Piecewise Constant Spectral Graph Neural Network
Authors: Vahan Martirosyan
Jhony H. Giraldo
Fragkiskos D. Malliaros
Venue: Accepted by TMLR ()

Abstract: Graph Neural Networks (GNNs) have achieved significant success across various domains by leveraging graph structures in data. Existing spectral GNNs, which use low-degree polynomial filters to capture graph spectral properties, may not fully identify the graph's spectral characteristics because of the polynomial's small degree. However, increasing the polynomial degree is computationally expensive and beyond certain thresholds leads to performance plateaus or degradation. In this paper, we introduce the Piecewise Constant Spectral Graph Neural Network(PieCoN) to address these challenges. PieCoN combines constant spectral filters with polynomial filters to provide a more flexible way to leverage the graph structure. By adaptively partitioning the spectrum into intervals, our approach increases the range of spectral properties that can be effectively learned. Experiments on nine benchmark datasets, including both homophilic and heterophilic graphs, demonstrate that PieCoN is particularly effective on heterophilic datasets, highlighting its potential for a wide range of applications.

Published in Transactions on Machine Learning Research (04/2025)
Piecewise Constant Spectral Graph Neural Network
Vahan Martirosyan
Université Poris-Saclay
CentraleSupélec, Inria, France
Jhony H. Giraldo
shony.giraldde booor.ggirrad@@tccccm.pppp....
LTCI, Télécom Paris Institut Polytechnique de Paris, France
Fragkiskos D. Malliaros
Université Poris-Saclayy
CentraleSupékec, Inria, France
Reviewed OpenReview: https: opppreveepeerevieu.. net/ formm? id= sdVVDDDOD
Abstract
Graph Neural Networks (GNNs) have achieved significant success across various domains by leveraging graph structures in data. Existing spectral GNNs, which use low-degree polynomial filters to capture graph spectral properties, may not fully identify the graph's spectral characteristics because of the polynomial's small degree. However, increasing the polynomial degree computationally expensive and, beyond certain thresholds, leads to performance platcaus or degradation. this paper, we introduce the Piccowise Constant Spectral Graph Neural Network (PieCoN) to address these challenges. PieCoN combines constant spectral filters with polynomial filters to provide more flexible way to leverage the graph structure. By adaptively partitioning the spectrun into intervals, our approach increases the range of spectral properties that can be effectively learned. Experiments on nine benchmark datasets, including both homophilic and heterophilic graphs, demonstrate that PieCoN is particularly offective on heterophilic datasets, highlighting its potential for a wide range of applications. The implementation of PieCoN available at htttts://ggttuu. com/vmart20/PicCn
1
Introduction
Graph Neural Networks (GNNs) [Wu al. 2021 Zhou al. 2020) have demonstrated remarkable performance across various application domains. They have boen succossfully applied in areas such social network
analysis (Panagopoulos et al. 2023). recommendation systems (Wu et al. 2019 Ying et al. 2018 drug
discovery (Jiang et al. 2021 Bongini et al. 2021), and materials modeling (ooley et al. 2019 9 Duval et al.
2023), where data can naturally be represented as graphs. GNNs can be broadly classified into two types:
spatial and spectral GNNs. Spatial GNNs (Kipf & Welling
2017 Veličković et al. 2018: Chien et al.) 2021
use a message passing approach to learn node representations by collecting information from neighboring nodes. This approach allows spatial GNNs to capture local structural information and adapt to varying neighborhood sizes. On the other hand, spectral GNNs (Wang & Zhang, 2022; Defferrard al.) 2016 He et al. 2021 Castro-Correa et al. 2024) use the graph's spectral characteristies, such the graph Laplacian's 's eigenvalues and cigenvectors, to transform node foatures. By leveraging the spectral domain, these models can capture global structural patterns and apply graph convolution operations in the frequency domain.
Many existing spectral GNNs use low-degree polynomial filters, which approximate the filtering functions by applying polynomials to the graph's Laplacian matrix or other graph-shift operators (Wang & Zhang 2022 Defferrard al. 2016 He al. 2021). One disadvantage these low-degree polynomial filters is that they continuous and, because of the low degree, may not give enough weight to specific eigenvalues, the
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change between closely spaced eigenvalues cannot vary significantly. This can be problematic, particularly in real-world graphs where certain eigenvalues like zem have large multiplicities [Lu et al. 2024 Lim et al. 2023
In this paper, we propose the Piecewise Constant Spectral Graph Neural Network (PieCoN) to overcome this limitation. Our approach combines constant spectral filters with polynomial filters to better capture the spectral properties of the graph. The constant filters are defined by setting the values in the diagonal eigenvalue matrix to ones within specific intervals and zeros elsewhere, effectively isolating different frequency bands.
By partitioning the spectrum into intervals, PieCoN ex- pands the range of spectral characteristics that can be learned, improving the model's performance. Figure] compares the response of a JacobiConv filter (Wang & Zhang 2022) versus the response of a PieCoN filter across the spectrum. The figure shows how polyno- mial filters produce smooth response, while piecewise constant filters can sharply focus selected intervals, capturing crucial spectral properties that polynomial fil- ters miss. For example, PieCoN can create a sharp drop at eigenvalue 0, which is impossible to achieve with low- degree polynomial filters like JacobiConv. This sharp discontimuity crucial because eigenvalue 0 often has high multiplicity in real-world graphs (as detailed in Table 1 and contains important structural information. Our theoretical analysis provides important insights about spectral GNNs by establishing error bounds for polynomial spectral filtering and proving that our model is invariant to eigenvector sign flips and basis changes. We validate our approach on standard benchmark datasets, showing improved performance, particularly when handling graphs with multiple zero eigenvalues.
10.0
6.0-
Filter Output
4.0
2.0-
0.0
-2.0
JacobiConv PieCoN
-4.0-
-6.0
-1.0
-0.8
-0.5 -0.2 0.0 0.2 0.5 0.8 1.0 Eigenvalues
Comparison of JacobiConv and PieCoN trained filters the Chameleon dataset.
Figure
Our main contributions as follows:
We introduce a novel spectral GNN model (PieCoN) that uses piecewise constant filters combined with polynomial filters to enhance learning from the spectral properties of graphs.
We propose a new method isolate frequency bands in the spectral domain by partitioning the eigenvalue spectrum into intervals, allowing the model to focus crucial spectral properties.
We demonstrate, through experiments, that PieCoN outperforms or shows competitive performance against spatial and spectral GNNs on real-world datasets.
2
Related Work
GNNs have evolved significantly since the early work by Bruna et al. 22014) who introduced the first modern spectral-based graph convolution network using the graph Fourier transform. Later, Kipf & Welling (2017) simplified this approach with the Graph Convolutional Network (GCN) model, which applies a first-order approximation of spectral filters to make GNNs more scalable. Other important advancements include Graph Attention Networks (GAT) /Velicković et al.; 2018), which use attention mechanisms to weigh neighboring nodes differently, and GraphSAGE (Hamilton et al. 2017), which focuses on inductive learning by generating node embeddings through sampling and aggregation techniques.
Message-passe pastnggg neural networks (MPNNs) Gilmer el al. 2017 are class of GNNs where nodes iteratively exchange and aggregate information with their neighbors. Besides GCN and GAT, several other message- passing approaches have emerged address specific challenges. For instance, the Graph Isomorphism Networks (GIN) model (Xu et al. 2019h is designed to be as powerful as the 1-WL test for distinguishing non-isomorphic graphs, thus improving expressiveness.
