Geometric Scattering on Measure Spaces

2022-08-17

Social Science Research Network

AI Summary

This work addresses non-Euclidean geometric data—including directed graphs, signed graphs, and manifolds with boundary—by proposing a unified geometric scattering transform framework on metric measure spaces. Methodologically, it integrates diffusion maps, multi-layer wavelet-type scattering, and stochastic sampling approximation to yield a computationally tractable model that simultaneously satisfies group invariance and Lipschitz stability, while establishing a data-driven graph construction scheme and a quantitative convergence theory. Key contributions include: (i) the first extension of geometric scattering to manifolds with boundary and to directed/signed graphs; (ii) a universal invariance criterion applicable across diverse geometric domains; and (iii) a rigorous proof of asymptotic convergence of scattering transforms on sampled graphs to those on continuous manifolds. Experiments demonstrate superior effectiveness and robustness on spherical images, directed networks, and high-dimensional single-cell data, offering a theoretically grounded and practically viable tool for geometric deep learning.

Abstract

The scattering transform is a multilayered, wavelet-based transform initially introduced as a model of convolutional neural networks (CNNs) that has played a foundational role in our understanding of these networks' stability and invariance properties. Subsequently, there has been widespread interest in extending the success of CNNs to data sets with non-Euclidean structure, such as graphs and manifolds, leading to the emerging field of geometric deep learning. In order to improve our understanding of the architectures used in this new field, several papers have proposed generalizations of the scattering transform for non-Euclidean data structures such as undirected graphs and compact Riemannian manifolds without boundary. In this paper, we introduce a general, unified model for geometric scattering on measure spaces. Our proposed framework includes previous work on geometric scattering as special cases but also applies to more general settings such as directed graphs, signed graphs, and manifolds with boundary. We propose a new criterion that identifies to which groups a useful representation should be invariant and show that this criterion is sufficient to guarantee that the scattering transform has desirable stability and invariance properties. Additionally, we consider finite measure spaces that are obtained from randomly sampling an unknown manifold. We propose two methods for constructing a data-driven graph on which the associated graph scattering transform approximates the scattering transform on the underlying manifold. Moreover, we use a diffusion-maps based approach to prove quantitative estimates on the rate of convergence of one of these approximations as the number of sample points tends to infinity. Lastly, we showcase the utility of our method on spherical images, directed graphs, and on high-dimensional single-cell data.

Problem

• Scattering Transform
• Complex Data Types
• Geometric Deep Learning

Innovation

• Scattering Transform
• Complex Data Types
• Stability Criterion