Title: HeroFilter: Adaptive Spectral Graph Filter for Varying Heterophilic Relations
Venue: NeurIPS 2025
Accepted: yes
Keywords: Graph heterophily

graph signal processing

model generalization
Abstract: Graph heterophily, where connected nodes have different labels, has attracted significant interest recently. Most existing works adopt a simplified approach - using low-pass filters for homophilic graphs and high-pass filters for heterophilic graphs. However, we discover that the relationship between graph heterophily and spectral filters is more complex - the optimal filter response varies across frequency components and does not follow a strict monotonic correlation with heterophily degree. This finding challenges conventional fixed filter designs and suggests the need for adaptive filtering to preserve expressiveness in graph embeddings. Formally, natural questions arise: Given a heterophilic graph $\mathcal{G}$ , how and to what extent will the varying heterophily degree of $\mathcal{G}$ affect the performance of GNNs? How can we design adaptive filters to fit those varying heterophilic connections? Our theoretical analysis reveals that the average frequency response of GNNs and graph heterophily degree do not follow a strict monotonic correlation, necessitating adaptive graph filters to guarantee good generalization performance. Hence, we propose HeroFilter, a simple yet powerful GNN, which extracts information across the heterophily spectrum and combines salient representations through adaptive mixing. HeroFilter's superior performance achieves up to 9.2% accuracy improvement over leading baselines across homophilic and heterophilic graphs.

HEROFILTER: Adaptive Spectral Graph Filter for Varying Heterophilic Relations

Abstract

Graph heterophily, where connected nodes have different labels, has attracted significant interest recently. Most existing works adopt a simplified approach using low-pass filters for homophilic graphs and high-pass filters for heterophilic graphs. However, we discover that the relationship between graph heterophily and spectral filters is more complex - the optimal filter response varies across frequency components and does not follow a strict monotonic correlation with heterophily degree. This finding challenges conventional fixed filter designs and suggests the need for adaptive filtering to preserve expressiveness in graph embeddings. Formally, natural questions arise: Given a heterophilic graph G , how and to what extent will the varying heterophily degree of G affect the performance of GNNs? How can we design adaptive filters to fit those varying heterophilic connections? Our theoretical analysis reveals that the average frequency response of GNNs and graph heterophily degree do not follow a strict monotonic correlation, necessitating adaptive graph filters to guarantee good generalization performance. Hence, we propose HEROFILTER , a simple yet powerful GNN, which extracts information across the heterophily spectrum and combines salient representations through adaptive mixing. HEROFILTER's superior performance achieves up to 9.2 %accuracy improvement over leading baselines across homophilic and heterophilic graphs.

1 Introduction

Graph Neural Networks (GNNs) have emerged as powerful tools for learning from graph-structured data across a broad range of domains [1, 2, 3, 4, 5, 6, 7, 8, 9]. Despite their empirical success, GNNs are known to suffer performance degradation when applied to graphs that exhibit heterophily , a structural property where connected nodes tend to have dissimilar features or labels. This stands in contrast to the homophily assumption, foundational to many GNN architectures, which presumes that adjacent nodes share similar attributes. A growing body of work [10, 11, 12, 13, 14] has shown that this mismatch in assumptions significantly limits the effectiveness of conventional GNNs, often rendering them less effective than simple multilayer perceptrons (MLPs).

To better understand and mitigate the limitations of GNNs under heterophily, researchers have adopted a spectral view grounded in graph signal processing (GSP) [15, 16, 17, 18]. In this framework, graph signals are decomposed into frequency components via the eigen decomposition of the graph Laplacian, where low-frequency components capture smooth variations across the graph, and high-

frequency components capture rapid, localized changes. Classic GNNs such as Graph Convolutional Network (GCN) [19] and Graph Attention Network (GAT) [20] can thus be interpreted as applying low-pass filters, effectively amplifying low-frequency components and suppressing high-frequency noise [15]. This design aligns well with homophilic structures, where relevant information is concentrated in the low-frequency spectrum. However, it is often inadequate for heterophilic graphs, where informative signals may lie in higher-frequency bands.

To address this, recent works have proposed filter-based GNNs that explicitly incorporate high-frequency information [22, 23], and empirical studies have supported the effectiveness of high-pass or mixed filters for heterophilic settings [24, 23]. Nevertheless, most existing approaches rely on a simplifying assumption: low-pass filters are used for homophilic graphs, and high-pass filters are used for heterophilic ones. This binary perspective assumes a monotonic relationship between the heterophily level of a graph and its optimal spectral filter, an assumption that we show is often violated in practice.

To investigate this assumption, we conduct controlled experiments on synthetic graphs with varying heterophily degrees. As shown in Figure 1, we observe that the frequency responses of trained spectral filters [21] exhibit complex, non-monotonic behavior across the spectrum. For example, in graphs with low heterophily (e.g., 0.1), the learned filters reveal significant activation in both low- and mid-frequency bands, contrary to the expectation of a purely lowpass response. Similarly, graphs with high het-

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erophily (e.g., 0.9) do not exhibit a purely high-pass response, retaining strong low-frequency components. These findings suggest that the relationship between heterophily and spectral response is more intricate than previously assumed, and that fixed filter types (low-pass or high-pass) are insufficient for capturing this complexity. These observations motivate two central research questions:

Q1 (Theoretical Understanding): How can we characterize the relationship between graph heterophily, spectral filters, and the prediction performance of GNNs?

Q2 (Adaptive Computation): How can we design adaptive filters and GNN models that perform robustly across graphs with varying and possibly mixed heterophily patterns?

To address Q1, we conduct a theoretical analysis that formally establishes the connection between the heterophily degree of a graph, its spectral representation, and the performance of GNNs. We introduce a novel spectral-domain measure of heterophily and derive an error bound that highlights the limitations of fixed filtering strategies and motivates the need for adaptivity in spectral design.

To address Q2, we propose HEROFILTER , a novel GNN architecture inspired by the MLPMixer [25, 26]. HEROFILTER consists of two key components: (1) a Patcher that identifies spectrally relevant neighbors for each node via adaptive polynomial filters, and (2) a Mixer that aggregates and transforms patch representations along both the patch and feature dimensions. This architecture allows HEROFILTER to modulate its spectral response based on the graph structure and heterophily level, while remaining computationally efficient. We further introduce Fast-HEROFILTER , a scalable variant that avoids eigen decomposition through efficient approximation.

Beyond conceptual proof and proposed theories, we conduct extensive experiments on homophilic, heterophilic, and large-scale real-world graphs. HEROFILTER consistently achieves state-of-the-art accuracy, improving over strong baselines by up to 9.2%. Data and code are available 2 .

2 Preliminaries

Notation. We denote scalars by regular lowercase letters (e.g., c ), vectors by bold lowercase letters (e.g., r ), and matrices by bold uppercase letters (e.g., X ). A graph is represented as G = ( V , E , X ) , where V is the set of nodes, E ⊆ V × V is the set of edges, and X is the node feature matrix. Let n = |V| be the number of nodes. We denote by A and L the adjacency and Laplacian matrices, respectively, and by ˜ A and ˜ L their normalized forms. A summary of key notations is provided in the Appendix B.