I Note that, for the Laplacan matrix, the multiplicity eigenvalue zero corresponds to the number connected components. This correspondence does not hold for the normalized adjacency matrix we employ here.
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Spectral GNNs leverage the eigenvalues and cigenvectors of the graph Laplacian to perform graph convolution in the spectral domain. ChebNet (Defferrard et al. 2016) introduced the use of Chebyshev poly polynomials for spectral filtering, improving the scalability of spectral GNNs. JacobiConv (Wang & Zhang 2022) further enhances this by employing orthogonal Jacobi polynomials to flexibly Iearn graph filters. Recent work by Maskey et al. (2023) introduces a novel ional graph Laplacian approach defined in the singular value domain address over-smoothing, providing theoretical guarantoes for both directed and undirected graphs. For heterophilic graphs with label noise, R*LP (Cheng et al. 2024 demonstrates that increasing graph homophily can help mitigate the impact of noisy labels.
While previous spectral GNNs have leveraged polynomial filters to approximate spectral properties of graphs, they are limited by their lack of flexibility in handling critical eigenvalues because of the low degree of the polynomial filters. DSF (Guo et al., 2023) addresses this by employing a shared network on positional encodings to learn unique polynomial coefficients per node, highlighting the advantages of nodo-specific filters over node-unified ones. NFGNN (Zheng et al., 2023) proposes a node-oriented spectral filtering approach that learns specific filters for each node, better adapting to local homophily patterns. PP-GNN (Lingam al., 2022) also explores piecewise spectral filtering, but our approach differs in several key aspects. While PP-GNN uses a fixed two-part partitioning with low-degrce polynomials and continuity constraints, PicCoN implements adaptive K- K-part partitioning with constant filters that allow for discontimuities between intervals. This design enables PieCoN to better capture critical eigenvalues with sharper responses. Additionally, our approach decomposes filters into positive and negative parts (Eq. (10)) and maintains invariance to eigenvector sign flips and basis changes (Proposition 2).
3 Background
We consider an undirected graph G = EV,E, X) with n nodes. Here, V = (u), . bp the set of nodes, & S V ((8)= = =m) represents the set of edges, and X E mxd is the node feature matrix. The adjacency matrix A (0, 1)mxn of the graph is defined as A. 1 if there is an edge between nodes i; and uy, and Aij 0 otherwise. The degree matrix D = diag(d),..... dd.) is a diagonal matrix with the i-th diagonal entry as d4 = E, Aij, representing the degree of node The normalized adjacency matrix Á is defined D à AD- The normalized adjacency matrix is symmetric and can be decomposed as Å UAUT, where A is diagonal matrix containing the eigenvalues A; of Å, and U is an orthogonal matrix whose columns are the corresponding eigenvectors u, Let s be the number of distinct eigenvalues of Å, denoted by X For an cigenvalue X, we define v(A) as the algebraic multippicity of X, which is the number of times appears in the eigenvalues. A graph signal is a function that assigns a scalar value to each node in the graph. Formally, a graph signal can be represented as a vector x E R", where each entry ø, corresponds to the signal value at node t;.
dn))
3.1 Graph Neural Networks
Graph Neural Networks (GNNs) are deep learning architectures designed to learn from graph-structured data. They can be broadly categorized into two lypes: spatial GNNs and spectral GNNs. Both approaches aim to learn node representations by leveraging the graph structure and node features but differ fundamentally in how they process graph information.
Graph Fourier Transform and Spectral GNNs. The graph Fourier transform OOtega et al. 2018 of graph signal æ E R is defined as à = U x, and its inverse is given by æ = U (hhmaan et al. 2013 This transform projects a graph signal from the spatial domain (node space) to the spectral domain (frequency space), similar to how the classical Fourier transform operates on time signals.
The spectral filtering of a signal æ with a kernel v is defined as:
u G x (6 UVU
(1)
where V- diag(01...... 0n) represents the spectral kernel coefficients and @ denotes element-wise multi- plication. Spectral GNNs leverage this graph Fourier transform to define convolution operations in the
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Initial filters
Trained filters
Constant filters
Trained constant filter
Input Graph
Eigenvalue segmentation Alg
Combined Filter
0.3
To. 7
Tx-1
ourTa
Polynomial filters
Trained polynomial filter
13
R--1
Σ
ouT2 > Aad
Ax-1
Σ
Z,
ou ToTToo
A of (XW
9
Figure 2: Overview of the PieCoN model. Our method processes an input graph through eigenvalue segmentation (Alg. I create constant filters, while separately applying polynomial filters. These filters are trained and combined to create the final spectral filter.
spectral domain. These models approach graph learning from a signal processing perspective, using the eigendecomposition of graph matrices to define filtering operations.
To avoid the computationally expensive cigen-decompositioon, polynomial functions h(Ā) are often used to approximate different kernels in spectral GNNs. Specifically, the spectral filter h(A) is parameterized as a polynomial of degree P:
M(A) Σ
3 p=0
(2)
where Bp are learnable coefficients. Consequently, the filtering process can be reformulated as:
P h(A)a - x.
p=0
which allows efficient computation the filtered signal using only matrix multiplications.
Spatial GNNs. Spatial GNNs, also known as MPNNs, operate directly on the graph structure by aggregating information from neighboring nodes (Gilmer et al. 2017). The general form of message passing can be expressed as:
h (++1) UPDATE (h,", AGGREGAAE (In : JEN()))
(4)
where h.")) represents the feature vector of node i at layer I, Wii is the set of neighboring nodes of i, AGGREGATE is permutalionn-nvariian function that combines information from the neighbors, and UPDATE is function that updates the node's representation.
Popular examples of MPNNs include the GCN model Kipf & Welling 2017 which can be formulated as:
H (+1)
(HHWW().
(5)
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where HD is the matrix of node representations at layer I, W( is a learnable weight matrix, and of-) is non-linear activation function.
4 Piecewise Constant Spectral Graph Neural Network (PieCoN)
Current spectral GNNs often have limited flexibility in how they process graph structures due to their reliance polynomial filters. We propose PieCoN, a model that combines different types of spectral filters to process graph data in ways that complement the capabilities of polynomial filters. The key steps of our methodology, illustrated Fig. 2 include: (1) partitioning the graph spectrum into intervals based on significant points identified through spectral analysis, (2) constructing constant spectral filters for each interval capture global and local spectral properties, and (3) combining constant spectral filters with polynomial filters. Below, describe the methodology in detail. All the proofs of theorems and propositions are provided in Appendix
4.1 Identifying Significant Points in the Spectrum
Understanding the spectral properties of a graph is crucial for analyzing its structure. A key challenge is identifying significant points in the eigenvalue spectrum, which can reveal important structural insights (Fig. 3). Algorithm [ addresses this by identifying large gaps in the cigenvalue spectrum, which can indicate distinct frequency bands in the graph's 's spectral rep- resentation and highlight structural changes regions of high
0.02
Eigenvalues, & derivatives
0.01
0.00
spectral variation (Von Luxburg 2007: Fiedler 1973 Chung 1997). By adaptively partitioning the eigenvalue space around these critical points, PicCoN can capture the most informative spectral features of the graph.
