Generalized Linear Graph Representation: A Compact Operator Space for Graph Signal Processing and Graph Neural Networks

Graph Signal Processing (GSP) and Graph Neural Networks (GNNs) rely fundamentally on the matrix representation of the underlying graph topology. This representation defines key operators such as the graph Fourier transform, spectral filtering, and convolution. Existing parameterized operator families interpolate only partial subsets of classical graph matrices, while broader formulations become non-compact when representing transition-type operators, limiting both theoretical analysis and stable learning. To address this issue, we propose the Generalized Linear Graph Representation (GLGR), denoted by $\mathbf{Q}_{α,l}$, as a compact two-parameter operator family defined on a bounded linear domain. GLGR unifies major classical operators together with transition-type operators without requiring asymptotic parameters. Theoretically, we show that $\mathbf{Q}_{α,l}$ admits a variational decomposition balancing local smoothness and global degree-weighted energy, derive spectral perturbation bounds, and establish graph-aware sufficient conditions for positive semi-definiteness. Building on this formulation, we develop Adaptive GLGR Convolution (AG-Conv), which makes the propagation operator itself learnable within end-to-end GNNs. Experiments on graph classification and node classification benchmarks show that GLGR improves both fixed-operator representation search and adaptive graph learning across multiple backbones.

A Unified Fractional Spectral Framework for Spatiotemporal Graph Signals: Bi-Fractional Transform and Geodesic Coupling

Graph signal processing extends spectral analysis to data supported on irregular domains. Existing fractional transforms for two-dimensional graph signals, including the two-dimensional graph fractional Fourier transform (GFRFT), typically impose a shared fractional order across dimensions, which limits adaptivity to heterogeneous spatiotemporal spectra. To address this limitation, we propose the two-dimensional graph bi-fractional Fourier transform, which assigns independent fractional orders to the factor graphs of a Cartesian product, enabling decoupled spectral control while preserving separability, unitarity, and invertibility. To further resolve the basis ambiguity in temporal fractional analysis, we develop a geodesic-coupled GFRFT by constructing a coupling path along the principal geodesic on the unitary manifold, thereby unifying graph-induced and discrete temporal bases with guaranteed unitarity and a closed-form inverse. Building on these transforms, we derive a differentiable Wiener-type filtering framework with a hybrid optimization strategy: the fractional orders are learned end-to-end from data, while the coupling parameter is fixed as a structural regularizer. Experiments on real-world time-varying graph datasets and dynamic image restoration tasks demonstrate consistent gains over state-of-the-art fractional transforms and competitive learning-based baselines.

FGFRFT: Fast Graph Fractional FourierTransform via Fourier Series Approximation

The graph fractional Fourier transform (GFRFT) generalizes the graph Fourier transform (GFT) but suffers from a significant computational bottleneck: determining the optimal transform order requires expensive eigendecomposition and matrix multiplication, leading to $O(N^3)$ complexity. To address this issue, we propose a fast GFRFT (FGFRFT) algorithm for unitary GFT matrices based on Fourier series approximation and an efficient caching strategy. FGFRFT reduces the complexity of generating transform matrices to $O(2LN^2)$ while preserving differentiability, thereby enabling adaptive order learning. We validate the algorithm through theoretical analysis, approximation accuracy tests, and order learning experiments. Furthermore, we demonstrate its practical efficacy for image and point cloud denoising and present the fractional specformer, which integrates the FGFRFT into the specformer architecture. This integration enables the model to overcome the limitations of a fixed GFT basis and learn optimal fractional orders for complex data. Experimental results confirm that the proposed algorithm significantly accelerates computation and achieves superior performance compared with the GFRFT.