Spectral Graph Neural Networks

• Spectral Graph Neural Networks are graph convolution models that perform frequency-domain filtering using Laplacian eigendecomposition to modulate signal components.
• They utilize polynomial and orthogonal filter bases (e.g., Chebyshev, Jacobi) to achieve scalability, well-conditioned optimization, and precise spectral adaptations.
• These architectures offer enhanced expressiveness and performance on heterogeneous graphs by capturing global structures and adaptive frequency responses.

Spectral Graph Neural Networks (Spectral GNNs) constitute a class of graph neural architectures that leverage the spectral decomposition of the graph Laplacian to perform convolutions in the frequency domain. Unlike spatial message-passing methods, which aggregate features locally among neighbors, spectral GNNs operate by learning graph filters that modulate signal components associated with different Laplacian eigenvalues and eigenvectors. This frequency-domain perspective confers advantages in modeling global structure, interpretability, and adapting to graph-specific phenomena such as homophily and heterophily.

1. Mathematical Foundations of Spectral GNNs

Spectral GNNs are grounded in spectral graph theory and graph signal processing. Given an undirected graph $G=(V,E)$ with adjacency $A$, degree matrix $D$, and normalized Laplacian $L = I - D^{-1/2}AD^{-1/2}$, one forms the eigendecomposition $L = U\Lambda U^\top$, where $U$ is orthonormal and $\Lambda=\mathrm{diag}(\lambda_1,\dots,\lambda_n)$ contains graph frequencies. The graph Fourier transform of a signal $x$ is $U^\top x$, with the inverse $U\hat{x}$. Spectral filtering is performed as $y = U h(\Lambda) U^\top x$, where $h:\mathbb{R}\to\mathbb{R}$ specifies the frequency response. In practical models, $h(\lambda)$ is parameterized as a polynomial or in an orthogonal basis, e.g., Chebyshev ($T_k$), Bernstein, Jacobi polynomials, or Trigonometric expansions (Chen, 2020, Bo et al., 2023, Li et al., 15 Apr 2024). For scalability and localization, polynomial filters are preferred, since $g_k(L)x$ can be computed via sparse matrix multiplications without explicit eigen-decomposition.

2. Expressive Power and Universality

The expressive power of spectral GNNs rests on their ability to approximate arbitrary real-valued graph signals under mild conditions. The universality theorem (Wang et al., 2022) establishes: for any target $Y$ and input $X$, if the Laplacian $L$ has no repeated eigenvalues and the node features $X$ possess nonzero support in every frequency ($U^\top X$ has no zero rows), then there exists a polynomial filter $g$ such that $Y = g(L) X W^*$ for some $W^*$. This universality is achieved even in completely linear spectral GNNs, which lack nonlinearities, provided that the spectral support is complete. Under these conditions, spectral GNNs can match or exceed the 1-WL (Weisfeiler–Leman) graph isomorphism test in distinguishing non-isomorphic structures and in approximating any node-wise signal. JacobiConv leverages Jacobi polynomial bases with built-in orthogonality to further optimize learning (Wang et al., 2022).

3. Filter Basis Design and Optimization

The choice of filter basis strongly conditions the numerical optimization landscape. When representing filters as $g(\lambda)=\sum_{k=0}^K \theta_k g_k(\lambda)$, the filter basis should be orthonormal with respect to the spectrum energy density $f(\lambda)$ ($$H_{ij} \approx \int g_i(\lambda)g_j(\lambda)f(\lambda)d\lambda$$), which guarantees well-conditioned Hessians and rapid convergence (Wang et al., 2022). Jacobi polynomials accommodate flexible weighting, outperforming monomials, Chebyshev, and Bernstein when the spectrum is non-uniform. JacobiConv implements a low-rank polynomial-coefficient decomposition for additional stabilization. Recent advances propose piecewise constant filters (PECAN), which partition the spectrum and learn constant responses per interval, resolving the inability of polynomials to fit discontinuous or narrow-band responses (Martirosyan et al., 7 May 2025). Trigonometric polynomial filters (TFGNN) further enhance band-limited approximation by decomposing filters into sinusoidal slices, parameterized through Taylor expansion for efficiency (Li et al., 15 Apr 2024).

4. Advanced Frameworks and Extensions

Spectral GNN research has diversified, introducing frameworks that generalize or augment classical spectral filtering:

• Diverse Spectral Filtering (DSF): Classical spectral GNNs are inherently homogeneous—they apply the same spectral filter globally. DSF introduces local-global decomposition by endowing each node with position-specific filter weights learned via spectral positional encoding, substantially improving performance and interpretability on regionally heterogeneous graphs (Guo et al., 2023).
• Automatic Propagation Discovery (AutoSGNN): Conventional model selection is labor-intensive and graph-type-specific. AutoSGNN automates the discovery of spectral filter architectures by unifying the search space and leveraging LLM-guided initialization plus evolutionary search, outperforming expert-designed baselines on diverse benchmarks (Mo et al., 17 Dec 2024).
• Spatially Adaptive Filtering (SAF): SAF highlights the intrinsic link between spectral filtering and spatial aggregation, showing that spectral filtering implies an adapted, non-local, signed-weight adjacency. SAF fuses explicit spectral and non-local spatial pathways, modeling both node similarity and dissimilarity to address long-range dependencies and heterophily (Guo et al., 17 Jan 2024).
• Hyperbolic-PDE Perspective: Hyperbolic-PDE GNNs interpret message passing as a discretized wave equation, constraining node features to the Laplacian eigenvector span. This formalism unifies polynomial filters and enables precise control of frequency modulations, with empirical accuracy gains on both smooth and band-pass graph tasks (2505.23014).
• Specformer: Scalar-to-scalar spectral filters limit expressivity. Specformer applies Transformer self-attention across the full eigenvalue sequence to parameterize set-to-set global filters, achieving permutation-equivariant and universal representations (Bo et al., 2023).
• State Space Models (GrassNet): GrassNet employs learned state space models indexed by the ordered spectrum, providing frequency-dependent modulation that can differentiate even for repeated eigenvalues, transcending the polynomial-induced identification bottlenecks (Zhao et al., 16 Aug 2024).