Graph Spectral Filters. Spectral GNNs leverage the graph signal processing framework to define convolutional operations in the spectral domain. The eigendecomposition of the normalized Laplacian ˜ L yields ˜ L = UΛU ⊤ , where U ∈ R n × n is the matrix of orthonormal eigenvectors forming the graph Fourier basis, and Λ is a diagonal matrix containing the corresponding eigenvalues. Given a graph signal x ∈ R n , its graph Fourier transform is defined as ˆ x = U ⊤ x , and the inverse transform is x = U ˆ x . The spectral graph convolution between a signal x and a filter g is defined as:

where g ( Λ ) is a learnable filter function applied in the spectral domain.

The GCN [19] can be interpreted as a spectral GNN that employs a first-order Chebyshev polynomial to approximate the filter function, yielding g ( Λ ) = ( I + Λ ) -1 . This corresponds to a low-pass filter, which suppresses high-frequency components and retains smooth signal patterns across the graph [15].

Graph Heterophily. Many GNN models implicitly assume homophily, where connected nodes are expected to have similar features or labels [15]. However, in practice, numerous real-world graphs violate this assumption and instead exhibit high heterophily-connections between nodes of differing labels or characteristics [27]. To quantify this, we adopt the notion of node-level heterophily [11], defined as follows:

Definition 1 (Node Heterophily) . Let N ( v i ) = { v j ∈ V : ( v i , v j ) ∈ E} denote the neighbors of node v i , and let y i denote its label. The node heterophily h i for node v i is defined as:

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The heterophily degree vector h = ( h 0 , . . . , h n -1 ) ∈ R n captures the heterophily level of each node, and the overall heterophily of the graph is computed as the average: 1 n ∑ n -1 i =0 h i .

High-heterophily graphs are common in domains such as social networks, citation graphs, and web hyperlinks. Benchmark datasets including Texas , Squirrel , Chameleon , Cornell , Wisconsin , and Actor [28, 29, 30] are representative examples. In such settings, conventional GCN-based message passing, which aggregates information from neighboring nodes, often leads to feature smoothing that can obscure useful class-distinctive signals, resulting in suboptimal performance.

3 Theoretical Bound in Terms of Graph Heterophily, Graph Filter, and Prediction Performance

In this section, we address Q1 by theoretically analyzing the relationship among the graph filter, the heterophily degree of a graph G , and the generalization performance of GNNs. We begin by demonstrating that the average spectral response of a filter and the heterophily level of the graph do not follow a simple monotonic relationship. This result, formalized in Proposition 1, motivates the need for adaptive filtering. We then establish that adaptive filters can, in principle, align closely with arbitrary label distributions (Proposition 2). Finally, we derive a generalization error bound (Theorem 1) that quantifies how graph heterophily and spectral filter design jointly affect learning performance. Full proofs of all results are provided in the Appendix C.

Spectral Characterization of Heterophily. While graph filters are naturally defined in the spectral domain, heterophily is typically measured in the spatial domain. To bridge this gap, we introduce a heterophily degree vector in the spectral domain via the graph Fourier transform.

Definition 2 (Heterophily Degree Vector in Spectral Domain) . Let ˜ A = UΛU ⊤ be the eigendecomposition of the normalized adjacency matrix ˜ A , where U contains the eigenvectors and Λ is the diagonal matrix of eigenvalues. Then the heterophily degree vector in the spectral domain ˆ h ∈ R n is defined as:

where h is the heterophily degree vector (spatial) and ˆ h = ( ˆ h 0 , . . . , ˆ h n -1 ) , ˆ h i is heterophily of the i -th element in the spectral domain.

Non-Monotonicity of Frequency Response and Heterophily. We next examine how the average spectral response of a graph filter relates to the heterophily in the spectral domain.

Proposition 1. Let ˜ A be the normalized adjacency matrix with eigendecomposition ˜ A = UΛU ⊤ , where λ 0 ≤ · · · ≤ λ n -1 = 1 . Then the average filter response is lower bounded by:

This result reveals that the average filter response 1 n ∑ i g ( λ i ) and the average heterophily in the spectral domain 1 n ∑ i ˆ h i do not follow a monotonic relationship. As shown in Figure 1, graphs with low overall heterophily in the spatial domain exhibit strong responses in mid-frequency bands, contradicting the assumption that low-pass filters are always optimal in such cases.

Need for Adaptive Filtering. The non-monotonic and graph-dependent nature of heterophily-frequency interactions suggests that fixed filters (e.g., purely low-pass or high-pass) are suboptimal. To accommodate diverse frequency needs, we consider an adaptive polynomial filter:

where σ is an activation function, w k ∈ R n are learnable weights, and ⊙ denotes the Hadamard product. GCN [19] is a special case where g ( Λ ) = ( I + Λ ) -1 corresponds to a low-pass filter [15].

We next show that such adaptive filters can match any label pattern in the frequency domain.

Proposition 2. Let Y = ( y 1 , . . . , y n ) be the label vector with y i ∈ { 0 , . . . , C -1 } , and let Λ be the eigenvalue matrix of ˜ A , assuming all eigenvalues are nonzero. Then there exist weights { w k } K k =1 such that:

where g ( Λ ) = ∑ K k =1 σ ( w k ⊙ Λ k ) , σ (0) = 0 , and Align ( · , · ) denotes cosine similarity.

This proposition highlights the expressive power of adaptive filters to align with the target signal, particularly under varying heterophily.

Generalization Error Bound. We now quantify how filter design and heterophily in the spectral domain affect generalization error.

Theorem 1. Consider a binary classification task on a graph G with n nodes. Let X = ( X 0 , X 1 ) be the filtered node features for nodes belonging to class 0 and class 1, respectively. Y = ( y 0 , y 1 ) be the label indicators for classes 0 and 1. Let δ , η be the spectral coefficients of the label and feature differences, respectively. Then the error is upper bounded as:

where c 1 is a constant, I δ = { i | δ i = 0 } , and ψ ( x ) = min { max { x, -1 } , 1 } .

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Downstream Tasks

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Theorem 1 formally characterizes how filter choice and heterophily in the spectral domain influence prediction performance. To explore this relationship, we synthesized nine graphs with heterophily degrees ranging from 0.1 to 0.9 as shown in Figure 2. For each graph, we employed five learnable spectral filters passing specific eigenvalue segments: 0-0.4, 0.4-0.8, 0.8-1.2, 1.2-1.6, and 1.6-2.0. Our synthetic experiments demonstrate that different frequency regions contribute variably to accuracy depending on the heterophily level. This further supports the need for flexible, graph-specific filtering strategies.

4 HEROFILTER Framework

The theoretical analysis in Section 3 highlights two key limitations of existing GNN approaches when applied to heterophilic graphs. First, the spectral behavior of real-world graphs with varying heterophily is complex and non-monotonic, rendering uniform low-pass or high-pass filters ineffective. Second, different frequency bands contribute differently to prediction performance depending on the graph's heterophily structure. These insights motivate the design of HEROFILTER, a graph learning framework that adaptively aligns its spectral processing with the graph's heterophily profile.

