Graph Fractional Fourier Transform: A Unified Theory

The fractional Fourier transform (FRFT) parametrically generalizes the Fourier transform (FT) by a transform order, representing signals in intermediate time-frequency domains. The FRFT has multiple but equivalent definitions, including the fractional power of FT, time-frequency plane rotation, hyper-differential operator, and many others, each offering benefits like derivational ease and computational efficiency. Concurrently, graph signal processing (GSP) extends traditional signal processing to data on irregular graph structures, enabling concepts like sampling, filtering, and Fourier transform for graph signals. The graph fractional Fourier transform (GFRFT) is recently extended to the GSP domain. However, this extension only generalizes the fractional power definition of FRFT based on specific graph structures with limited transform order range. Ideally, the GFRFT extension should be consistent with as many alternative definitions as possible. This paper first provides a rigorous fractional power-based GFRFT definition that supports any graph structure and transform order. Then, we introduce the novel hyper-differential operator-based GFRFT definition, allowing faster forward and inverse transform matrix computations on large graphs. Through the proposed definition, we derive a novel approach to select the transform order by learning the optimal value from data. Furthermore, we provide treatments of the core GSP concepts, such as bandlimitedness, filters, and relations to the other transforms in the context of GFRFT. Finally, with comprehensive experiments, including denoising, classification, and sampling tasks, we demonstrate the equivalence of parallel definitions of GFRFT, learnability of the transform order, and the benefits of GFRFT over GFT and other GSP methods.(1)(1) The codebase is available at https://github.com/koc-lab/gfrft-unified.