Graph Scattering beyond Wavelet Shackles

Christian Koke, Gitta Kutyniok

Keywords: Geometric Deep Learning, Graph Convolutional Networks, Scattering, Wavelets, Stability Guarantees, Rigorous Proofs, Quantum Chemistry

Abstract: This work develops a flexible and mathematically sound framework for the design and analysis of graph scattering networks with variable branching ratios and generic functional calculus filters. Spectrally-agnostic stability guarantees for node- and graph-level perturbations are derived; the vertex-set non-preserving case is treated by utilizing recently developed mathematical-physics based tools. Energy propagation through the network layers is investigated and related to truncation stability. New methods of graph-level feature aggregation are introduced and stability of the resulting composite scattering architectures is established. Finally, scattering transforms are extended to edge- and higher order tensorial input. Theoretical results are complemented by numerical investigations: Suitably chosen scattering networks conforming to the developed theory perform better than traditional graph-wavelet based scattering approaches in social network graph classification tasks and significantly outperform other graph-based learning approaches to regression of quantum-chemical energies on QM$7$.

TL;DR: We substantially generalize graph-wavelet based scattering networks, develop a corresponding theory and outperform competing models in predicting quantum chemical energies.