel Gram matrix K. In particular, we evaluate an eigengap-basd based method widely used in spec- tral dimensionality selection, in which ro is selected at the index corresponding to the largest rel- ative decay between consecutive eigenvalues (Ng et al., 2001). As result, To computed as To arg maxigi<N X, et al., 2024). As shown in Table 20, the performance of LR-FGDM and BRA-LR-FGDAA with the automatically selected rank ro is only marginally different from that achieved via cross-validation.
X.-X.11 The study is performed using the base FSNC model, COLA (Liu
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The deviations remain consistently marginal, typically smaller than 0.2%. The results suggest that the automatic selecceettion strategy based on the eigenvalue decay rate provides an efficient alter- native to tuning the rank To. without sacrificing performance.
Table 20: Ablation study on the selection strategy of ro for the low-rank regularization within LR-
FGDM and LRA-LR-FGDM. Results are reported under the 5-way 5-shot FSNC setting.
Method
Selection Method of
CoraFull
ogbn-arxiv
Coauthor-C3
DBLP
COLA (Liu 2024)
87.83
67.52
87.34
87.23
COLA (LR-FGDM)
Cross-Validee Validation
89.66
69.63
89.83
89.51
COLA (LR-FGDM)
Automatic 89.56
69.68
89.74
89.55
COLA (LRA-LR- LR-FGM
Cosss-Validation
90.32
70.22
90.39
90.07
COLA (LRA-LR-FGDM)
Automatic 90.36
70.04
90.21
90.11
G.11 T-SNE VISUALIZATION COMPARING THE SYNTHETIC NODES AND REAL NODES IN THE NOVEL CLASSES
To provide a more intuitive assessment of the semantic correctness of the generated nodes, we perform t-SNE visualization study in this section. The visualization results illustrate the spatial distribution of the node embeddings of the synthetic nodes generated by LR-FGDM compared to the node embeddings of the real nodes in the corresponding few-shot classes. For each of the 5-way 5-shot novel classes, 20 real nodes are randomly sampled, and 20 synthetic nodes are generated by FGDM for comparison. As illustrated in Figure embeddings the synthetic nodes produced by LR-FGDM form compact clusters aligned closely with the real node clusters, demonstrating that the generated nodes preserve both semantic structure and class-level discriminativeness. These qualitative results complement the quantitative FND/FED metrics and provide additional evidence that LR-FGDM generates semantically meaningful node representations.
Nade (Ctac 1)
10
20-
Systhcti Node (Cise
Heat Node (Clase
Real Mode (Class 3)
-10
Spatheti Node (Clss 3)
-20
-10
-10
Ical Node (Lise 5)
Syafacii Noáe (Clas S)
-12
20
(5)
Coanthor:CS
(c) oghn-arxiv
Figure 7: t-SNE visualization of the embeddings of real nodes and synthetic nodes generated by FGDM. For each of the 5-way 5-shot novel classes, 20 real nodes are randomly sampled, and 20 synthetic nodes are generated by FGDM for comparison.
G.12 ABLATION STUDY ON THE LDM
To demonstrate the contribution of the proposed generative mechanism, we perform an additional ablation study in which the LDM LR-FGDM is replaced with two alternative generative models, including a Variational Autoencoder (VAE) (Kipf & Welling, 201 6a) and generic diffusion model, DDPM (Ho et al., 2020). The study is performed using the base FSNC model, COLA (Liu et al., 2024). The ablation models are denoted as COLA (DDPM) and COLA (VAE). It is observed in Ta- ble 21 that COLA (LR-FGDM) significantly outperforms the two ablation models, COLA (DDPM) and COLA (VAE), demonstrating the better quality of the augmented graphs generated by FGDM.
Table 21: Ablation study on the generative model used in LR-FGDM. Results are reported under the 5-way 5-shot FSNC setting.
