Frames and vertex-frequency representations in graph fractional Fourier domain

Abstract

Vertex-frequency analysis – particularly the windowed graph Fourier transform (WGFT) – captures the correspondence between vertices and frequencies in graph signal processing. However, existing methods often struggle to accurately extract vertex-frequency features from sparse graph signals encountered in real-world applications, such as those derived from COVID-19 datasets. To address this limitation, we propose the multi-windowed graph fractional Fourier transform (MWGFRFT), a novel framework that enhances vertex localization through multi-windowed analysis and improves (fractional) frequency resolution via fractional-order transforms. Theoretically, MWGFRFT generalizes several existing transforms, including the WGFT, as special cases. To improve computational efficiency, we further develop a fast algorithm, FMWGFRFT. Experimental results demonstrate that FMWGFRFT effectively captures vertex-frequency features in the graph fractional Fourier domain and exhibits strong robustness in practical scenarios. Applications include anomaly detection and adaptive learning of graph fractional Laplacian bases, highlighting its potential for analyzing complex signals over irregular graph domains.