HEROFILTER consists of two core modules: (1) The HEROFILTER Patcher , which dynamically selects spectrally relevant neighbors for each node by learning adaptive polynomial filters. This step enables the model to attend to contextually important nodes across spectral bands, beyond simple local neighborhoods, and (2) The HEROFILTER Mixer , which processes each node's patch using a dual-axis MLP architecture that mixes across both spatial (patch) and feature dimensions. This ensures that the model can effectively combine diverse signals arising from the selected patches.

Together, these components allow HEROFILTER to flexibly adapt its receptive field and frequency sensitivity, providing an interpretable and scalable mechanism for handling graphs with varying or mixed heterophily.

4.1 HEROFILTER Patcher

The goal of the patcher is to construct, for each node, a set of contextually informative neighbors that reflect spectral, rather than purely topological, similarity. Traditional GNNs aggregate over immediate neighbors, implicitly assuming local homophily. This assumption breaks down in heterophilic settings.

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Thus, we propose to learn a filter g ( Λ ) that scores nodes based on their spectral alignment, enabling a more adaptive notion of neighborhood.

We define the filter using a learnable polynomial function over the eigenvalues of the normalized adjacency matrix:

where σ is a non-linear activation function and w k are learnable frequency-specific weights. This parameterization allows the model to selectively emphasize different frequency bands based on training data, consistent with our theoretical observation that fixed low/high-pass filters are suboptimal.

To apply this filter in the graph domain, we transform it via the graph Fourier basis U , resulting in the node-node relevance matrix:

Here, R ij measures the spectral relevance of node j to node i under the learned filter. For each node, we sort all other nodes by their relevance scores and select the topp most aligned ones to form the patch. Mathematically:

The resulting patches are structurally diverse and frequency-aware, allowing the model to capture high-order or cross-cluster interactions often present in heterophilic graphs.

For each selected patch, we extract the corresponding feature vectors from X , yielding a patch tensor:

where p is the patch size and d is the feature dimension. These patches now serve as the input for the next step of the model.

4.2 HEROFILTER Mixer

Once patches are constructed, we need to aggregate and transform them into useful node-level representations. Unlike traditional GNNs that rely on fixed aggregation schemes (e.g., mean, sum), our objective is to allow richer, learnable interactions both across patch elements (spatial mixing) and across features (feature mixing). This is motivated by the observation that information relevant to classification may lie in combinations of both positional and attribute-level patterns.

We adapt the MLP-Mixer architecture to this setting by defining two mixing layers:

Patch-Mixing Layer. For each node's patch P v ∈ R p × d , we apply an MLP across the patch dimension:

This operation allows the model to reason over relationships between the selected nodes in the patch, independently for each feature. It enables the capture of role-based, cross-hop, or structurally asymmetric dependencies.

Feature-Mixing Layer. Next, we apply an MLP across the feature dimension:

This allows the model to learn joint representations over node features, enabling flexible composition and transformation of feature information within each patch.

The output of the mixer is a tensor ˜ P ∈ R n × p × d . We then apply a global aggregation function (e.g., mean, sum, or flatten) over the patch dimension to produce a final node representation. This aggregated output is passed to a classification layer.

4.3 Why Mixing Both Dimensions

Patch-mixing captures the heterophily in node context by modeling which other nodes influence the representation of a given node. Feature-mixing captures intra-node complexity by transforming raw input features into abstract representations. Therefore, the combination is essential: patch mixing enables adaptivity to structural irregularities (e.g., in heterophilic regions), while feature mixing enables expressive node-level reasoning.

4.4 Fast-HEROFILTER Patcher

Although the HEROFILTER Patcher is expressive, its reliance on eigen decomposition can be prohibitive for large-scale graphs. To improve scalability, we propose Fast-HEROFILTER, an efficient variant that approximates the patcher using iterative proximity ranking, inspired by personalized PageRank and diffusion-based similarity.

We define the following objective for each node v :

where r v is a ranking vector and e v is a one-hot vector. This objective balances global smoothness and local personalization.

Minimizing J ( r v ) yields the recurrence:

which has the closed-form solution:

To avoid matrix inversion, we approximate this with a truncated Neumann series:

This expansion can be interpreted as diffusing information K steps away from v , with attenuation c k . Since ˜ A k = UΛ k U ⊤ , this operation also approximates spectral filtering.

Finally, we construct patches by selecting the topp entries in each r v , with the Fast-HEROFILTER patching rule:

Since R can be efficiently precomputed and patch extraction is fully parallelizable, this formulation enables scalable and adaptive patch construction, serving as a practical alternative to spectral filtering. Full pseudocode is provided in the Appendix D.

5 Experiment

We empirically evaluate HEROFILTER to validate the key claims made in this work: (1) that adaptive spectral filtering enables robust performance across both homophilic and heterophilic graphs; (2) that our proposed patching and mixing modules contribute significantly to model generalization; and (3) that HEROFILTER offers a scalable, interpretable, and architecture-agnostic foundation for future graph learning models. Due to space constraints, we include additional experiments (scalability, sensitivity, and runtime) in the Appendix G and Appendix H.

We evaluate HEROFILTER on 16 node classification benchmarks encompassing diverse graph heterophily degrees. The homophilic graphs include Cora, CiteSeer, PubMed, and OGBN-Arxiv, where neighboring nodes tend to share similar labels. The heterophilic graphs consist of Texas, Squirrel, Chameleon, Cornell, Wisconsin, Actor, Arxiv-Year, and Snap-Patents, which feature connections between dissimilar nodes and present greater challenges for conventional GNNs. Notably, Arxiv-Year and Snap-Patents also serve as large-scale datasets, with up to 3 million nodes and 14 million edges, enabling assessment of the model's scalability and efficiency in high-volume scenarios.

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We compare against classical GNNs (GCN, GAT, GraphSAGE, ChebNet), spectral models (BernNet, GPRGNN), heterophily-aware methods (H 2 GCN, BM-GCN, G 2 GCN), and graph transformers (GOAT, NAGphormer, Exphormer, PolyFormer, VCR-Graphormer). Detailed dataset and model descriptions are provided in the Appendix E.

5.1 Overall Performance

Tables 1 and 2 summarize classification accuracy across all datasets. Three core insights emerge:

(1) HEROFILTER achieves strong performance across both homophilic and heterophilic regimes. On homophilic graphs, HEROFILTER matches or exceeds state-of-the-art performance (e.g., 70.5% on OGBN-Arxiv). More importantly, HEROFILTER substantially outperforms baselines on heterophilic datasets. For instance, on Squirrel and Snap-Patents, HEROFILTER improves over the next-best model by 9.0% and 9.2%, respectively. This validates the theoretical claim that a fixed low-pass or high-pass filter is insufficient for generalization across heterophily levels and underscores the effectiveness of adaptive spectral filtering.

(2) HEROFILTER remains effective on large and noisy graphs. While several baselines fail on large graphs (marked OOM), HEROFILTER maintains strong accuracy (e.g., 64.2% on Snap-Patents) without sacrificing runtime scalability. This highlights the value of patch-based processing and FastHEROFILTER for memory efficiency-an increasingly critical concern in modern GNN deployments.