Eigenvalnes derivative
-0.01
Eigenvalues Significant Points
-0.02
500
1000
1500
2000 2500
Algorithm D identifies significant points in the eigenvalue spec- trum by analyzing local patterns in the distribution of eigen- values. For each position i in the sorted sequence of eigenval- ues, the algorithm quantifies how much eigenvalue X; deviates from the statistical properties of its neighboring eigenvalues. Specifically, it computes the deviation of the discrete derivative
Eigenvalue Index
Figure 3: Using the derivative of eigenval- ues to identify significant points which show relatively high changes in the spectrum.
de = A - X from the mean of derivatives within windows before and after position i, normalized by their respective standard deviations. This normalization produces a significance score 8; that measures how abnormal each eigenvalue gap is relative to its local context. Positions with the highest scores represent points where the spectrum exhibits sudden changes, effectively identifying natural boundaries between different structural components of the graph. The algorithm also handles eigenvalue multiplicity by assigning zero significance to positions where consecutive eigenvalues are identical, ensuring that clusters of repeated eigenvalues remain intact. The identified significant points serve as adaptive partitioning boundaries for the spectrum, allowing PieCoN to focus computational resources on the most informative regions of the cigenvalue distribution. This partitioning preserves sign and basis invariance (as proven in Section 5.2), making our approach robust to different eigenvector calculation methods. A parameter sensitivity analysis for Alg. is presented in Appendix B
4.2 Construction of Constant Filters
For cach interval (ak, 0k+1), construct spectral filter Tk. k, shown in Fig.
The filter is defined as:
TU=UE.T = (. "
U hhoaarel)
where Eg binary diagonal matrix with non-zero entries corresponding the cigenvalues within the k-th interval and U [10000441] is the submatrix of U with columns as a++1 1. Specifically, the matrix E, for
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Algorithm Thresholding Algorithm for Identifying Significant Eigenvalue Gaps
function IDENTIFY SSEEPEIIANSIIIINTTTAAAC (AAEIICAATTGA((((d X, w, K) d: Discrete derivative of cigenvalues, A: Sorted eigenvalues, Window size for averaging, K: Number of top indices (Number of spectral intervals) > A very small positive value to avoid division by zero > Significance of each index
2: 6 E small constant
3: $ s
4: for + w to n -
1p: Op +- mean(dh- std(d;-ws)
Mean and standard deviation before Mean and standard deviation after b Sum of normalized distances to adjacent means
Mn: On f meann(du std l1t
8i t- d- + Idd-unl
7:
opte
onte
8:
if A A-1 then
8; 4-0 0
Set to zero if no gap exists
end if 11: end for
10:
12:
6g = 0, 0K- = n
13:
01,02,.... a-- t indices of the largest K values in
14: return ao, ,
15: end function
an interval (ak, ak+1) defined as:
Eg = diag(0,
1 .)....).
This construction ensures that Eς caplures the eigenvectors in the specified interval, allowing the filter TΛ to encapsulate the corresponding spectral properties. Our approach differs from traditional polynomial- spectral filters by enabling more flexible and tailored filtering of the graph's 's spectral components.
4.3 Polynomial Filters
In addition the spectral filters Tx: polynomial filters of the form Å" are used (Fig. 5). These filters provide way to incorporate local neighborhood information into the model. By adjusting the polynomial degree p, capture varying scales of locality in the graph.
Using the distinct eigenvalues of Å we can get:
(A-
X
(8)
This implics that polynomial filters have at most s free parameters because any polynomial of degree higher than can be reduced to polynomial of degree 8. The inclusion of polynomial filters complements the constant spectral filters by providing smooth interpolation between different spectral components. This combination allows PieCoN to capture both sharp and gradual changes in the graph's spectral properties.
4.4 Combining Constant and Polynomial Filters
The final embedding matrix of the nodes is computed by combining the constant spectral filters and polynomial filters. As in Jacobi convolutions (Wang & Zhang 2022), we apply independent filtering on each of the channels in X simultaneously:
K- Z.
Σ k=0
on o XW + Σ Spt A o W
(9)
p=0
Constant Filters
Polynomial Filtors
where:

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1.0
1.0
0.8
0.5
Filter Output
Filter Output
0.6
0.0
0.4
-0.5-
0.2
0.0
-1.0
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0 Eigenvalues
0.5
1.0
Eigenvalues
Figure 4: Constant filters.
Figure 5: Polynomial filters.
Matrix, the 1th column of the Matrix;
X is the imput feature matrix, representing the initial features of the nodes;
w and 2 are weight matrices to be learned during training. W maps the input features to an intermediate space, and W maps the intermediate representations to the embedding space;
(-) is a non-linear activation function applied element-wise, introducing non-linearity into the model;
Og are learnable coefficients associated with the spectral filters T₃ for each dimension I;
3pt are learnable coefficients associated with the polynomial filters A" for each dimension L.
Each element of T& represents similarity between two nodes in some range of frequencies. When perform the matrix multiplication MX for some similarity matrix M = Tk, cach entry Mij in the similarity matrix M represents the weight or importance of node in contributing to the feature vector of node i. If M is non-negative, it means each node contributes either positively or not at all to the feature aggregation. Negative values, on the other hand, would imply subtracting features from neighbors, which typically not meaningful in most graph-based learning contexts, where the goal is to aggregate features enhance node representations. Therefore, we separate TĄ into positive and negative parts, follows:
K-1
K-1 Og o +
Z. =
Oz (Tx) T (XW"
Σ B A (XW p= 0
(10)
K=0
Positive Part
Negative
Polynomial Filters
where d'l and Oi are learnable coefficients associated with the spectral filters Tx and T for each dimension I. The superscripts + and indicate coefficients for the positive and negative parts of the spectral filters, respectively.
It is important to note that while our method could technically classified as piecewise polynomial approach, we do not construct separate polynomial functions for each interval. Instead, our approach combines constant functions within specific spectral intervals with global polynomial filters.
4.5 Computational Complexity
The computational complexity of our method is broken down as follows:
Figendecomposition (precomputation): O(n3) for computing spectral components.
Filter construction and sparsification (precomputation): On 2Kn log(m)) O(33) for con- structing and sparsifying Tk filters, where K is the number of spectral intervals, and is the edge count.

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3. Model propagation: O[(K + P)md) during training and inference, where P is the polynomial degree, and d is the feature dimension. This matches the theoretical time complexity of JacobiConv (Wang & Zhang 2022) and GPRGNN (Chiee et al., 2021). while being more efficient than BernNet's 's (He et al, 2021 md). An empirical comparison the computational efficiency between our approach and The polynomial filtering method JacobiConv is provided Appendix C
It worth noting that while spectral methods like ChebNet (Defferrard et al. 2016 BernNet (He et al., 2021), and JacobiConv (Wang & Zhang 2022) avoid explicit eigendecomposition, they still face challenges when using higher-degree polynomials (needed for better approximation), as the recursive computation of A becomes computationally intensive in practice.
5 Theoretical Analysis
This section provides theorctical and empirical analyses establish the advantages of piccewise constant spectral filters over polynomial filters and validate the design choices of PieCoN. For the theoretical part, explain an error bound from polynomial approximation theory, which shows the limits of polynomial spectral filtering when dealing with sharp changes in functions. We discuss the challenges with eigenvector representations, such as their invariance under sign flips and basis shifts, which can alfect the generalization of graph learning models. Additionally, in Appendix D we analyze how specific graph structures influence the eigenvalue spectrum, with a focus on the eigenvalue 0, and explain why separating constant filters into negative and positive parts helps improve model performance and reduces approximation errors.
