Geometric Scattering on Measure Spaces

TL;DR

Geometric scattering on measure spaces generalizes Mallat's Euclidean scattering to broad domains including directed/signed graphs and manifolds with boundary. The authors introduce two scattering variants, windowed $S$ and non-windowed $\overline{S}$, and develop an invariance/equivariance framework under a group of bijections $\mathcal{G}$, along with stability analyses for both wavelet and scattering transforms. They connect graph-based scattering to manifold scattering via diffusion maps, providing convergence rates as sample size grows and offering data-driven schemes to implement the manifold transform from point clouds. Empirically, the approach yields strong results on spherical images, single-cell data, and directed graphs, illustrating its versatility for geometric deep learning tasks on irregular domains.

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.

Figures (3)

• Figure 1: The MNIST dataset projected onto the sphere.
• Figure 2: Wavelets on the FAUST dataset with $g(\lambda) = e^{-0.0005\lambda}$, $j = 1, 3, 5$ from left to right. Positive values are red, while negative values are blue.
• Figure 3: Meta-graphs for synthetic datasets (Reproduced from zhang2021magnet).

Theorems & Definitions (65)

• Remark 1
• Proposition 1
• Remark 2
• Proposition 2
• Remark 3
• Theorem 1
• Theorem 2
• Remark 4
• Remark 5
• Lemma 1