Graph Fourier transforms on directed product graphs

TL;DR

This paper introduces two novel Graph Fourier Transforms for directed product graphs, enabling efficient representation and denoising of spatial-temporal data on directed networks, extending existing undirected graph methods.

Contribution

The paper proposes new GFTs based on SVD of graph Laplacians for directed Cartesian product graphs, unifying and extending spectral methods for directed graphs.

Findings

Effective representation of spatial-temporal data on directed networks.

Successful denoising of temperature data using spectral bandlimiting.

In undirected case, the proposed GFTs coincide with joint GFTs in literature.

Abstract

Graph Fourier transform (GFT) is one of the fundamental tools in graph signal processing to decompose graph signals into different frequency components and to represent graph signals with strong correlation by different modes of variation effectively. The GFT on undirected graphs has been well studied and several approaches have been proposed to define GFTs on directed graphs. In this paper, based on the singular value decompositions of some graph Laplacians, we propose two GFTs on the Cartesian product graph of two directed graphs. We show that the proposed GFTs could represent spatial-temporal data sets on directed networks with strong correlation efficiently, and in the undirected graph setting they are essentially the joint GFT in the literature. In this paper, we also consider the bandlimiting procedure in the spectral domain of the proposed GFTs, and demonstrate its performance to denoise the temperature data set in the region of Brest (France) on January 2014.