5.1 Error Analysis for Polynomial Approximation
To analyze the fundamental limitations of polynomial approximations in spectral filtering, we establish a theorem characterizing the error bounds.
Theorem 1 (Approximation error for e-dense eigenvalues). Let Å E RRax be normalized adjacency matrir with spectrum { where -1 λ s... An, S 1. Assume that these eigenvalues are e-dense on [-1,1] and that de < 1. Let f [-1,1] + R be filter function with || supzel-1.1] [f(x) = 1. For any polynomial p E Phhe space of pobyomialls of degree at most d), the approrimation error is:
n
E(p f) f)=)) = Σ Ip(Ae) f(A) 11 de)
(11)
Generally, as increases, the cigenvalues become more densely packed in [-1,1], causing to approach zero. This means d'e will also approach zero, making the lower bound on the approximation error converge to I|pll= - 1. Therefore, to minimize the error bound while maintaining sufficient approximation power, setting [| < is a natural choice since the target function satisfies ||/ = 1. This normalization allows for fair comparison between different polynomial approximations and simplifies the analysis. Hence, in the following theorem, we specifically focus on polynomials with || S to analyze the particular challenge of approximating functions with jump discontinuities.
Thcorem (Approximation error for functions with jump discontinnities). Let Å E Ruxn be a normalized adjucency malrir with spectrum {>i}i1 where -1 Àι .... S An 1. Let f 1-11,11] be some filter function with || 1 that the model needs to find and suppose it has fump discontinuity of magnitude h > O between consecutive eigenvalues AR and ARt1 (If(AR+1) (((((+))-((A))) h). For any polynomial p E Pa (the space of polymomials of degree at most d), satisfying l|Plloo S 1, the approcimation error is:
E(p, Σ El())) - f(0) h- h -
(12)
This result demonstrates key limitation: For small d when DR41 Arl < de the approximation error Elp 2 & z h. This is particularly problematic for spectral filtering where: (1) sharp transitions in the
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Table 1: Statistics of the datasets used for node classification. v(0)
the multiplicity of eigenvalue
Dataset
Nodes
Edges
Classes
Homophily Ratio
Chameleon
2,277
31,396
0.23
0.52
Squirrel
5,201
198,423
0.22
0.37
Actor
7,600
26,705
0.22
0.15
Amazon- Amazon-Ratings Ratings
24,492
93,050
0.38
0.17
Texas
183
168
0.11
0.35
Cora
2,708
5,278
0.81
0.11
Citescer
3,327
4,614
0.74
0.14
Amazon-Photo
7,650
71,831
0.83
0.02
Pubmed
19,717
44,324
0.80
0.61
filter response often desired (h is large), (2) some eigenvalues may be very close together (lAR+1 A is small), and (3) using high-degree polynomials is computationally expensive.
In contrast, by using piecewise constant filters, we can add constant filter only at point AR+1 with the value h and eliminate the jump entirely. The ablation study in Table demonstrates that adding constant filters to polynomial filters improves performance. This theoretical result justifics combining polynomial filters with constant filters in our approach.
5.2 Sign and Basis Invariance
Eigenvectors corresponding given eigenvalue can have multiple representations. For example, if X 0 is cigenvalue of the normalized adjacency matrix Á, any eigenvector u, associated with À = 0 can be replaced with its opposite =4 any linear combination of eigenvectors for the same eigenvalue. This variability introduces sign and basis ambiguity problems, which can lead to inconsistent or unpredictable results in learning lasks.
Proposition (This follows directly from (Lm et al. 2023 changes and basis choices.
Polynomial filters invariant to sign
The invariance of polynomial filters stems from the stability of eigenvector products U,U under different basis representations. This property ensures that matrix powers Å" maintain consistent behavior regardless of the specific eigenbasis chosen, making polynomial filters reliable for spectral graph operations.
Proposition 2. Constant fillers invariant to sign changes and basis choices.
Both polynomial and constant filters are robust to different eigenvector representations, ensuring that learned representations are not affocted by arbitrary sign changes basis choices, thus improving model stability and generalization. By leveraging filters with this characteristie, our model produce consistent outputs despite the inherent ambiguities in eigendecomposition.
6
Experimental Evaluation
6.1
Datasets
We evaluate PieCoN on seven diverse node classification datasets with varying graph structures and homophily ratios (Table D. Cora, Citeseer, and Pubmed are citation networks where nodes are rescarch papers and edges represent citations. Photo is a product co-oceurrence graph with nodes as products and edges representing co so-purchase relationships. Actor a graph where nodes are actors and edges denote co-occurrence in films, Chamclcon and Squirrel are graphs derived from Wikipodia pages. Nodes represent web pages, and edges denote mutual links. Texas is an academic web graph where nodes are webpages from the University of Texas and edges represent hyperlinks between pages. Amazon-Ratings is product co-purchasing network

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Table 2: Results on real-world node classification tasks.
Heterophilic
Homophilic
Model
Autazon- AuszonnRattups Ratings
Artigon-Phood Phota
Chameleon
Squirrel
CH
Pubmed
Actor
Texts
Cora
Spatial-bused GNNs
GCN
68.1011.a0
50.1127
34.6520.os
48.50200.a
78.6921.aa
87.1820.57
81.04go 57
85.87200.5
87.31 20.31
GAT
63.1321.a
44.4925.ss
33.932214
50.28250ss
77.5420.os
88.032a.79
80.52+0.71
90.94.0.ee
87.292n..
H₂GCN
67.1121.a
36.42111
35. 35.8621os
48.1715.ss
88.36=2.22
86,9221.s7
77.0741
93.02 93.0216..
88. 88,9326,ss
GCNII
63. 63.4120.ss
11. 11.901122
36.8 36.8920.ss
16.00111.a0
89.18
88, 2..
79.97
89 89.911.0.s
89.6820.3e
Spectral based GNNs
Free eigenvalues
69.581T.s1
59. 19.7011.on
41.6116es
44.2811.
88.20
81.91000
7T.3910
86.0810.4
86.UT16.0
LanczueNes
64.8111.
48.6411m
38.16
48.35
76.39 14.45
87.77
80.05 ss
93.21 2,0.0.
84.41220000
ChebyNet
59.28 1.225
40.5510.42
37.61 20.00
50:20300se
77.21
86.67
79.11
93.77 1.0..2
90.11 11
GPR-GNN
67.281T.e
50. 50.1511
39.92_..m
19.37710
88,53_3..
88.57 (..57_o.m
80.12
93.88510.2
91.361...4
BernNet
68 68.2911
51.3516m
41.7921.nn
48.82
89.02
88.52
80.09 m5
93.63100.s
88.9810
PPGNN
69.45211s
48.4712
39.65_00.00
47.90200ss
85.57=2.62
88.3220.00
80.98
95.099...3t
90.11
ChebNetll
71.37 11.3
ST. 72.72200.00 so
41.76_1.07
48.79
89.11
88.71
80.53+0.m
94.92 94.92
80.76:00,2
JacobiCoue
73.9221.07
$7.3816
40.43_0.81
48.53g0
89.02_2.77
88.6922100
81.65
95.36220.4
87 87.83326 43
DaP-JacobiRR
72.1725.m
55, $5.8428
39.80
48.6820.s
89.18 89.18
88.31 20.1
81.1 81.1 1...
