Angular Graph Fractional Fourier Transform: Theory and Application

2025-11-20

AI Summary

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.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Agentic search
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Graph spectral representations are fundamental in graph signal processing, offering a rigorous framework for analyzing and processing graph-structured data. The graph fractional Fourier transform (GFRFT) extends the classical graph Fourier transform (GFT) with a fractional-order parameter, enabling flexible spectral analysis while preserving mathematical consistency. The angular graph Fourier transform (AGFT) introduces angular control via GFT eigenvector rotation; however, existing constructions fail to degenerate to the GFT at zero angle, which is a critical flaw that undermines theoretical consistency and interpretability. To resolve these complementary limitations - GFRFT's lack of angular regulation and AGFT's defective degeneracy - this study proposes an angular GFRFT (AGFRFT), a unified framework that integrates fractional-order and angular spectral analyses with theoretical rigor. A degeneracy-friendly rotation matrix family ensures exact GFT degeneration at zero angle, with two AGFRFT variants (I-AGFRFT and II-AGFRFT) defined accordingly. Rigorous theoretical analyses confirm their unitarity, invertibility, and smooth parameter dependence. Both support learnable joint parameterization of the angle and fractional order, enabling adaptive spectral processing for diverse graph signals. Extensive experiments on real-world data denoising, image denoising, and point cloud denoising demonstrate that AGFRFT outperforms GFRFT and AGFT in terms of spectral concentration, reconstruction quality, and controllable spectral manipulation, establishing a robust and flexible tool for integrated angular fractional spectral analysis in graph signal processing.

Problem

• Existing graph fractional Fourier transforms lack angular control mechanisms
• Angular graph Fourier transforms fail to degenerate properly at zero angle
• No unified framework exists for integrated angular fractional spectral analysis

Innovation

• Unified framework integrating fractional-order and angular spectral analyses
• Degeneracy-friendly rotation matrix ensuring exact GFT degeneration
• Learnable joint parameterization enabling adaptive spectral processing