Orthogonal Transforms for Signals on Directed Graphs

TL;DR

This paper introduces a novel graph signal transform for directed graphs using Schur decomposition, enabling invariant subspaces and orthogonal bases, especially for defective graphs where diagonalization isn't possible.

Contribution

The paper proposes a Schur decomposition-based transform for directed graph signals, offering a flexible approach to generate invariant subspaces with approximate orthogonality, improving over diffusion wavelets.

Findings

Provides a new transform for directed graphs with defective operators

Enables construction of invariant subspaces with orthogonal bases

Offers more flexible subspace generation than diffusion wavelets

Abstract

In this paper we consider the problem of defining transforms for signals on directed graphs, with a specific focus on defective graphs where the corresponding graph operator cannot be diagonalized. Our proposed method is based on the Schur decomposition and leads to a series of embedded invariant subspaces for which orthogonal basis are available. As compared to diffusion wavelets, our method is more flexible in the generation of subspaces, but these subspaces can only be approximately orthogonal.