94.901
88.97
OptBasiNNNN
74.402..00
63. 63.9811.12
48.8020..1
87.38-2.79
87.962..71
80.79
94.712ii.s
87.36200
Specformer
74.92:20.8
64. 26
41.50 11.562122
00M
83.602a.27
87. 87.052n.87
80 20.98
96. 96.2920.50
OOM
Unilfilter
74.11=108
63.522
40. 40. 11.11
50.0220070
86.72_a.77
80. 80.1071
81.21
94.9620.74
91 01.36.0.45
PicCoN
75.75.0.00 06
65. 65.67=0.0
39.79-0.56
52.37
89.342a.11
89.16g0.d
80.98
95.65
91.39
"p < 0.05 (significant difference from PieCoN)
< 0.001 (highly significant difference from PicCoN)
where nodes are products and edges indicate frequent co-purchases, with the task of predicting product rating classes. The ratio of eigenvalue O multiplicity to the number of nodes is shown in the last column of Table [
All datasets were randomly split into 60% training, 20% validation, and 20% test sets for 10 different seeds. For each dataset, we report the average performance along with the 95% confidence interval. Details about hyperparameter optimization and the rumning environment are provided in Appendix E
6.2 Baseline Models
We compare PicCoN against several bascline models categorized into different groups based on their underlying graph learning methodologies:
Spatial-based GNNs: Graph Convolutional Network (GCN) (Kipf & Welling 2017). Graph
Attention Network (GAT) (Veličković et al. 2018 Higher-order GCN (HaGCN) (Zhu et al. 2020 and GCNII (Chen et al.
2020).
Spectral-based GNNs: UniFilter (Huang al. 2024), LanczosNet (Liao al. 2019). Net
(Defferrard al.
2016).
Generalized
PageRank GNN (GPR-GNN) (Chien el al. 2021). BernNet (He
et al.
2021 PP- OP-GNN
[Lingam et al.
2022), ChebNetll (He et al. 2022). DSF
Jacobi-R (Guo et al.
2023
JacobiConv
(Wang & Zhang
2022h. OptBasisGNN (Guo & Wei 2023 and Specformer Bo
et al.
2023
Free eigenvalues: A graph neural network that learns a spectral filter by directly parameterizing the eigenspectrum --UUU where contains trainable eigenvalues.
6.3 Results
Table 2 shows the node classification accuracy of PieCoN compared baseline models across various datasets. We observe that PieCoN achieves the highest performance seven datasets. Notably, the largest.
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Table 3: Ablation study results.
Amazoot- Ra Patings
Amazon- Amazon-Photo Photo
Pos. Part
Nog. Part
Chameloon
Squirrel
Citosner
Pabued
Poty:
Actor
Texas
Cora
66.33 s6
18.391107s
40.31
10.21100.7
88.69 15..891...2
68.74.0
80.33,g..0 ss
p...7
91.24_n.10
x
K
67.26 so
$4.8525
32.0721.rr
49.0610de
90.66.2..0
84.3011 os
75.4516
92.7010
90.722...
X
73.6120.:1
80.50210s
38. 28.9
90.00
86.8011
94.90220.as
90.742...8
X
74.77:21.0
65 15.00
39. 39.02.0.54
19. 00.2811to
89 89.67.2.4
87 57.222
81 81.54goo.a
94.7015.a
90.83..u44
76.76....
65.67 .0.42
30.79_..56
52.37 _0.50
83.341a.m
89.16_004
80:98
95.65_0.
91.38_...0
"p 0.05 (significant difference from full model)
"*p < 0.001 (highly significant difference from full model)
improvements are observed the heterophilic datasets Chameleon, Squirrel, and Amazon-Ratings, with gains of 0.83%, 1.41%, and 2.09%, respectively. This may be linked to the high multiplicity of the eigenvalue in the normalized adjacency matrix of these graphs (see Table D to which our method gives more importance.
For homophilic datasets, such as Cora, Amazon-Photo, and Pubmed, PieCoN also demonstrates competitive performance, achieving slight improvements over existing methods. The smaller gains suggest. that tradi- tional GNNs already perform well in these settings, as they inherently align with homophilic assumptions. Nonetheless, PicCoN remains robust, indicating that its spectral filtering approach does not hinder its ability to learn from homophilic graphs. These results indicate that PieCoN is effective in both heterophilic and homophilic settings.
We also run t-tests comparing PieCoN with each bascline method to verify the statistical significance of our results. Stars in Table 2 show when a baseline performs significantly different than PieCoN (* for p< 0.05, *** for p < 0.001). Most baselines show significant differences on heterophilic datasets, confirming that our gains are meaningful. For example, all methods except Specformer and UniFilter show significantly lower performance Chameleon. On homophilic datasets, several methods show no significant difference from PieCoN, indicating competitive but not always superior performance.
6.4 Ablation Study
We have performed ablation study using Eq. 10 to evaluate the contribution of each component on model performance. The results in Table reveal several key findings. First, the full model incorporating all three components (positive part, negative part, and polynomial filters) achieves the best performance on 6 out of 9 datasels, with notable improvements on Chameleon (75.5%%), Squirrel (65.67%), and Amazon-Ratings (52.37%). The combination of positive and negative parts without polynomial filters also shows strong performance, suggesting that these components capture complementary spectral information. For instance, on Squirrel, adding the negative part to the positive part improves accuracy from 60.50% to 65.00%. Interestingly, the Actor dataset, using only polynomial filters yields the best performance (40.31%) 1)), while Citeseer, the positive part alone achieves optimal results (81.72%).
In separate experiment, we also create simple spectral method with the eigenvalues as parameters. The results of this experiment are presented in Table 2 with the model name "Free eigenvalues". However, this approach may be less effective because the method docs not receive any explicit structural information associated with the eigenvalues.
7 Limitations
Our work has some limitations. The compulational complexity of O(n") for eigendecomposition presents scalability challenges for large-scale graphs. However, this preprocessing step occurs only once and has become increasingly practical with modern hardware, taking approximately 8 minutes for datascts with 25,000 nodes on an NVIDIA A100 GPU. It is worth noting that our approach is designed for graphs of reasonable size (up to tens of thousands of nodes), where the eigendecomposition is worthwhile one-time preprocessing cost.
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given the expressivity benefits. Also, our work does not address the problem of scaling GNNs to massive graphs with millions of nodes, which is separate research area requiring specialized techniques like graph sampling or partitioning beyond the scope of this paper. Furthermore, the model's performance is highly dependent on how we partition the eigenvalue intervals, and our current approach using hard thresholding to identify significant spectral changes may not be optimal.
8 Conclusion
In this paper, we presented the Piecewise Constant Spectral Graph Neural Network (PieCoN), a new approach to graph prediction tasks. Our method aims to address some limitations of cxisting spectral GNNs by combining constant spectral filters with polynomial filters to caplure a broader range spectral characteristics in real-world graphs. We introduced an adaptive spectral partitioning technique that analyzes the derivative of sorted eigenvalues to identify significant spectral changes. This helps focus on the most informative regions of the spectrum. PieCoN expands the search space of possible eigenvalue filters beyond traditional polynomial-based filters, allowing for more tailored capture of graph spectral properties. This is particularly useful when dealing with graphs that have high eigenvalue multiplicity. By integrating spectral filters with polynomial filters, our approach attempts to model both global graph structure and local neighborhood information. Our experiments on nine benchmark datasets, covering both homophilic and heterophilic graph structures, suggest that PieCoN performs well on both types of datascts.