(3) HEROFILTER bridges gaps between GNNs, spectral methods, and transformers. Transformer models with strong inductive biases underperform on small or irregular graphs (e.g., Texas, Wisconsin). In contrast, HEROFILTER's simpler design generalizes well without architectural complexity. Compared to BernNet or GPRGNN, HEROFILTER does not require fine-tuned frequency designs and yet achieves superior results, highlighting its robustness and ease of use.

5.2 Ablation Studies

To better understand HEROFILTER's internal design, we conduct controlled experiments analyzing the contributions of the Patcher, Mixer, and positional structure.

Effectiveness of Patchers. We compare HEROFILTER's adaptive polynomial patcher against several alternatives: heat filter, bandpass filter, and a shared-parameter variant. As shown in Table 3, our method consistently outperforms others, especially on heterophilic graphs (e.g., 57.3% on Chameleon). This highlights the importance of a learnable and flexible filter form, consistent with our theoretical insights from Section 3. While bandpass and heat filters capture fixed spectral bands, they lack the adaptability needed for general graphs.

Importance of Patch Order. We assess the role of node order within each patch. As shown in Table 4, randomly shuffling the patch degrades performance on all datasets, with up to 6.3% accuracy loss on Chameleon. This demonstrates that patch structure encodes meaningful spectral and positional information, and that the model effectively leverages this structure, an important design signal for future permutation-sensitive architectures.

Patch-Induced Graphs. To isolate the value of the patcher, we create graphs by linking each node only to its selected patch neighbors, then apply existing GNNs (GCN, FAGCN). Results in Table 5

show significant performance gains, especially for GCN on Chameleon (+16%), validating that HEROFILTER Patcher successfully surfaces informative neighbors even for GNNs not explicitly tuned to heterophily. Conversely, slight degradation on Cora for GCN confirms that fixed low-pass assumptions may conflict with spectrally diverse patches.

Component-Level Analysis. We remove or replace individual layers in the HEROFILTER Mixer to evaluate their contribution (Table 6). Removing the patch-mixing or feature-mixing layers causes consistent drops in performance across datasets, confirming their complementary roles. Notably, the patch-mixing layer provides the largest standalone gain (e.g., +2.02% on Snap-Patents), suggesting its importance in modeling structural diversity within patches.

6 Related Work

Learning on Heterophilic Graphs. Traditional GNNs, including GCN [19], GAT [20], and APPNP [31], are grounded in the homophily assumption, neighboring nodes tend to share similar labels. However, real-world networks often violate this assumption, exhibiting heterophily where connected nodes belong to different classes [32, 27]. This has motivated a range of models that enhance GNN performance under heterophily. H 2 GCN[32] and BM-GCN [33] extend message passing to multi-hop neighborhoods or learn block-level compatibility. Other methods like FAGCN [34] and BernNet [22] leverage spectral insights, using learnable filters to balance low- and high-frequency information. ChebNetII [35] revisits Chebyshev polynomials for deeper spectral modeling, while G 2 GCN [36] introduces gradient gating mechanisms. These works highlight the importance of highfrequency components in heterophilic settings but often rely on heuristic assumptions or task-specific filter tuning.

Spectral GNNs and Frequency-Adaptive Filters. A complementary line of work focuses on understanding GNNs from the spectral perspective. The view of GNNs as graph filters in the spectral domain, applying transformations g ( Λ ) via eigen decomposition of the Laplacian, has been formalized in studies such as [15, 37]. Spectral GNNs such as ChebNet [21] and GPRGNN [37] design polynomial filters to capture information across the graph spectrum. While these methods enhance theoretical interpretability and offer flexibility, most assume a global, fixed filter shape (e.g., low-pass or band-pass), which limits adaptivity. As recent theoretical works note [17, 16], the relationship between graph heterophily and optimal spectral response is non-monotonic and dataset-dependent, suggesting the need for instance-level adaptivity in filter design.

Graph Transformers and Global Attention. Transformer architectures have been adapted to graphs to address the limitations of locality in message passing. Approaches such as NAGphormer [38], GOAT [39], Exphormer [40], VCR-Graphormer [41], and PolyFormer [42] integrate global attention mechanisms, often augmented with positional encodings or spectral bias terms. While effective on large-scale and heterophilic graphs, these models tend to be computationally expensive and architecturally complex. Moreover, their reliance on learned positional encodings can make generalization brittle, especially on smaller graphs or graphs with evolving structure.

Our proposed HEROFILTER bridges the gap between the spectral flexibility of filter-based GNNs and the architectural simplicity of transformer-free encoders. Instead of hard-code frequency biases or depending on hand-tuned spectral forms, HEROFILTER introduces an adaptive polynomial filter that dynamically aligns its frequency response with the graph's heterophily structure. Theoretical results provide the first formal link between graph heterophily, spectral filter response, and generalization error. This connection not only strengthens empirical observations from prior work [27, 22], but also introduces a framework for future frequency-aware and architecture-agnostic GNN design.

7 Conclusion

In this paper, we make three key contributions: (1) a theoretical analysis that, for the first time, formally connects graph heterophily, spectral filter response, and generalization error-challenging the prevailing assumption of monotonic filter-heterophily correlation; (2) a modular architecture that integrates adaptive polynomial filters with a lightweight MLP-Mixer backbone, enabling interpretable and efficient spectral reasoning across diverse graph structures; and (3) extensive empirical validation across 16 benchmark datasets, where HEROFILTER consistently outperforms state-of-the-art GNNs and graph transformers on both homophilic and heterophilic graphs, including large-scale settings.

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NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: The abstract and introduction clearly state the core contributions of HEROFILTER, including its theoretical analysis, adaptive filter design, and experimental improvements. These claims are supported throughout the paper with theoretical bounds, algorithmic innovation, and extensive empirical evaluation.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: The paper discusses the limitation on computation efficiency which they alleviate this limitation by introducing an efficient model variant.

3. Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: The paper provides full assumptions and proofs in Appendix C for Proposition 1, Proposition 2, and Theorem 1, which establish the theoretical grounding for the proposed adaptive filtering mechanism.

4. Experimental result reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: The paper details datasets, baselines, metrics, and experimental setups, and refers to the appendices for parameter settings and runtime details, making reproduction feasible.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification: The paper provides an anonymous open repository link (https://anonymous.4open.science/r/HeroFilter-221C/) that includes code and datasets for reproducing results.

6. Experimental setting/details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: Detailed descriptions of training settings, data splits, patch sizes, optimizers, and hyperparameter tuning procedures are included in the appendices.

7. Experiment statistical significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: All reported results include standard deviations (e.g., 86.8 ± 1.5), indicating that experiments were repeated with multiple random seeds.

8. Experiments compute resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: The runtime and scalability analysis are included in Appendix G. FastHEROFILTER is proposed specifically for scalable deployment, and compute efficiency is discussed.

9. Code of ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: The research uses only public benchmark datasets, poses no human or environmental risk, and follows ethical machine learning practices.

10. Broader impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [NA]

Justification: The impact statement notes that the work is intended to advance machine learning but does not specifically highlight societal impacts.

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: The work does not involve any high-risk models or data types.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: All datasets used are publicly available and either CC BY 4.0 or MIT-licensed. Proper citations are provided.

13. New assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [NA]

Justification: No new datasets or models requiring separate documentation are introduced beyond what is presented in the paper and repository.

