Understanding Graph Neural Networks with Generalized Geometric Scattering Transforms
This work provides a theoretical foundation for future deep learning architectures on graph-structured data, but it is incremental as it builds upon prior generalizations of scattering transforms.
The authors tackled the problem of understanding graph neural networks by introducing generalized geometric scattering transforms for graphs using asymmetric wavelets, showing that these transforms unify and extend theoretical guarantees for existing architectures with provable stability and invariance.
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.