Acknowledgements
Vahan Martirosyan is the recipient of a PhD scholarship from the STIC Doctoral School of Université Paris-Saclay.
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A Proofs
Theorem 1 (Approximation error for e-dense eigenvalues). Let Å E Ru be a normalized adjacency matrir
with spectrum {A where -1 < < ... I X, < 1. Assume that these eigenvalues are e-dense on [-1, and that e' < 1. Let f [-1, + be a filter function with llf = supzel-1.11 If(x)] = 1. For any polynomial p E (the space of polynomáals of degree most d) the opprorsmation ervor
n
(v, f) - Σ
- - (1
(13)
Proof. Let To E 1]], be a point where |p(xo) = llploo i.e., the point where the polynomial p attains its
maximum absolute value on the interval [-1, 1]. Since the eigenvalues {No}==1 are e-dense on [-1,1], there
exists eigenvalue A3 such that (xo >y1 By Markov's polynomial inequality (Sahoo & Riedel 1998
|p'l I d P
(14)
Using the mean value theorem
Achiezer
1992 there exists & E (wo, (or [)g, xo] if X3 > mo) such that:
(p(x0) p(),)) = (E)1 I00 - x S llp'llo 1 - >ul - d 111l000 -E.
(15)
Therefore,
1p(Ag)1 2 [p(xa)] - - 1p(2)) - p(A))1 2 1p1l - dl1pllaa 111l (1 ddee.
(16)
Since || 1, we know that | < 1. Then,
(m(A,) - f(A)) )1 2 - Iepllx 110 -dc) 1. Since (p(A)) f(A.)) 1p(2)) f(Ar)), have:
(17)
Elp, 2 |1lll dee) -1
(18)
This establishes the claimed lower bound the approximation error.
Theorem 2 (Approximation error for functions with jump discontinuities). Let Å Ruxn be a normalized adjacency matrix with spectrum {i where -1 S >1 S ... S >n 1. Let f: [[-1,1] R be some filter function with || = that the model needs to find and suppose it has a jump discontinuity of magnitude h > O between conseculive eigenwalues AR and AR+1- For any polynomial E Pa(lhe space of polynominls of degree at most d), satisfying l|plloo I 1, the approzimation error is:
E(p fp. = E - s(a)) - Aal- d.
(19)
Proof. By Markov's polynomial inequality
(Sahoo Riedel 1998):
Ip' /lx S d 1111ax S d. Vp E Pa.
(20)
Next using the mean value theorem Achiezer 1992), we find that 3 Ery E [2s, v) such that:
pp(x) p(v) Ip (f2w)) -2- vl S d vl. V E
(21)
Now, we define the local error consocutive cigenvalues as:
& = |p(A)) - f(AR)) + Ip(A++)) f(A++1))
(22)
Applying the triangle inequality, we obtain:
&g 2 (/(AR++)) - -
1p(Ar+1) - p(AR))
(23)
2 DAR+1 - .dl.d?. d?.
Since it holds that fpp f) 2 &2, the result. follows.
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Proposition (This follows directly from (Lim et al. 2023 ). Polynomial filters invariant to sign changes and basis choices.
Proof. The key property use is that the product UN u where Un is matrix of eigenvectors associated with eigenvalue μ4, remains invariant under sign changes and different choices of basis (Lim et al. 2023). This implics that regardless of which orthonormal basis is chosen for given cigenspace, the product U U stays the same.
Consider two orthonormal bases U and V for the eigenspace corresponding to eigenvalue μ., There exists an orthogonal matrix Q such that:
=
(24)
Using this, we show the invariance:
V
U
(25)
where we used the fact that QQ T since Q is orthogonal. This confirms that U U is invariant under the change of basis.
Now, consider the matrix power A", which can be expressed as:
n X u = UUNUUU
A"
(26)
where X; are the distinct eigenvalues, and Ux are the corresponding eigenvector matrices.
Since each term UN X is invariant to basis choices and each term u.u (-u:) u.) is invariant
sign changes, follows that A" also invariant to both.
Proposition 2. Conslant fillers are invariant to sign changes and basis choices.
Proof. Consider the constant filter T& defined over the interval [ak, ok+1) for some k. Let tk be the index such that X₁ = Aoa and thus M Aax-i-1-
Suppose 1 are indices such that d + = A141 =.... -.. * A,. According to Alg, no significant point will be selected with I <i<r, ensuring that constant eigenvalue intervals are not split. Consequently, X1, * Aag-1 when a 1, and N * dazts when ax+1 S
Ap-1
The distinct eigenvalues of Å within the interval [Mag A0R41-1] are:
X No 1.... No. 1-1-
(27)
Using Equation we express Ta
4411-1
TB = U Upapaa (acart1)
Σ 1=0L
Ux
ux = Σ UNU
(28)
s=
Since each term U is invariant to basis choices and each term u.u, = (-u;)(-u:)) is invariant to sign changes, follows that Tr also invariant to both.
Parameter Sensitivity Analysis of Algorithm
We further investigate the impact the window size (w) and the number of limits (K) from Alg. on model performance. To isolate the effect of each parameter, we conduct controlled experiments where we fix one parameter its optimal value while varying the other. Specifically, when examining window size effects, fix K at its optimal value for each dataset, and when studying the number of spectral intervals,
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76.0
76.0
Test Accuracy (%)
Test Accuracy (%)
74.0
74.0
72.0
72.0
70.0
70.0
68.0
68.0
Chameleon Accuracy
Chameleon Accuracy
66.0
66.0
10
20
30
40
50
60
5
10 15
20
25
30
Window Size (W)
Number of Spectral Intervals (K)
Figure 6: Effect w on Chameleon dataset per- formance with optimal K.
Figure 7: Effect K on Chameleon dataset per- formance with optimal w.
90.0
90.0
Test Test Accuracy (%)
88.0
Test Accuracy (%)
88.0
86.0
86.0
84.0
84.0
82.0
82.0-
Cora Accuracy
Cora Accuracy
80.0
80.0
10
20
30
40
50
60
5
10 15
20
25
30
Window Size (W)
Number of Spectral Intervals (K)
Figure 8: Effect of on Cora dataset performance with optimal K.
Figure 9: Effect of K Cora dataset perfor- mance with optimal w.
we fix w at its optimal value. As shown in Fig. and Fig. 7 the Chameleon dataset exhibits distinctive parameter sensitivities. Performance peaks at w = 20 with an accuracy of 76.30%, followed by a sharp decline and stabilization around 74.70% for larger window sizes. For the number of spectral intervals, we observe a dramatic performance increase from K to K 10, with optimal performance in the range 10 K < 15 (reaching 75.84%), followed by a gradual decreasc K increases further. In contrast, Cora (Fig. and Fig. 9 shows more stability across different window sizes with optimal value at = 40 reaching 69.13% accuracy. For spectral intervals, Cora demonstrates a similar pattern to Chameleon but with less pronounced differences, showing peak performance at 10 < K < 12.
These results indicate that the number of spectral intervals (K) significantly influences model performance, whereas window size (w) has a more limited effect. Too few partitions fail capture important spectral characteristics, while too many may introduce noise.
C Computational Efficiency Comparison
To evaluate the computational efficiency of our approach, we conduct a running time analysis comparing the constant filter component of PieCoN with JacobiConv's polynomial filters.
