Understanding Graph Neural Networks with Generalized Geometric Scattering Transforms

TL;DR

This paper introduces a generalized geometric scattering transform for graphs using asymmetric wavelets, providing theoretical guarantees and unifying existing graph scattering architectures to advance graph neural network understanding.

Contribution

It proposes a broad class of asymmetric wavelet-based graph scattering transforms with proven stability and invariance, extending and unifying prior models.

Findings

The proposed transforms have similar theoretical guarantees as symmetric counterparts.

The construction unifies various existing graph scattering architectures.

Lays groundwork for future graph neural networks with provable properties.

Abstract

The scattering transform is a multilayered wavelet-based deep learning architecture that acts as a model of convolutional neural networks. Recently, several works have introduced generalizations of the scattering transform for non-Euclidean settings such as graphs. Our work builds upon these constructions by introducing windowed and non-windowed geometric scattering transforms for graphs based upon a very general class of asymmetric wavelets. We show that these asymmetric graph scattering transforms have many of the same theoretical guarantees as their symmetric counterparts. As a result, the proposed construction unifies and extends known theoretical results for many of the existing graph scattering architectures. In doing so, this work helps bridge the gap between geometric scattering and other graph neural networks by introducing a large family of networks with provable stability and invariance guarantees. These results lay the groundwork for future deep learning architectures for graph-structured data that have learned filters and also provably have desirable theoretical properties.