Haar-Laplacian for directed graphs
This addresses the need for effective spectral methods in directed graph analysis, which is incremental as it builds on existing graph theory.
The paper tackles the problem of enabling spectral convolutional networks and signal processing for directed graphs by introducing a novel Laplacian matrix based on a Haar-like transformation, resulting in better performance in applications such as weight prediction and denoising.
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.