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0.12
ZacobiConv (Degree)
Averuge Epoch Time (Seconds)
PicCoN (Number Spectral Intervals (K))
0.10
0.08
0.06
0.04
0.02
0.00
5
10
20
30
40
50
60
70
80
90
100
Parameter Value (Degree/ Number of Spectral Intervals (K))
Figure 10: Computation time of PieCoN vs. JacobiConv.
Figure 10 illustrates the average epoch time (in seconds) as we increase the number of spectral intervals (K) in PieCoN compared increasing the polynomial degree in JacobiConv. For fair comparison, we disabled the polynomial part in the PieCoN implementation for this experiment.
The results show that PieCoN's constant filters require less computation time than high-degree polynomial filters. At parameter value K = 100, PieCoN takes approximately 0.05 seconds per epoch compared to JacobiConv's 0.13 seconds. This efficiency comes from our direct spectral interval filtering approach, which applics filters to specific cigenvalue intervals rather than performing sequential matrix multiplications needed for higher-degree polynomials.
Analysis of Graph Structure and Eigenvalue Zero
The presence of cigenvalue in graph spectra reveals important structural propertics that many polynomial-based GNN methods overlook. Real-world datasets often exhibit high multiplicity of eigenvalue (Table D yet methods like JacobiConv (Wang & Zhang 2022 Bernnet (He et al.] 2021), and Chebynet (Defferrard et al. 2016 which use low-degree polynomials of the normalized adjacency matrix Å, do not adequately capture these properties.
Thcorem 3 [Bancrjce (2008)). Let be a graph and H be subgraph of J. with eigenvalue O. Then V(H) p1,P2,..-. Pm. VJJ and E(H) S F.J) consists of edges between vertices in V(H). We define IHl to be the graph obtained from J by attaching copy of Hl follows: (1) add new set of vertices 91, 92...., 4m to J, where each qs corresponds to Pe in H; (2) add edges between vertices in q1 .1........ 4m that mirror the edges H; (3) for each 1,2, ,,...,m and ench verlez r E VJ) VVH), add an edge ger r if there exists an edge P&r in J.
Figure 11: Simple graph with dupli- cate subgraphs. After adding the du- plicates, the multiplicity of eigenvalue O increases from 0 to 3.
Then, the graph JH has eigenvalue with associated eigenvector that is nonzero only al the nodes Pt and qu. Furthermore up. -
Theorem rrevaals that when a graph contains duplicate substructures, it leads to eigenvalue O with eigenvector localized to specific node sets. This localization property is particularly relevant for node classification and community detection tasks, as nodes with similar structural roles often share the same labels (Fig. 11
In analyzing how the negative and positive parts of Tş affect and change the structure of the graph, we consider simple graph with duplicate subgraphs, illustrated in Fig. II In this graph, nodes with the same labels duplicates. According to Theorem g these duplicates create eigenvalues equal to 0 in the
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Figure 12: Simple graph with added positive edges.
Figure 13: Simple graph with added negative edges.
Table Hyperparamcter ranges used for optimization.
Hyperparameter
Values
Learning Rate (1r)
0.0005, 0.001, 0.005, 0.01, 0.05 0.0, 5e-5, 1e-4, 5e-4, 1e-3
Weight Decay (weight_decay)
Feature Dropout (feat_dre) (deat_dropout)
0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9
Number of Layers (nlayer)
1, 2, 4,
Hidden Dimension (hidden dim)
16, 32,
Window size for Algorithm wwindow___ size)
5, 10, 15, 20, 25, 30, 35, 40, 45,
50,
55, 60, 65, 70, 75, 80, 85, 90, 95, 100
Number of spectral intervals [num_ limita)
1, 4, 5, 6, 7, 8,
9,
10,
11,
12, 14, 16, 17, 18, 19,
eigendecomposition of the normalized Adjaceney matrix. Let Un denote the eigenvectors corresponding the eigenvalue We can decompose Rg = U into negative (RT) and positive parts (R²). Using these parts, we construct two graphs by choosing the edges with the highest score in these matrices. The graphs resulting from this process, including original and added edges, shown in Figures 1 and 13
From these graphs, we observe that both negative and positive edges identify connections between duplicate motifs. The negative edges also reveal connections between duplicate nodes within these duplicate motifs, highlighting their role in capturing structural similarities.
Another intuition split T& that, for example, the second eigenvector provides a direction that best separates the graph into two groups while minimizing the connections cut. between them. The positive and negative values show two clusters that are internally connected but separated from each other (Fiedler 1973:
Von Luxburg 2007
E
Hyperparameter Optimization and Running Environment
All experiments carried out Linux machine with NVIDIA A100 GPU, Intel Xcon Gold 6230 CPU (20 cores @ 2.1GHz), and 24GB RAM. Hyperparameter tuning was performed using the Hyperopi Tree of Parzen Estimators (TPE) algorithm (Bergstra et al. 2011 with the hyperparameter ranges shown in Table
The Adam optimizer was used for training with 2,000 epochs. Hyperparameters were selected to achieve the best performance on validation set.
19


Reviews:
[{"review_id": "gD1uVESigW", "reviewer": "Reviewer_VxFY", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n1. The paper is clear, and despite many concepts that are introduced, it is easy to follow.\n2. The solution idea is simple, even though the specific algorithm proposed for partitioning the graph spectrum into different intervals is not well motivated in the writing.\n\n\n### Weaknesses\n1. The contribution of the paper appears limited as the method is very similar to PP-GNN (Lingam et al. 2022). Indeed, PP-GNN fits a low degree polynomial in each interval, while PieCoN sums a polynomial to a filter that is constant in each interval. Please clarify the implications of this difference. \n1. The paper claims that polynomial filters may not *``fully identify graph's spectral properties''* (e.g., in the abstract), without actually introducing what this means in practice, nor discussing how including constant filters fixes this problem. From the theoretical analysis, I believe that the authors mean that polynomial filters cannot approximate functions with sharp changes if the polynomial degree is small, while constant filters can. Being more concrete on this point can significantly improve the paper.\n2. The theoretical results are not very surprising, especially Proposition 1 (which follows directly from Lim et al., 2023) which I think can be presented in words.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes PieCoN, a spectral GNN which includes constant spectral filters alongside polynomial filters to capture both sharp and gradual changes in the graph’s spectral properties. Theoretically, the authors provide an error bound for spectral polynomial approximation, showing that commonly used polynomial filters (which have small degree) are limited when approximating functions with sharp changes. Empirically, the proposed method outperforms existing polynomial-filters based spectral GNNs and spatial GNNs on 5 out of 7 datasets.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Please discuss the difference with PP-GNN (Lingam et al. 2022) in more detail.\n- Please explain what is the problem of polynomial filters when dealing with large multiplicity (said in the introduction). Indeed, while it is clear that when using low degree polynomials the change between closely spaced eigenvalues cannot vary significantly, I do not see how it connects to large multiplicity (where the eigenvalues are not closely spaced, but identical). \n- Can you explain why we need the polynomial filters instead of relying on constant filters only? I think this is especially relevant because we could impose additional constraints to maintain continuity between contiguous intervals, as in PP-GNN.\n- Algorithm 1, which is used to partition the graph spectrum, should be explained and not just referenced in the text. I think the algorithm just identifies eigenvalues that deviate significantly from the surrounding values by comparing them to local mean values, but please expand.\n- **The paper needs to include the empirical comparison with the related work also targeting the limited flexibility of polynomial filtering**, such as  DFS (Guo et al., 2023), NFGNN (Zheng et al., 2023) and PP-GNN (Lingam et al. 2022). Indeed, as these works are presented in the related work section they should serve as a direct baseline. \n- Related to the above, please add the missing datasets that are present in DFS (Wisconsin Cornell Texas Twitch-DE Pubmed Computers Photo), and explain why the reported numbers on the datasets that are present are different from those reported in DFS.\n- What is the impact of the window size W and the impact of the number of partitions K? Please add an ablation study.\n\n\nAdditional minor changes:\n1. Please move the proofs to the appendix and provide only an high level proof overview in the main paper.