14. Crowdsourcing and research with human subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: The paper does not involve any crowdsourcing or human subject research.

15. Institutional review board (IRB) approvals or equivalent for research with human subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: The research does not involve human participants or require IRB approval.

16. Declaration of LLM usage

Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research?

Answer: [NA]

Justification: Large Language Models (LLMs) were used only for writing assistance (e.g., editing and formatting) and had no role in the technical contributions of the research.

A Appendix Content

Appendix B: Key symbols and notations in this paper.

Appendix C: Detailed proofs of Proposition 1, Proposition 2, and Theorem 1.

Appendix D: Algorithm descriptions including HEROFILTER Framework and FastHEROFILTER Framework.

Appendix E: Dataset statistics, including node count, edge count, number of classes, feature dimensions, and heterophily scores for both homophilic and heterophilic graphs. Details of baseline models used for comparison and implementation details.

Appendix F: Parameter settings and tuning procedures, including hyperparameter search spaces.

Appendix G: Scalability and runtime analysis of Fast-HEROFILTER and baseline models showing the inference speed of each model.

Appendix H: Parameter sensitivity studies showing the impact of patch size and filter numbers.

B Symbols and Notations

C Proofs of the Theorems

Proposition 1. Let ˜ A be the normalized adjacency matrix with eigendecomposition ˜ A = UΛU ⊤ , where λ 0 ≤ · · · ≤ λ n -1 = 1 . Then the average filter response is lower bounded by:

Proof. According to the weighted AM-GM inequality [43], we have

As we have valid polynomial filer g ( λ i ) ∈ [0 , 1] for all i [44], then

Since both sides of the inequality are greater than 0, we take the logarithm

Hence, it completes the proof.

Proposition 2. Let Y = ( y 1 , . . . , y n ) be the label vector with y i ∈ { 0 , . . . , C -1 } , and let Λ be the eigenvalue matrix of ˜ A , assuming all eigenvalues are nonzero. Then there exist weights { w k } K k =1 such that:

where g ( Λ ) = ∑ K k =1 σ ( w k ⊙ Λ k ) , σ (0) = 0 , and Align ( · , · ) denotes cosine similarity.

Proof. To match g ( Λ ) with Y , we require g ( Λ ) = c Y , where c = 0 is a scalar. This requires:

̸

If Y i = 0 , then we need g i ( Λ ) = 0 , ∀ i ∈ { 1 , . . . , n } . Let w k,i = 0 for all k , since σ (0) = 0 , the conclusion holds.

If Y i > 0 . Case 1: σ ( x ) is bounded. Suppose σ ( x ) ∈ [0 , M ] for some M > 0 . Since Y i ∈ [0 , C -1] , we have c Y i K ∈ [0 , M ] by choosing c large enough. Since σ is continuous and monotonic, we can always find w k,i to satisfy σ ( w k,i (Λ i ) k ) = c Y i K .

Case 2: σ ( x ) is unbounded. In this case, no scaling constraints are required. The left term can be arbitrarily large if w k,i is chosen large enough. So we can find w k,i , s.t. σ ( w k,i (Λ i ) k ) = c Y i K .

Thus, we have for all i ∈ { 1 , . . . , n } . Then we have

The cosine similarity is

Hence, it completes the proof.

Theorem 1. Consider a binary classification task on a graph G with n nodes. Let X = ( X 0 , X 1 ) be the filtered node features for nodes belonging to class 0 and class 1, respectively. Y = ( y 0 , y 1 ) be the label indicators for classes 0 and 1. Let δ , η be the spectral coefficients of the label and feature differences, respectively. Then the error is upper bounded as:

where c 1 is a constant, I δ = { i | δ i = 0 } , and ψ ( x ) = min { max { x, -1 } , 1 } .

̸

Proof. Let ψ be the clamp function defined as

and for x ∈ [ -1 , 1] , the first-order Taylor expansion of 1 1+ e x is 1 2 -1 4 x . Denote R ( x ) as the remainder term, that is, R ( x ) = 1 1+ e x -1 2 + 1 4 x . since we have

Therefore,

According to the above conclusion,

z = g ( I -˜ A )( X 1 -X 0 ) l , noting that C ≤ ∥ ψ ( z ) ∥ 6 6 ≤ ∥ ψ ( z ) ∥ 4 4 ≤ ∥ ψ ( z ) ∥ 3 3 ≤ ∥ ψ ( z ) ∥ 2 2 ≤ n , then we have

where c 1 is a constant. For any η , we construct ˜ η i = ψ 1 g ( 1 -λ i ) δ i ( η i ) such that | ˜ η i g (1 -λ i ) δ i | ≤ 1 and ∑ i ∈I g,δ,η ψ ( η i g (1 -λ i ) δ i ) = ∑ i ∈I g,δ,δ, ˜ η ˜ η i g (1 -λ i ) δ i . We define m g ≜ min i ∈I g,δ,η ˜ η i δ i .

From the proof of Proposition 1, for any g ( · ) and δ , we have

According to Eq. 29 and Eq. 30, we have

D Algorithm

Algorithm 1 HEROFILTER Framework

Require: source graph ( , , X ) ; adjacency matrix A ; patch size p ; layer number m

1: Preprocessing:

G V E Ensure: Y

2: ˜ A ← Normalization ( A )

4: HEROFILTER Patcher ϕ :

3: Λ , U ← EigenDecomposition ( ˜ A )

5: R ← U g (Λ) U T

6: P ← X [ topp col ( R , p )] 7: P 0 ← P 8: HEROFILTER Mixer:

9:

for

l

1

to

m

do

10:

ˆ

P

l

σ

(

MLP

(

σ

(

MLP

(

LayerNorm

LayerNorm

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l

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)

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patch

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▷

▷

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▷

P

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R

n

×

p

×

d

Mixing along patch dimension

Mixing along feature dimension

13: Y ← MLP ( ¯ P , θ predict ) ▷ Aggregate along patch dimension via summation function and predict node label.

11: ˜ P l ← 12: ¯ ˜ P ← Aggregate ( P m )

14: return Y

In the standard HEROFILTER framework, we generate patches denoted by P = { P v : v ∈ V} for all nodes using the patcher ϕ . The patches P undergo multiple layers of HEROFILTER Mixer operations to yield P m , which is subsequently employed for node label prediction to obtain Y . Initially, we preprocess the data by normalizing matrix A , followed by the extraction of Λ and U from the modified adjacency matrix ˜ A . Applying the patcher ϕ on X to our graph, we obtain patches of size [ n × p × d ] , where n denotes the number of nodes, p the patch size, and d the dimension of the input node features. Each patch, represented by a [ p × d ] matrix, encapsulates the extracted features from a node and its top p neighboring nodes. These patches are then processed by

p

a patch-mixing layer, which mixes information along the patch dimension p by transposing its feature dimension with its patch dimension and passing it through an MLP layer. This process is followed by Layer Normalization and an arbitrary activation function, after which another transpose operation is conducted between the feature and patch dimensions, resulting in a tensor of size [ n × p × d ] . This tensor is further processed by a feature-mixing layer, utilizing an MLP to operate on the d -dimension. By passing P through m layers, the final P m is constructed. Finally, we aggregate along the patch dimension p of P m using an arbitrary aggregation function, yielding a final representation of size [ n × d ] . An additional MLP is applied to this final representation for classifying the label of each node, resulting in a matrix of size [ n ×|Y| ] .

