graph-frequency decomposition

To address the theoretical limitations of existing graph spectral transforms—namely, the lack of angular controllability in the Graph Fractional Fourier Transform (GFRFT) and the failure of the Angle-based Graph Fourier Transform (AGFT) to reduce to the standard Graph Fourier Transform (GFT) at zero angle—this paper proposes the Angle-based Graph Fractional Fourier Transform (AGFRFT). Methodologically, AGFRFT unifies fractional-order and angular parameters within a single spectral framework, constructs a rigorously defined family of rotation matrices ensuring exact GFT recovery at zero angle, and introduces two learnable variants (I-AGFRFT and II-AGFRFT) enabling joint optimization of angular and fractional-order parameters. Experimental results demonstrate that AGFRFT consistently outperforms GFRFT and AGFT on real-world graph signal denoising, image processing, and point cloud analysis tasks. It achieves superior spectral concentration, higher reconstruction fidelity, and enhanced controllability over spectral operations.

This study critically examines whether spectral graph neural networks (spectral GNNs) genuinely leverage spectral properties of graphs to achieve performance gains in node classification tasks. Through theoretical analysis grounded in graph signal processing, examination of Vandermonde systems, and demonstration of equivalence with message-passing neural networks (MPNNs), the work reveals that existing spectral GNNs—such as MagNet and HoloNet—fail to correctly implement the graph Fourier transform. Their reported performance advantages are instead attributable to implicit MPNN-like architectures or implementation artifacts. Rigorous reimplementation and ablation studies confirm that when spectral GNNs strictly adhere to spectral theory, their performance deteriorates significantly, thereby challenging the theoretical foundation underpinning their use in node classification.

Existing graph spectral convolution methods suffer from limited flexibility in spectral basis selection and kernel parameterization, hindering effective modeling of large-scale spatial signal distributions and restricting the expressivity of spectral filters. To address these limitations, we propose WaveGC—a novel graph convolutional model that integrates multi-resolution graph wavelet bases with matrix-valued spectral filter kernels for efficient graph convolution. Our key innovation lies in constructing general-purpose graph wavelets satisfying the admissibility condition via odd-even Chebyshev polynomial decomposition, enabling theoretical decoupling of short-range and long-range information and supporting their adaptive fusion. Extensive experiments demonstrate that WaveGC consistently outperforms state-of-the-art graph wavelet neural networks on both short-range and long-range graph learning tasks, validating its enhanced representational capacity and superior generalization performance.

Conventional diffusion models for graph generation suffer from O(n²) computational complexity in the node space, hindering scalability. Method: This paper proposes GGSD, the first model to jointly integrate graph Laplacian spectral decomposition with denoising diffusion probabilistic modeling, establishing a novel spectral-space diffusion paradigm. GGSD employs spectral truncation for efficient low-dimensional representation and introduces a linear-complexity, permutation-invariant Transformer architecture that supports node feature fusion. Crucially, it generates graph structures directly in the node space while achieving theoretical O(n) complexity. Contribution/Results: Extensive experiments demonstrate that GGSD significantly outperforms state-of-the-art methods on both synthetic and real-world graph datasets, achieving superior trade-offs among generation speed, structural fidelity, and scalability.

Spectral graph neural networks (GNNs) suffer from limited expressive power and degraded fitting performance due to repeated eigenvalues of the normalized Laplacian matrix. This work theoretically establishes, for the first time, that the number of *distinguishable* eigenvalues fundamentally bounds the expressive capacity of spectral GNNs. To address this, we propose an unsupervised eigenvalue correction mechanism that applies controlled perturbations and eigenvalue redistribution to break eigenvalue clustering and enhance eigenvalue distinguishability. We further design a compatible polynomial graph convolutional filter leveraging the corrected spectrum. Extensive experiments on synthetic graphs and multiple real-world benchmark datasets demonstrate that our method significantly improves node classification accuracy, while enhancing model generalization and robustness—thereby overcoming spectral GNNs’ inherent dependence on favorable eigenvalue distributions.

This work addresses the challenge that traditional graph filters struggle to simultaneously achieve interpretability, scalability, and cross-graph generalization when processing heterogeneous graph signals. The authors propose an adaptive spectral shaping framework that learns reusable baseline spectral kernels and modulates them with a small number of Gaussian components to construct multi-peak, multi-scale filters. By employing Chebyshev expansions instead of explicit eigendecomposition, the method enables efficient computation. The introduced TASS mechanism facilitates few-shot adaptation across graphs using a fixed set of baseline kernels, significantly enhancing transfer stability. Experiments demonstrate that the proposed approach achieves substantially lower reconstruction errors than fixed wavelet and linear filter banks on multiple synthetic graph benchmarks, while maintaining both interpretability and strong cross-graph generalization capabilities.

Traditional spectral graph neural networks struggle to effectively model long-range dependencies due to the high computational cost of full-graph eigendecomposition and the lack of vertex-domain locality. This work proposes a local–global spectral graph neural network that circumvents full-graph decomposition by performing localized spectral analysis on subgraphs and efficiently integrating global information through Cauchy-structured matrices. Driven by graph topology, the method operates with only quadratic computational complexity while preserving both local structural awareness and the capacity to capture global dependencies. On benchmark tasks emphasizing non-local relationships, the approach achieves state-of-the-art performance with orders of magnitude fewer parameters than existing spectral methods.

This work addresses the challenge that conventional graph signal processing methods struggle to effectively model node data in heterogeneous networks due to disparities in dimensionality, modality, and geometric structure. To overcome this limitation, the authors propose a unified framework termed Layered Signal Processing (SSP), which characterizes heterogeneous local signal spaces through network layers and the linear mappings between them, thereby generalizing fundamental operations such as spectral analysis, filtering, and sampling. Key contributions include the first formal definition of the Layered Fourier Transform (SFT), whose frequency basis is constructed from topological and restriction mappings; the introduction of representation layers that accommodate diverse bases, dictionaries, or embeddings while preserving spectral properties; and the design of polynomial layered filters along with a joint node-component sampling strategy. Experiments on synthetic, motion capture, and financial datasets demonstrate significant performance gains over classical baselines, and the framework establishes conditions for perfect reconstruction of bandlimited signals.

This work addresses the challenge of denoising signals defined on directed graphs, overcoming the limitation of conventional approaches that are restricted to undirected graphs. It extends Wiener filtering theory to the directed graph setting for the first time, deriving optimal denoising solutions under both correlated and uncorrelated noise assumptions based on distinct notions of graph stationarity. The proposed framework integrates concepts from graph signal processing, classical Wiener filtering, and models of stationarity tailored to directed graphs. Experimental results on real-world temperature graph data demonstrate the effectiveness of the method, achieving significantly improved denoising performance for directed graph signals. This study thus establishes both a theoretical foundation and a practical toolset for signal processing on directed graphs.

This work addresses the challenge in graph signal processing where spectral-based filtering methods are often inapplicable due to incomplete knowledge of the full graph topology. To overcome this limitation, we propose the first data-driven algebraic framework for subgraph filtering, constructing a distance-aware Laplacian-based subgraph filtering algebra that defines a structured and controllable class of filters capable of approximating full-graph filters. Leveraging statistical learning theory, we establish risk bounds on the approximation performance under least-squares loss, providing rigorous theoretical guarantees. Empirical evaluations demonstrate that our approach significantly outperforms polynomial filters, distribution-agnostic operators, and end-to-end numerical learning baselines on real-world datasets.