\n2. I think using W for both window size and for the weights is confusing. Maybe you can denote the window size by w instead of W.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "b69c1zetWq", "reviewer": "Reviewer_vcVp", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths \n\n- The error bound analysis clearly motivates the approach, with low degree polynomial filters unable to account for jump discontinuity between eigenvalues which are close.\n\n- The proposed method is shown to perform well on a range of tasks, providing some evidence for the efficacy of the approach.\n\n- Whilst polynomial filters are sign and basis invariant, the authors highlight that their full method is also sign and basis invariant. This demonstrates a more detailed evaluation of their approach.\n\nMajor Weaknesses\n\n- The error in the bound of Section 5.1 can be reduced using a high degree polynomial. The authors suggest that the limitation of this is that they are computational expensive. However, the method proposed by the authors also increases the computation as outlined in 4.5. Given that the method is motivated because of the computational limitation of high degree polynomials, more work should be done to compare their complexity and the relative benefits of your approach. A timing comparison of standard polynomial filter, high degree polynomial filter and your method would help a lot in this regard.\n\n- The authors highlight that “the model’s performance is highly dependent on how we partition the eigenvalue intervals”. Adding the results which show this are important to provide a better understanding of the method. The method provides additional hyperparameters within the thresholding algorithm so showing how robust the results are over changes to these parameters and what the optimal values are, will better highlight the practical value of the approach.\n\n- At multiple points in the paper, the authors claim that their method would improve for cases where there exists eigenvalues with large multiplicities. This is not clear from the paper as you have the same filter output for a given eigenvalue (shown in Figure 1). From the bound and the analysis, it seems the method would help when eigenvalues are closely spaced but not when they are identical.\n\nMinor Weaknesses\n\n- The bold text used in the experimental results is quite misleading. All top results that are statistically equivalent should be highlighted in bold. For example, in the ablation on Chameleon in Table 3 using positive+negative without the polynomial looks not significantly worse than positive+negative+polynomial. Both should really be in bold. This should be done for all statistically equivalent results.\n\n- PP-GNN is mentioned in the paper to be the most similar method [Lingam et al, 2022]. However, I don’t see any performance comparison. This should be done to show why your method is better than only using two frequency bands.\n\n- You say that “real-world graphs where certain eigenvalues like zero have **large** multiplicities”. This multiplicity is given by the number of connected components in the graph, so I am surprised this would be large in many practical scenarios. In Table 1, you claim to list this multiplicity (Lamba(0)/Nodes), but this looks like it should be labelled (Lamba(1)/Nodes). For example, on Cora (Lambda(1)/Nodes) is 0.11, as in the table, but (Lambda(0)/Nodes) is much less.\n\n[Lingam et al, 2022] Vijay Lingam et al. A piece-wise polynomial filtering approach for graph neural networks. In ICLR Workshop\non Geometrical and Topological Representation Learning, 2022.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors provide an error bound for spectral polynomial approximation and show that using piecewise constant filters can remove this error. This motivates combining constant spectral filters with polynomial filters to increase the range of spectral properties that can be effectively learned without using computationally expensive high degree polynomials. The authors show that this combination performs well on a range of homophilic and heterophilic tasks.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "My requested changes go some way to reduce the weaknesses discussed in the previous section. \n\n- I strongly encourage timing experiments of the approach with baseline comparisons to different degrees of polynomial filters (weakness 1).\n- Results should be shown in order to better understand the effects of Hyperparameters in the Thresholding Algorithm (weakness 2).\n- The authors should be clearer on the benefits of the approach with regards to the multiplicity of eigenvalues. The quoted multiplicity in Table 1 should be revisited and explained in more detail (weakness 3).\n- PP-GNN could be added as a baseline in Table 2.\n- The bold numbers should reflect statistical significance in the results tables.", "broader_impact_concerns": "none required", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "stYPAdRnFy", "reviewer": "Reviewer_Dk7G", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The article studies an important problem, the design of GNNs for learning problems on graphs. This problem is of practical importance as GNNs have many variants.\nThe article has the merit to contain extensive numerical experiments to compare their baseline to existing variants of GNNs, for node classification on various datasets.\n\nHowever the article has several major weaknesses and issues as outlined below:\n\n(i) The article does not give precise presentations of what a GNN or spectral GNN is (and a comparison thereof), in particular Section 3 is relatively poor and does not connect the rest of the paper well. A detail: Section 3 contains only one subsection (3.1). This would be useful to state it formally to clarify ideas.\n\n(ii) The authors do not present very clearly the scientific problem addressed by the paper. For example, the choice of Figure 1 is not very informative. The figure only displays the behavior of a polynomial vs. a piecewise polynomial. The authors mention that the piece can ``focus''  on selected intervals, but this is very informal, some more explanation should be given on what this precisely means in the context of GNNs, and how it connects to some of their properties (for example, expressivity). \n\n(iii) The theoretical analysis' content. \n- Section 5.1 The Theorem 1 supposes that the polynomial has infinity norm 1 on [-1, 1]. Why is it the case? To my understanding, if this assumption is removed than the lower bound can get negative (hence uninformative) by simply scaling the coefficients of a polynomial, without increasing its degree. The authors should at least justify this assumption.\n- Section 5.2: This section would be more complete if one would highlight specificities of the polynomial / constant filters. Is it true that only the piecewise/constant filters have those properties? Otherwise, it is not clear what is the theoretical gain of PieCON vs the polynomial ones. Furthermore, to my understanding, PieCON is not polynomial nor constant, it is piecewise polynomial (it has finitely many polynomial pieces).  \n\nOn a minor note, the writing in general is uneven throughout the article to convey the important messages about the above points, and would gain from proofreading. \n\nTherefore, I believe the article is not ready for publication under this form. It would benefit from major changes, and repositioning. Please see below the suggestions that would strenghten the article.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In this article, the authors propose a design of a spectral GNN to circumvent the limitations of existing spectral GNNs that use polynomial activations. After a brief overview of the existing types of spectral GNNs, including the polynomial ones, the authors present the design of their Spectral GNN PieCON. After showing some properties of PieCON, the authors compare experimental performance of PieCON to existing GNNs on node classification tasks.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I would suggest the authors to:\n\n- Give a clear mathematical definition of what a Spectral GNN is, how it differs from a standard GNN, and how PieCoN matches the definition with the appropriate choice of filters.\n\n- Enrich the motivation of the study, for example by additional small experiments that would further support the messages of Figures 1 and 3.\n\n- Reposition and complete the theoretical analysis of Section 5. For example, how does PieCON combines the best of both worlds (constant and polynomial) -- if it actually does.", "broader_impact_concerns": "Not applicable.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: Please carefully take into account any residual discussion with the Reviewers -- especially Reviewer Dk7G -- and incorporate any and all relevant takeaways into the camera-ready version. Once this is done, the paper should be ready for publication in TMLR.
Decision: Accept (poster)