Algorithm 2 Fast-HEROFILTER Framework

In the Fast-HEROFILTER framework, we generate patches P = { P v : v ∈ V} for all nodes using patcher ϕ fast, mix P using multiple layers of patch-mixing operations to obtain P m , and predict node labels Y using ¯ P m . The patcher process involves minimizing an objective function, as described in Eq. 12, to obtain the patcher scores r v for each node v . These scores are then stacked to form the patcher score matrix R . The dangling scalar c controls how the patcher adapts to different degrees of heterophily during score computation. When c is lower, the focus of the patcher score matrix R focuses on relationships between nodes with similar spectral properties. On the other hand, when c is higher, the patcher score matrix R captures relationships across nodes with varying spectral characteristics, allowing the model to adapt to more diverse heterophily patterns. The choice of c enables the framework to flexibly adjust its spectral response based on the underlying heterophily structure of different graphs. The rest of the framework remains the same as the standard HEROFILTER framework.

E Experiment Setup

Data Splits. In our experiments, we are strictly using the same training and testing environment across different baselines. Following the official data split provided by [45], heterophilic graph datasets (Texas, Squirrel, Chameleon, Cornell, Wisconsin, Actor) and homophilic graph datasets (Cora, CiteSeer, PubMed, OGBN-Arxiv) are split roughly by 48 : 32 : 20 . Two large-scale heterophilic graph datasets (arxiv-year and snap-patents) are following [46], using a 50 : 25 : 25 random data split. All datasets are provided with masks in the newest version of Pytorch Geometrics.

Implementation Details. We employ the Adam optimizer [47] with a learning rate of 0.01 and a weight decay of 5e-4 for training the model. The default training setting for all models is performed using torch-geometric default masks (train, validation, and test) and trained for a maximum of 500 epochs, a hidden dimension of 64, a dropout rate of 0.5, and a number of layers of 2. Early stopping is applied with a patience of 50 epochs, which monitors the validation loss and terminates the training if

there is no improvement observed within the specified patience. We employ each baseline method as a representation learner, subsequently concatenating their outputs with a two-layer MLP to perform the node classification task. To assess the performance of the models, we use the standard accuracy metric, which is commonly adopted in node classification tasks.

Datasets. We evaluate our model using several standard graph datasets originated from [48, 30, 28, 46, 49, 50]: Cora, Citeseer, PubMed, OGB-Arxiv, Snap-Patents, Arxiv-Year, Texas, Squirrel, Chameleon, Cornell, Wisconsin, and Actor. These datasets are diverse in terms of their sizes and degrees of heterophily. Specifically, homophilic graph datasets are Cora, Citeseer, and PubMed, while heterophilic graph datasets include Texas, Squirrel, Chameleon, Cornell, Wisconsin, and Actor. To demonstrate the effectiveness of HEROFILTER on larger scale graph datasets, we also evaluate our model on Penn94 [46], Arxiv-Year, OGBN-Arxiv [49], and patent networks [50] datasets.

Comparison Methods. We benchmark our model against a variety of established methods. For homophilic graphs, we consider: GCN [19], GAT [20], GPRGNN [37], ChebNet [21]/ChebNetII [35], APPNP [31], GCN-JKNet [51], GraphSage [52], and FAGCN [23]. For heterophilic graphs, we explore: H 2 GCN [32], BM-GCN [33], BernNet [44], and G 2 -GCN. For graph transformer based methods, we compare with: NAGphormer [38], VCR-Graphormer [41], Exphormer [40], Polyphormer [42] and GOAT [39]. Lastly, we incorporate MLP.

F Details of Parameter Settings and Tuning

We perform a grid search to find the optimal hyperparameters for our model, including the learning rate, patch size, and filter number. The search space is defined as follows: the learning rate ∈ { 0 . 01 , 0 . 008 , 0 . 005 , 0 . 003 , 0 . 001 } , the patch size ∈ { 8 , 16 , 32 , 64 , 96 } , and the filter number ∈ { 10 , 50 , 100 , 150 , 200 } . For Fast-HEROFILTER, we set c as 0.5 to balance the spectral response across different frequency components. The optimal hyperparameters are determined based on the performance of the validation set.

G Scalability and Runtime Analysis

Scalability Analysis on HEROFILTER Mixer. We analyze the runtime of the algorithm as the number of nodes and patch size varies to assess the scalability of our proposed approach in Figure 4a. We perform experiments on synthetic datasets with varying numbers of nodes from {10, 100, 1,000, 10,000, 100,000}. We also consider different patch sizes, specifically {8, 16, 32, 64}. The runtime results are reshaped into a matrix for visualization purposes. Figure 4a shows the runtime in seconds on a logarithmic scale for the different patch sizes as a function of the number of nodes. As expected, the runtime increases with the number of nodes and patch size. Nevertheless, Fast-HEROFILTER demonstrates reasonable scalability as the growth in runtime is sub-linear, indicating that our approach

Chameleon

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Patch size 8

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Patch size 32

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(a) Scalability analysis of Fast-HEROFILTER. The x-axis shows the number of nodes pro- cessed, and the y-axis shows the running time per iteration.

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10

can handle large-scale graph datasets effectively. The results also highlight the importance of selecting an appropriate patch size to balance the trade-off between computational efficiency and performance.

Runtime Analysis on HEROFILTER Mixer. We further evaluate the runtime performance of HEROFILTER against existing filter-based and graph transformer methods across four datasets: Squirrel, Chameleon, Arxiv-year, and Snap-patents. Since HEROFILTER patcher can be computed offline as a preprocessing step, we only record the inference time of HEROFILTER mixer. Shown from Figure 4b, our findings reveal that HEROFILTER consistently achieves the fastest inference time across all datasets. Specifically, on Squirrel and Chameleon, HEROFILTER executes in 0.92 and 0.75 milliseconds, respectively, outperforming the next fastest method (Exphormer) by 8 times. For larger datasets like Arxiv-year and Snap-patents, HEROFILTER maintains its efficiency (45.60 and 71.66 milliseconds), while transformer-based methods like VCR-Graphormer either require significantly more time (1832.73 milliseconds) or fail to complete due to out-of-memory (OOM) errors.

