Haar-Laplacian for directed graphs

TL;DR

This paper proposes a new Hermitian Laplacian for directed graphs inspired by Haar transformations, enabling spectral convolutional networks and improved signal processing while preserving directionality and weight information.

Contribution

Introduces a Haar-Laplacian matrix for directed graphs that maintains directionality, is theoretically sound, and supports new spectral convolutional network architectures.

Findings

HaarNet outperforms existing methods in weight prediction.

The approach improves denoising on directed graphs.

The Haar-Laplacian exhibits desirable properties like robustness and sensitivity.

Abstract

This paper introduces a novel Laplacian matrix aiming to enable the construction of spectral convolutional networks and to extend the signal processing applications for directed graphs. Our proposal is inspired by a Haar-like transformation and produces a Hermitian matrix which is not only in one-to-one relation with the adjacency matrix, preserving both direction and weight information, but also enjoys desirable additional properties like scaling robustness, sensitivity, continuity, and directionality. We take a theoretical standpoint and support the conformity of our approach with the spectral graph theory. Then, we address two use-cases: graph learning (by introducing HaarNet, a spectral graph convolutional network built with our Haar-Laplacian) and graph signal processing. We show that our approach gives better results in applications like weight prediction and denoising on directed graphs.