# Spectral Projector-Based Graph Fourier Transforms

## TL;DR

This paper introduces a spectral projector-based graph Fourier transform that uniquely defines graph signal harmonics, especially useful for defective or non-diagonalizable adjacency matrices common in real-world large sparse graphs.

## Contribution

It proposes a novel GFT framework using spectral projectors that handles defective matrices and provides a coordinate-free, unique spectral decomposition.

## Key findings

- GFT satisfies a generalized Parseval inequality
- Spectral components can be ordered by total variation
- Illustrative application on urban traffic data

## Abstract

The paper presents the graph Fourier transform (GFT) of a signal in terms of its spectral decomposition over the Jordan subspaces of the graph adjacency matrix $A$. This representation is unique and coordinate free, and it leads to unambiguous definition of the spectral components ("harmonics") of a graph signal. This is particularly meaningful when $A$ has repeated eigenvalues, and it is very useful when $A$ is defective or not diagonalizable (as it may be the case with directed graphs). Many real world large sparse graphs have defective adjacency matrices. We present properties of the GFT and show it to satisfy a generalized Parseval inequality and to admit a total variation ordering of the spectral components. We express the GFT in terms of spectral projectors and present an illustrative example for a real world large urban traffic dataset.