H Parameter Sensitivity Analysis

We conduct a parameter sensitivity analysis to investigate the impact of patch size and the number of filters in Figure 5. The analysis was performed on two datasets, namely the Cora and Texas datasets. We varied the patch size from {8, 16, 32, 64} and the number of filters from {10, 50, 100, 200}. We measure the accuracy of the model for each combination of parameters. Figure 5 shows the results of the parameter sensitivity analysis. The plots reveal that for both datasets, increasing the patch size and the number of filters generally leads to higher accuracy, although the improvements tend to plateau beyond certain values. This indicates that our model can effectively handle different patch sizes and filter configurations while highlighting the importance of selecting appropriate parameters to achieve optimal performance.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a viewpoint on the issue of homophilic or heterophilic graphs. There exists a nonlinear relationship between the degree of graph heterogeneity and high and low-frequency filters rather than a simple linear one. Therefore, a single high or low-frequency filter cannot accomplish the task, and an adaptive filter needs to be introduced. A HERO filter framework is proposed. The first part first constructs a polynomial filter by weighting frequency domain information, establishes a node correlation matrix based on this, and finally sorts the correlation degrees and reconstructs the node neighbors. The second part adopts a dual MLP hybrid structure, obtaining patch dimensions from the reconstructed node neighbors and new feature dimensions, thereby enhancing the model's ability to express complex heterophilic structures. In addition, a fast HERO filtering method is proposed for large-scale graphs. Experiments show that it works well in heterophilic graphs.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n1. The proposed approach is novel, linking the heterogeneity of images, spectral filter responses, and propagation errors. Single high-frequency and low-frequency filters, and even some composite filters, are considered from the perspective of heterogeneity or homogeneity, necessitating an adaptive filter to address the issue.\n2. The first part of the model, which re-establishes node relationships based on frequency domain information, is a novel concept that can capture the relationships between heterophily nodes.\nWeaknesses\n1. From the perspective of error bound, only theoretical analysis is provided without proof, which seems somewhat vague.\n2. In the second part, the use of a two-layer MLP structure lacks interpretability, and I do not believe it can effectively address the issue of node feature representation. From the ablation experiments, the results of removing the mixing layer are also not significantly different.\n3. There is a crucial low-level error. There are only 34 references in the main text, but references 35-39 are mentioned in the attachment, and there are no references in the attachment. I am not sure what references 35-39 are.\n\nQUESTIONS:\n1.For the second part of the model, could other hybrid methods be improved and compared in experiments to enhance versatility?\n2. Regarding the effectiveness of the two-layer MLP structure, Table 5 only shows the results of two large-scale data sets. Could you provide more ablation experiment results from heterophily datasets?\n3. The first part of the model reconstructed the neighbors. Could you compare the heterophily degree of the graph after reconstruction? This would also support the proposed theory.\n\nLIMITATIONS:\nNo. The paper does not discuss limitations.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-24"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates the heterophily of graphs and observes that the frequency response varies with the degree of heterophily and the graph's eigenvalues. Building on this insight, the authors propose HEROFILTER, a method that incorporates neighborhood information based on spectral affinity instead of relying solely on topological connectivity. The approach is evaluated on both homophilic and heterophilic graph datasets for node classification tasks.\n\nSTRENGTHS AND WEAKNESSES:\n## Strength \n\nSpectral filter design is an important area in graph neural networks. This paper makes a valuable study by focusing on heterophilic graphs and highlighting the limitations of standard spectral methods that rely solely on low-pass or high-pass filtering. The proposed architecture introduces an adaptive polynomial filter based on spectral affinity, aligning the filter design with the node heterophily. This alignment allows the method to better capture relevant neighborhood information. The empirical results further support the effectiveness of the approach, showing improved performance on node classification tasks, especially in heterophilic settings.\n\n## Weakness\n- The authors use the heterophily to define the heterophily degree in spectral domain (Definition 2) and analyze this quantity. However, the motivation for focusing on node heterophily rather than edge heterophily is not discussed. The paper would benefit from a theoretical or empirical justification of why edge-level heterophily is not considered, and what insights could be gained by analyzing heterophily along edges instead.\n- It is not clear what is the definition of heterophily degree, is it $\\mathbf{h}$ or $h_i$ or $\\hat{\\mathbf{h}}$ or $\\hat{h}_i$?\n- The proof for Property 1 in Appendix C is hard to follow. The authors jump to fast between lines without clear explanation. When the equality holds, what insight or interpretation can be drawn from this result?\n- The theoretical results are inconsistently presented. The main text refers to “Property” and “Theorem” without formally stating them or clearly labeling their locations. For example, there is no “Theorem 1” in the main paper, and the proof of Theorem 3 in the main text appears as “Theorem 1” in the appendix, creating confusion. Additionally, the use of the term “Property” lacks formal mathematical rigor. The paper would benefit from standard terminology such as “Theorem,” “Proposition,” “Lemma,” and “Remark,” along with clearer cross-referencing and structure.\n- Lines 136–140 assert that the align function highlights the expressive power of adaptive filters under heterophily. However, the paper does not explain why computing cosine similarity with the label vector is appropriate or effective, particularly in heterophilic graphs. \n- The paper is missing spectral GNN models such as [1-9].\n- The paper does not specify how the baselines are trained, nor does it report the hyperparameters used for these methods. \n- The dimensions involved in the Patch-Mixing Layer (Equation 10) and Feature-Mixing Layer (Equation 11) are not specified. It is unclear what inputs LayerNorm and MLP act on and what outputs they produce.\n- It is not clear whether the results in the experiments are produced using HEROFILTER or its faster variant, Fast-HEROFILTER. Furthermore, while Fast-HEROFILTER is proposed for scalability, there is no analysis of the trade-off between runtime efficiency and performance.\n- The generalization error bound in lines 141–146 is difficult to interpret. First, it is unclear what $\\mathbf{X} = (\\mathbf{X}_0, \\mathbf{X}_1)$ represents, what are filtered node features? The dimensions of $\\mathbf{X}_0$ and $\\mathbf{X}_1$ are unspecified. Second, it is not clear why the analysis is limited to binary classification or how it generalizes to multi-class settings.\n\n[1] How powerful are spectral graph neural networks\n\n[2] Graph neural networks with convolutional ARMA filters\n\n[3] Specformer: Spectral graph neural networks meet transformers\n\n[4] Cayleynets: Graph convolutional neural networks with complex rational spectral filters\n\n[5] Equivariant machine learning on graphs with nonlinear spectral filters\n\n[6] Graph neural networks with learnable and optimal polynomial bases\n\n[7] Rethinking spectral graph neural networks with spatially adaptive filtering\n\n[8] Piecewise constant spectral graph neural network\n\n[9] SLOG: An inductive spectral graph neural network beyond polynomial filter\n\n### Other \n- Several references ([35]–[42]) are cited in the appendix but not listed in the reference section. \n- The legend text in Figure 3 is too small and difficult to read.\n- Algorithm 1 and Algorithm 2 in the appendix both carry the same title \"Fast-HEROFILTER Framework\"\n\nQUESTIONS:\n- The performance on the Chameleon and Squirrel datasets, which have been shown to contain duplicate nodes [1]. The results may therefore be biased. What are results on the cleaned versions of these datasets? \n- What is $\\mathbf{R}\\_v$ in Equation 9\n- Please clarify how these quantities are connected in the figure 1a. Does each point in the figure corresponding to a eigenvalue $\\lambda_i$, a heterophily degree w.r.t i, and frequency response $g(\\lambda_i)$?