Method
CoraFull
ogbn-arxiv
Coauthor-CS
DBLP
COLA (Liu et al., 2024)
87.83
67.52
87.54
87,23
COLA (DDPM)
88.85
68.35
88.42
88.12
COLA (VAE)
88.47
68.26
88.30
88.07
COLA (LR-FGDM)
89.56
69.68
89.74
89.55
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G.13 TRAINING TIME COMPARISON AGAINST BASELINE FSNC METHODS
We report the training time of LR-FGDM and LRA-FGDM when integrated with COSMIC and COLA on the augmented graphs in Table 22. The evaluation is performed on a single 80G NVIDIA A100 GPU. It is observed that LR- LRFFGDM introduces additional computational overhead due to the training of the HGAE, the LDM, and the subsequent low-rank transductive classifier. However, the increase in training time is moderate relative to the performance improvements it delivers. While lightweight metric-based methods such as ProtoNet and NormProp remain the fastest, their accu- racy is substantially lower across all benchmarks. In contrast, strong meta-learning and contrastive methods already incur significantly higher training costs. LR-FGDM adds only a modest overhead on top of COSMIC and COLA while consistently improving accuracy all datasets. Across all datasets, the increase in training time introduced by LR-FGDM and LRA-LR-FGDAA is in the range of 3.4% 4%.4%%%%2%. 6.2% compared the corresponding baselines, COSMIC and COLA.
Table 22:
Training time comparison (in seconds) and corresponding 5 5-way 5-shot accuracy (%) under the 5-way 5-shot setting. The training time for the HGAE and LDM FGDM and the generation time of the synthetic graph structures by FGDM are reported in Table 3.
CoraFull
Coauthor-CS
ogbn-arxiv
DBLP
Method
Time
Acc
Time
Acc
Time
Acc
Time
Acc
ProtoNet
19.4
51.54%
15.8
49.25%
40.6
53.31%
22.9
52.26%
Meta-GNN
83.3
70.50%
68.7
68.59%
125.1
31.52%
92.4
72.15%
GPN
53.0
72.82%
42.4
81.79%
171.8
50.50%
63.7
75.46%
G-Meta
662.5
75.84%
542.1
74.18%
598.4
47.16%
328.9
75.82%
TENT
58.9
10.10%
47.8
76.90%
84.2
61.38%
71.6
74.84%
KD-FSNC
74.6
85.89%
60.8
84.42%
112.3
64.91%
83.5
83.75%
NormProp
22.3
85.47%
18.1
84.61 %
41.9
64.28%
25.8
84.12%
STAR
267.4 4
87.31%
217.6
87.60%
398.5
66.98%
302.7
87.10%
DoG
850.5
86.47%
690.2
87.35%
1225.3
65.69%
1042.4
87.59%
COSMIC
824,9
86.34%
668.2
85.47%
1221.7
65.42%
1015.3
85.31
COLA
781.6
87.83%
632.3
87.54%
1155.1
67.52%
958.6
87.23%
COSMIC (LR-FGDM)
854.2
88.81%
691.3
87.72%
1266.5
67.59%
I056.3
87, 97.44%
COSMIC (A-LR-FFGDM)
875.3
89.47%
704.4
88.21%
1295.1
68.
1078.
87.96%
COLA (LR-FGDM)
810.5
89.66%
652.3
89.83%
1200.2
69.63%
992.4
89.51%
COLA (LRA-LR-FGDM)
829.2
90.32%
670.4
90.39%
1221.5
70.22%
1008.2
90.07%
G.14 STRUCTURAL PROPERTY EVALUATION OF THE AUGMENTED GRAPH
To demonstrate that the synthetic edges preserve key graph properties, we evaluate two quantita- tive graph structural properties, including the small-world property (Telesford et al., 2011) and the power-law degree distribution property (Clauset et al., 2009; Zhao al., 2024b). The smald-world property is measured by the small-world coefficient o C/Cuua T. where C and L denote the clustering coefficient and characteristic path length of the augmented graph, and Crand, Lrand are the clustering coefficient and characteristic path length a Pegree-preservvng random graph. The power-law de- gree distribution property is measured by the power-law exponent and the Kolmogorov-Smingg (KS) statistic between the empirical degree distribution and a fitted power-law. To evaluate the two structural properties above of the augmented graphs obtained by LR-FGDM, we compute the small-world coefficient a and the KS statistic for both the edges in the original graphs and their LR- counterparts across CoraFull, ogbn-arxiv, Coauthor-CS, and DBLP. The study is performed for the 5-way 5-shot FSNC setting. is observed in Table 23 that the synthetic graph structures generated by LR-FGDM exhibit small-worl coefficient and power-law exponent that closely match those of the original graph, with only marginal deviation in the KS statistic. These results confirm that the hierarchical edge reconstruction in HGAE successfully preserves both the small-world characteristics and the heavy-tailed degree patterns of real-world graphs.