5. Empirical Performance and Benchmarking

Spectral GNNs demonstrate strong empirical performance across a range of node-classification, signal filtering, and graph-level tasks, especially on graphs with significant heterophily or complex spectral structure.

• JacobiConv achieves tenfold lower MSE than prior spectral GNNs in synthetic filter recovery and consistently outperforms baselines on node classification across real-world datasets, despite being fully linear and forgoing MLPs (Wang et al., 2022).
• DSF boosts accuracy by up to 4.92% over existing spectral GNNs for heterophilous graphs, with interpretability conferred by local filter diversity (Guo et al., 2023).
• PECAN and TFGNN obtain superior accuracy for hard graphs by capturing discontinuous or band-pass spectral features (Martirosyan et al., 7 May 2025, Li et al., 15 Apr 2024).
• ChebNet2D introduces two-dimensional convolution, generalizing all filter paradigms and achieving universally zero construction error for arbitrary outputs. ChebNet2D is maximal in expressivity and computationally efficient (Li et al., 6 Apr 2024).
• Specformer outperforms polynomial-based GNNs and spatial models on narrow-band and comb-filter tasks, and on both node- and graph-level benchmarks (Bo et al., 2023).
• SAF attains top performance with up to +15.4% accuracy gain on heterophilous datasets, facilitated by explicit non-local signed-weight graphs (Guo et al., 17 Jan 2024).

Recent comprehensive benchmarking (Liao et al., 14 Jun 2024, Dong et al., 10 Dec 2024) reveals that variable and filter-bank models are essential for heterophily, while fixed filters (e.g., PPR, HK) suffice for homophily. Mini-batch training and sparsification techniques (SGNN-LS (Ding et al., 8 Jan 2025)) now scale spectral GNNs to million-node graphs.

Typical Effectiveness Table (accuracy gains from key papers)

Model	Best Domain	Accuracy Gain (vs Baseline)
JacobiConv	Real-world	Up to +12%
DSF	Heterophily	Up to +4.92%
ChebNet2D	Universal	1–3 pts on hardest benchmarks
PECAN	Heterophily	+0.8–2.1%
SAF	Heterophily	Up to +15.4%
Specformer	Band/Heterophily	Up to +6–7%
SGNN-LS	Scalability	No loss, enables ~100M nodes
GrassNet	Band/Discontinuity	+1–2 pts over best polynomials

6. Interpretation, Limitations, and Future Directions

Spectral GNNs offer interpretability through the explicit modulation of graph frequencies, revealing task-relevant mechanisms such as smoothing (homophily), sharpening (heterophily), and band-pass filtering. Canonical spectral filters are well-understood in terms of smoothness, separation, and multi-frequency behavior. However, challenges remain:

• Scalability: Exact eigendecomposition is limiting for large graphs; polynomial approximations and sparsification are effective but may sacrifice flexibility.
• Incompleteness: On simple-spectrum graphs, invariant spectral GNNs may fail to distinguish certain non-isomorphic graphs; sign- or rotation-equivariant augmentations (equiEPNN) can close this gap (Hordan et al., 5 Jun 2025).
• Optimization: Poorly conditioned parameter landscapes (block Hessians) can hinder training, especially for heterophilic graphs. Asymmetric gradient preconditioning is recommended (Liu et al., 16 Dec 2024).
• Interplay with Nonlinearities: Empirical analysis demonstrates that nonlinear and residual components in GNNs can re-introduce or create frequency components absent from input features (Dong et al., 10 Dec 2024).

Active directions include efficient eigen-computation, adaptive polynomial and piecewise bases, automated propagation mechanism search (AutoSGNN), spectral attention architectures, integration of spatial and spectral aggregation, and generalization to dynamic, directed, or signed graphs. The interpretable paradigm shift is that spectral filters implicitly induce a new spatial graph, and bridging both views consistently yields state-of-the-art, adaptable GNNs.

7. Taxonomy of Spectral GNN Filter Designs

A precise taxonomy organizes spectral filter types and models by their mathematical construction and spectral coverage (Bo et al., 2023, Liao et al., 14 Jun 2024):

Category	Filter Design	Representative Models
Fixed (low-pass)	Pre-set	GCN, S²GC, APPNP, PPR, HK
Variable (polynomial)	Learnable Poly	ChebNet, BernNet, JacobiConv, TFGNN
Filter-bank	Multiple bands	GNN-LF/HF, FAGCN, OptBasis, FiGURe
Attention-based	Global pattern	Specformer, GrassNet
Adaptive (node-wise)	Local-Global	DSF framework
Piecewise-constant	Spectrum partition	PECAN
Two-dimensional	2D conv	ChebNet2D
SSM-based	State-space	GrassNet

The selection of spectral filter type is dictated by graph size, frequency profile (homophilic vs. heterophilic), desired interpretability, computational budget, and target bandwidth. The majority of state-of-the-art spectral GNNs now employ adaptive, orthogonal, or non-local methods engineered for modern large-scale graph applications.