\n- In Figure 1b, what type of accuracy is here? \n\n\n[1] A critical look at the evaluation of GNNs under heterophily: Are we really making progress?\n\nLIMITATIONS:\nThe authors discuss the scalability issue and proposed a fast variant of the method.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposed an method to adaptively learn the context ( i.e., multi-hop neighbors ) of central node, that leverage the PPR importance scoring to sample the neighbors within K-order neighborhoods, aiming at addressing the heterophily diversity among nodes. The experiments across the full heterophily spectrum of graphs have been conducted to validate the effectiveness of the proposed method.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The paper is well-written and easy to follow.\n2. Theoretical analysis of the non-monotonic relationship between average filter response and the spectral heterophily is present.\n3. The proposed method consistently shows competitive performance on homophilic and heterophilic graphs.\n\nWeaknesses:\n1. The goal of this work is to learn a filter that scores nodes based on their spectral alignment so as to enable an adaptive notion of neighborhood, which means that specific parameter vector $\\omega_{ki}$ (in Eq.(6)) will be assigned for  $\\lambda_i$ of different order $k$. I think it is  the core of this paper. However, an alternative approach based uniform diffusion (with constant $c$) is used. While functionally effective, this method shares notable commonalities with established attention or diffusion mechanisms, offering limited novelty.\n2. The claimed novelty is undermined by insufficient comparisons with relevant works (e.g., [1-5]), including adaptive filtering approaches , diffusion/pageRank based and attention based methods.\n3.  No performance evaluation of the proposed HETEROFIlTER Patcher(not the fast implementation). Despite of its high time complexity, it is worthwhile to show the performance of HETEROFIlTER on the small/mediate heterophilic graphs.\n\nRef. [1]. Dong, et al. *Graph Neural Networks with Adaptive Frequency Response Filter*. CIKM 2021.\n\n[2].  J Guo, et al. *Rethinking spectral graph neural networks with spatially adaptive filtering*. arXiv preprint arXiv:2401.09071, 2024. 4, 2024.\n[3].  A Saha, et al. *Learning Adaptive Neighborhoods for Graph Neural Networks*. ICCV 2023.\n[4].  J Wang, et al. *Heterophily-Aware Graph Attention Network*. (arXiv:2302.03228) PR 2024. --HA-GAT\n[5].  E. Chien, J. Peng, P. Li, O. Milenkovic, *Adaptive universal generalized pagerank graph neural network*,  ICLR, 2021.\n\nQUESTIONS:\n1) It is unclear why let $g(\\Lambda)$ align with $\\mathbf{Y}$ (the label vector)\n2) Do the datasets Squirrel and Chamelon are duplicate removed ones?\n3) Table 5 is the ablation study on two large datasets (where most of comparison methods are OOM), what about other smaller heterophilic graphs?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents HEROFILTER, an adaptive spectral graph filter framework designed to address performance challenges of Graph Neural Networks (GNNs) on heterophilic graphs. Through theoretical analysis and experiments, the authors demonstrate that the relationship between graph heterophily and optimal filter response is not simply monotonic but varies complexly across frequency components. This finding challenges the conventional fixed filter design paradigm. Based on this observation, the authors propose a framework capable of adaptively capturing information across different frequency ranges. Experimental results show that HEROFILTER performs excellently on both homophilic and heterophilic graphs.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n1. The paper establishes a rigorous theoretical connection between graph heterophily, spectral filters, and GNN prediction performance, introducing a spectral domain measure of heterophily and deriving error bounds.\n\n2. The authors challenge the simplified assumption in existing research that \"homophilic graphs use low-pass filters, heterophilic graphs use high-pass filters,\" demonstrating through controlled experiments that this relationship is actually more complex and non-monotonic. This insight is crucial for understanding GNN behavior on heterophilic graphs.\n\n3. The authors conduct extensive experiments on 12 datasets with varying degrees of heterophily, showing that HEROFILTER outperforms existing methods on both homophilic and heterophilic graphs. \n\n**Weaknesses**\n\n\n1. Although the theoretical analysis in this paper effectively supports the main motivation of this paper, the design of specific solutions is still heuristic, resulting in a disconnect between theoretical analysis and practical solutions.\n\n2. Motivation has actually been explored in previous studies. [1] systematically discussed Q1 in its work, and [1-3] provided solutions to Q2. Authors should reasonably cite these references and discuss the similarities and differences between their work and these studies.\n\n3. As a key module of this work, the design of HEROFILTER Mixer is very similar to that of [4]. This requires clarification and explanation of the similarities and differences, as well as explicit references to [4].\n\n4. Unclear details. The author did not specify whether the experimental results were obtained using the original version of herofilter or the fast version. In addition, the number of steps when implementing the fast version was not mentioned. It is important to note that the number of passes is critical for the patcher, as it determines the size of the subgraph over the target node on which the filter is estimated [5].\n\n5. Fast-HEROFILTER actually follows the practice of APPNP, so it may not be suitable as a contribution point.\n\n[1] Node-oriented spectral filtering for graph neural networks. TPAMI 2023\n\n[2] How powerful are spectral graph neural networks. ICML 2022\n\n[3] Graph neural networks with diverse spectral filtering. WWW 2023\n\n[4] A Generalization of ViT/MLP-Mixer to Graphs. ICML 2023\n\n[5] The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. IEEE signal processing magazine 2013\n\nQUESTIONS:\nIn addition to the issues mentioned in “Weaknesses,” there are the following questions. It would be greatly appreciated if the authors could respond.\n\n1. What is the significance of the spectral correlation matrix mentioned in Section 4.1? Please elaborate and explain further.\n\n2. How is the specific operation of selecting top-k performed, and how is gradient backpropagation implemented?\n\n3. It is recommended that the results of [1] and [2] be added to the experimental comparison, and that a qualitative analysis of the patcher and mixer be added.\n\n4. In the implementation details section of the supplementary materials, the authors mention adding an additional two-layer MLP as a classifier for all baselines. I do not think this is reasonable, as this practice actually modifies the original model architecture, such as APPNP and BernNet. The authors need to provide an explanation for this practice.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper investigates graph heterophily, where connected nodes in a graph have different labels, challenging the conventional use of fixed spectral filters in Graph Neural Networks (GNNs). Traditional approaches apply low-pass filters for homophilic graphs and high-pass filters for heterophilic graphs, but the study reveals that the relationship between heterophily and filter response is complex and not strictly monotonic. This suggests that fixed filter designs are insufficient, as optimal filter responses vary across frequency components. Through theoretical analysis, they confirm that GNN performance does not correlate monotonically with heterophily, necessitating adaptive filters for better generalization. They propose HEROFILTER, a novel GNN that extracts and combines salient representations across the heterophily spectrum using adaptive mixing. HEROFILTER outperforms leading baselines by up to 9.2% in accuracy across both homophilic and heterophilic graphs, demonstrating its effectiveness and versatility.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n\n1. **Well-Motivated Approach:** The study clearly articulates the limitations of conventional fixed filter designs and effectively motivates the development of adaptive filtering to address varying heterophily degrees.\n\n2. **Robust Theoretical Analysis:** The paper provides a thorough theoretical foundation, demonstrating that the relationship between graph heterophily and GNN frequency response is not strictly monotonic, justifying the need for adaptive filters.\n\n3. **Strong Performance:** The proposed HEROFILTER achieves up to 9.2% accuracy improvement over leading baselines across both homophilic and heterophilic graphs, showcasing its effectiveness and versatility.\n\n## Weaknesses:\nThe paper appears comprehensive, with no clear weaknesses noted in the provided context.\n\nQUESTIONS:\nN.A.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-05"}