# Graph Fourier Transform Based on $\ell_1$ Norm Variation Minimization

## TL;DR

This paper introduces a new graph Fourier transform based on $
$ norm variation minimization, offering a computationally efficient alternative to the traditional Laplacian eigenvector approach, with demonstrated effectiveness.

## Contribution

It proposes an $
$ norm-based Fourier transform, provides a necessary condition for the basis, and develops a fast greedy algorithm for approximation.

## Key findings

- The greedy algorithm effectively approximates the $
$ Fourier basis.
- The $
$ Fourier transform shows similar decay rates to the Laplacian basis.
- Numerical experiments validate the approach's efficiency and effectiveness.

## Abstract

The definition of the graph Fourier transform is a fundamental issue in graph signal processing. Conventional graph Fourier transform is defined through the eigenvectors of the graph Laplacian matrix, which minimize the $\ell_2$ norm signal variation. However, the computation of Laplacian eigenvectors is expensive when the graph is large. In this paper, we propose an alternative definition of graph Fourier transform based on the $\ell_1$ norm variation minimization. We obtain a necessary condition satisfied by the $\ell_1$ Fourier basis, and provide a fast greedy algorithm to approximate the $\ell_1$ Fourier basis. Numerical experiments show the effectiveness of the greedy algorithm. Moreover, the Fourier transform under the greedy basis demonstrates a similar rate of decay to that of Laplacian basis for simulated or real signals.