SVD-Based Graph Fractional Fourier Transform on Directed Graphs and Its Application

TL;DR

This paper introduces two novel graph fractional Fourier transforms for directed graphs using singular value decomposition, enabling effective analysis of spatial-temporal data, with demonstrated denoising performance on weather data.

Contribution

The paper proposes new GFRFTs on directed graphs via SVD of fractional Laplacian matrices, extending the theory to multiple graphs and demonstrating practical denoising applications.

Findings

Effective representation of spatial-temporal data on directed graphs.

Improved denoising performance on temperature data.

Extension of GFRFT theory to generalized Cartesian product graphs.

Abstract

Graph fractional Fourier transform (GFRFT) is an extension of graph Fourier transform (GFT) that provides an additional fractional analysis tool for graph signal processing (GSP) by generalizing temporal-vertex domain Fourier analysis to fractional orders. In recent years, a large number of studies on GFRFT based on undirected graphs have emerged, but there are very few studies on directed graphs. Therefore, in this paper, one of our main contributions is to introduce two novel GFRFTs defined on Cartesian product graph of two directed graphs, by performing singular value decomposition on graph fractional Laplacian matrices. We prove that two proposed GFRFTs can effectively express spatial-temporal data sets on directed graphs with strong correlation. Moreover, we extend the theoretical results to a generalized Cartesian product graph, which is constructed by $m$ directed graphs. Finally, the denoising performance of our proposed two GFRFTs are testified through simulation by processing hourly temperature data sets collected from 32 weather stations in the Brest region of France.