H ALGORRITMS FOR THE TRAINING OF LR-FGDM
We present the training algorithm of the FGDM in Algorithm 1, which comprises two steps. The first step, which is from Line I to Line 5 in Algorithm 1, describes the training of the HGAE, The second
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Table 23: Structural property evaluation of the augmented graph. We report the small-world coefii-
cient and KS statistic for both the original graph and the LR-FGDM LR-FGIDM-augmented graph.
Metric
Graph
CoraFull
ogbn-arxiv
Coauthor-CS
DBLPP
Oniiinal
3.27
2.56
4.11
2.99
Augmented
3.12
2.48
3.99
2.94
Original
0.067
0.054
0.062
0.049
KS
Augmented
0.068
0.056
0.061
0.052
Algorithm Training FGDM (Training the HGAE and the LDM)
Input:
The input attribute matrix X, adjacency matrix A, the training epochs of the HGAE IIGAE, the labels base training set, training epochs of the LDM fLDM, and
Yhse of the labeled nodes Vuse
learning rate
Output: The parameters of the HGAE w and the parameters of the LDM 0
1: Obtain the inter-cluster neighbor map and the intra-cluster neighbor map M by applying balanced K-
Mcans clustering X
Initialize the parameter w of the HGAE
3: for f +- 1 to TAGAF do
4: Update by f- Vw with LhgaF from Eq.(1)
5: end for
Compute cluster prototypes [p:] from latent representations Z using semi-supervised K-means clus- tering (Bair, 2013)
7: Assign each node to its cluster prototype based on the clustering result
8: Initialize the parameter 0 of the LDM
Map the node attributes X and the adjacency matrix to the latent space using the encoder Je of the HGAF as = 9e (X, A)
10: Train the LDM with CPG (Ho & Salimans, 2022) on latent pairs (Z:, Pa(a)) where Pa(a) is the prototype associated with node
11: return The parameters of the HGAE and the parameters of the LDM 0
Algorithm 2 Generation of the Augmented Graph Gong
Input:
The input attribute matrix X. the adjacency matrix A, the set of support nodes Vun (vI, (vi...., vs), number of synthetic nodes per support node q, and the prototype assignments (Pe(s) hiz Output: The augmented graph Gau (V U Vsyn, X A
I: Let M q (aupl total number of synthetic nodes
Initialize counter m <-
3: for i 4- to Vupl do
4: for < to q do
5: Sample noise (0,
Condition on prototype P+(s) of support node
Generate latent feature Z... using LDM conditioned on Pr(()
8: m <- m +
9: end for
10: end for
Decode 2 = (2. MMM to Xsyn and Asyn with the decoder the HGAE (X.yn, Asy)) = g2(
12: Form augmented attribute matrix: X = [X: X.yn]
13: Form augmented adjacency matrix: Aaug = [A Agni Aggn A]
14: return Gong = (V UVS Veyn. Xang, A
step, which is from Line to Line 10, describes the training of the LDM. Algorithm 2 describes the generation process for the augmented graph Gmgg.
I EFFICIENT COMPUTATION OF THE TNN K
To approximate the top-ro eigenvectors, (rr) E R Nxro of the gram matrix K using the Nystrom method (Kumar et al., 2012), first sample M' landmark points from the training set, indexed by
I C [N] with И M' & N. Let Fz R xd be the features corresponding to the landmark
32

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Published as a conference paper at ICLR 2026
set. We define C= FF7 RN the cross-covariance matrix and W FzF, R RM'XM' xM' as the gram matrix on the landmarks. Next, we compute the top-ro sigendeccompoittion of W as W = where Q E TR x contains the top-ro eigenvectors and E RRuox is the diagonal matrix corresponding eigenvalues. The Nystrom approximation of K then given by K = CWiCT, and the approximate top-ro cigenvectors are computed as Ú(T)) = CQA 1/2 RNxro, which serves as an efficient approximation to U(ro) with significantly reduced computational cost.
