# A Digraph Fourier Transform With Spread Frequency Components

## TL;DR

This paper introduces a new directed graph Fourier transform (DGFT) that spreads frequency components across the spectral domain, using a heuristic approach with theoretical guarantees, demonstrated on synthetic and real-world graphs.

## Contribution

It proposes a novel DGFT construction method based on spectral dispersion and matroid constraints, with a scalable greedy algorithm and performance guarantees.

## Key findings

- Effective spread of frequency components demonstrated
- Scalable algorithm with provable guarantees
- Successful tests on synthetic and real-world graphs

## Abstract

We study the problem of constructing a graph Fourier transform (GFT) for directed graphs (digraphs), which decomposes graph signals into different modes of variation with respect to the underlying network. Accordingly, to capture low, medium and high frequencies we seek a digraph (D)GFT such that the orthonormal frequency components are as spread as possible in the graph spectral domain. This specification gives rise to a challenging nonconvex optimization problem, so we resort to a simple yet efficient heuristic to construct the DGFT basis from Laplacian eigenvectors of an undirected version of the digraph. To select frequency components which are as spread as possible, we define a spectral dispersion function and show that it is supermodular. Moreover, we show that orthonormality can be enforced via a matroid basis constraint, which motivates adopting a scalable greedy algorithm to obtain an approximate solution with provable performance guarantee. The effectiveness of the novel DGFT is illustrated through numerical tests on synthetic and real-world graphs.