Graph Fourier transform based on singular value decomposition of directed Laplacian

TL;DR

This paper introduces a new graph Fourier transform for directed graphs using singular value decomposition of the Laplacian, aligning with classical transforms in special cases and efficiently capturing signal variations.

Contribution

It proposes a novel GFT based on SVD of the directed Laplacian, extending the classical GFT to directed graphs with computational efficiency.

Findings

Consistent with classical GFT in undirected graphs

Matches classical DFT on directed circulant graphs, up to rotation and permutation

Efficiently represents graph signals with different variation modes

Abstract

Graph Fourier transform (GFT) is a fundamental concept in graph signal processing. In this paper, based on singular value decomposition of Laplacian, we introduce a novel definition of GFT on directed graphs, and use singular values of Laplacian to carry the notion of graph frequencies. % of the proposed GFT. The proposed GFT is consistent with the conventional GFT in the undirected graph setting, and on directed circulant graphs, the proposed GFT is the classical discrete Fourier transform, up to some rotation, permutation and phase adjustment. We show that frequencies and frequency components of the proposed GFT can be evaluated by solving some constrained minimization problems with low computational cost. Numerical demonstrations indicate that the proposed GFT could represent graph signals with different modes of variation efficiently.