We let Uno Ü(ro), then the sum of the top-ro eigenvalues of K is approximated by Uro TU,)) Since (K() Σ xxx, Ei=1 di and tr (Uro T "KU)))
N
C Σ [Urolst
1 K [Uroked) we can approximate the TNN K
| by K
tr (K) - tr Urr KU,,) which is separable. In particular, K E X,),,,,,
( Urola
K de Urolks) We remark that | is ready be optimized by standard
SGD algorithms because it is separable and expressed as the summation of losses individual train- ing data points. We perform the optimization for low-rank transductive few-shot node classification
in (2) with the approximate TNN [[],, instead of the original TNN ||]].,
33


Reviews:
[{"review_id": "WZhYIu9xlA", "reviewer": "Reviewer_dzx3", "summary": "This paper proposes Low-Rank Few-Shot Graph Diffusion Model (LR-FGDM) for few-shot node classification (FSNC). LR-FGDM augments the limited support set by generating realistic synthetic nodes and edges through a node-level graph diffusion generator (FGDM), which combines a Hierarchical Graph Autoencoder (HGAE) with prototypical regularization and a prototype-conditioned Latent Diffusion Model (LDM). To mitigate noise from diffusion generation, a low-rank regularization inspired by the Low-Frequency Property (LFP) is applied to the transductive classifier, with theoretical guarantees on reduced kernel complexity and generalization error. Extensive experiments across multiple graph benchmarks demonstrate that LR-FGDM consistently outperforms state-of-the-art FSNC methods in both accuracy and structural fidelity.", "questions": "1. On the Theoretical Scope of Theorem 3.1.\nCould the authors explicitly state the assumptions underlying Theorem 3.1 and clarify under what conditions the generalization bound holds? Specifically, does the theorem rely on boundedness, Lipschitz continuity, or i.i.d. sampling assumptions, and is it theoretically valid for all graph structures or only for homophilic/community-structured graphs?\n\n2. On the Missing Diffusion Ablation and Efficiency Evidence.\nWhile Table 2 ablates the low-rank and prototypical regularizations, the LDM module itself is never directly examined. Could the authors include a variant without the diffusion process (e.g., using only the VAE) or with a generic diffusion without the VAE, and report runtime and memory comparisons with COSMIC and DoG to clarify whether the generative component justifies its computational overhead?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The paper introduces a novel and well-motivated framework for few-shot node classification that combines prototype-conditioned diffusion-based augmentation with low-rank regularization. \n- The idea of performing node-level diffusion generation, as opposed to graph-level synthesis, is original and provides a meaningful contribution to generative graph learning. \n- The theoretical analysis grounding the low-rank transductive classifier adds rigor and clarity, while the empirical results demonstrate consistent improvements over strong baselines across multiple benchmarks.", "weaknesses": "- The pipeline figure could be substantially improved. It is visually cluttered and does not clearly separate the HGAE, diffusion, and low-rank classification stages, making it difficult to follow the data flow.\n\n- The length allocation of the paper is also unbalanced: the main text devotes extensive space to model formulation while leaving only two pages for experiments, which contain no figures and limited analysis. Important aspects such as efficiency comparisons and runtime scalability should be included in the main text rather than deferred to the appendix.\n\n- Table 2 suggests that the performance gain from the model’s architectural design may be relatively small, and Table 6 shows that the p-values for COSMIC (LR-FGDM) remain relatively high, raising concerns about the statistical robustness of the reported improvements.\n\n- The paper lacks an ablation study for the diffusion (LDM) component, making it difficult to assess how much of the improvement originates from the proposed generative mechanism rather than from low-rank or prototypical regularization.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FgPfpvHDRX", "reviewer": "Reviewer_678A", "summary": "This paper proposes a framework for few-shot node classification that uses a node-level graph diffusion model to generate synthetic support nodes and edges for augmenting the support set, and introduces low-rank regularization to mitigate diffusion-induced noise, with theoretical guarantees for transductive classification; extensive experiments on 8 datasets show LR-FGDM outperforms sota FSNC methods.", "questions": "See Weaknesses", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- LR-FGDM solves FSNC data scarcity by generating synthetic support nodes and edges.\n- The low-rank regularization in LR-FGDM fixes diffusion-induced noise, makeing the quality of synthetic data more reliable for model training.\n- The paper provides theoretical proof for the low-rank regularization, supporting the method’s validity with mathematical generalization bounds.\n- The proposed model outperforms state-of-the-art FSNC methods consistently, showing real effectiveness across 8 benchmark datasets.", "weaknesses": "- For the hierarchical edge reconstruction, \\(d_i\\) (number of connected clusters per node) is bounded by node degree, but what if nodes have extremely high degrees (e.g., hub nodes in scale-free graphs)?\n- The low-rank regularization relies on truncated nuclear norm (\\(\\|K\\|_{r_0}\\)), but \\(r_0\\) is selected via cross-validation—why not use adaptive rank selection (e.g., based on eigenvalue decay rate of K) to avoid manual tuning?\n- When generating synthetic edges, the HGAE decodes intra-cluster neighbor maps, but how to ensure synthetic edges follow real graph structural properties (e.g., small-world, power-law)?\n- The LDM is conditioned on prototypes instead of labels, but prototypes are learned from base classes, does prototype shift (between base and novel classes) reduce synthetic node quality for novel classes, and how to mitigate this?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "NZ6zQ7ZFer", "reviewer": "Reviewer_Kffe", "summary": "This isn't a paper in my field, but it appears to be a very solid piece of work. It includes derivations, well-conducted experiments, comparisons with recent baselines, and open-sourced code, a good paper.\n\nThe paper proposes LR-FGDM, a novel and well-motivated framework for few-shot node classification that combines node-level graph diffusion with low-rank transductive classification. Its key strengths include: (1) a prototype-conditioned graph diffusion model (FGDM) that generates both synthetic nodes and realistic edges without relying on unavailable class labels during test time; (2) an efficient hierarchical edge reconstruction mechanism that reduces decoding complexity; (3) a theoretically grounded low-rank regularization based on the Low Frequency Property, supported by a generalization bound; and (4) comprehensive experiments across diverse datasets—including heterophilic graphs—showing consistent improvements over state-of-the-art methods, along with new evaluation metrics (FND/FED) to assess generation fidelity.", "questions": "see above.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "The paper proposes LR-FGDM, a novel and well-motivated framework for few-shot node classification that combines node-level graph diffusion with low-rank transductive classification. Its key strengths include: (1) a prototype-conditioned graph diffusion model (FGDM) that generates both synthetic nodes and realistic edges without relying on unavailable class labels during test time; (2) an efficient hierarchical edge reconstruction mechanism that reduces decoding complexity; (3) a theoretically grounded low-rank regularization based on the Low Frequency Property, supported by a generalization bound; and (4) comprehensive experiments across diverse datasets—including heterophilic graphs—showing consistent improvements over state-of-the-art methods, along with new evaluation metrics (FND/FED) to assess generation fidelity.", "weaknesses": "Despite its strengths, the work has several limitations. First, the comparison with DoG is potentially unfair, as DoG is adapted using pseudo-labels that may not reflect its optimal use in few-shot settings. Second, the computational cost of the truncated nuclear norm (TNN) regularization—particularly the need for SVD—is not adequately addressed, raising scalability concerns. Third, while FND/FED quantify distributional similarity, the semantic correctness of generated nodes lacks intuitive validation (e.g., via embedding visualization). Fourth, the reason for strong performance on heterophilic graphs like Roman-Empire is not deeply analyzed. Finally, certain implementation details (e.g., SVD approximation, prototype initialization) remain unclear, which could affect reproducibility.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lWz8IwmN7m", "reviewer": "Reviewer_ocH3", "summary": "This paper focuses on few-shot node classification (FSNC) tasks with generative model-based data augmentation method. Specifically, the proposed LR-FGDM method adopts the diffusion model-based few-shot learning paradigm and serves as a plug-in module to enhance existing FSNC frameworks. The FGDM model utilizes a hierachical graph autoencoder to obatin label prototypes and synthesize graph attributes for original graph augmentation. A low-rank few-shot classification module is designed to reduce inductive noise from diffusion generation process. Experiments and ablation studies verify the effectiveness of LR-FGDM.", "questions": "Please refer to the weaknesses.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The proposed LR-FGDM is a plug-in-and-play framework that can be integrated into multiple FSNC models.\n2. It is a reasonable idea to apply low-frequency filters to avoid additional noise from generative model.\n3. Comprehensive experimental studies on the model's FSNC performance.", "weaknesses": "1. The overall contribution appears somewhat incremental, as the proposed framework mainly combines elements from prior works on diffusion on graphs (DoG) and graph augmentation–based few-shot learning.\n2. Although the authors claim that the low-rank regularization ensures an upper bound on the Frobenius norm of the predicted labels, the theoretical bound presented in Eq 3 is not illustrative enough to reflect its convergence characteristics.\n3. While the execution time for training and generation in FGDM is reported, the comparative training efficiency against baseline models remains unexplored.\n4. The paper would benefit from a more detailed introduction to the overall FSNC algorithm and a broader discussion on the range of FSNC frameworks to which LR-FGDM can be applied.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)