Geometric Scattering on Measure Spaces.

Area of Science:

• Geometric deep learning
• Signal processing
• Mathematical analysis

Background:

• The scattering transform models convolutional neural networks (CNNs), explaining their stability and invariance.
• Geometric deep learning extends CNNs to non-Euclidean data like graphs and manifolds.
• Existing scattering transform generalizations cover specific non-Euclidean structures (graphs, Riemannian manifolds).

Purpose of the Study:

• To introduce a general, unified geometric scattering model applicable to broad measure spaces.
• To establish a new criterion for desirable invariance properties in representations.
• To develop methods for data-driven graph construction for scattering transforms on sampled manifolds.

Main Methods:

• Developed a unified framework for geometric scattering on general measure spaces.
• Proposed a novel criterion for group invariance, proving its sufficiency for stability and invariance.
• Introduced two methods for constructing data-driven graphs for approximating manifold scattering transforms.
• Utilized diffusion maps to analyze convergence rates of graph scattering approximations.

Main Results:

• The proposed framework unifies existing methods and extends to directed graphs, signed graphs, and manifolds with boundary.
• The new invariance criterion guarantees desirable stability and invariance properties.
• Data-driven graph construction enables accurate approximation of scattering transforms on sampled manifolds.
• Quantitative convergence estimates were derived for graph scattering approximations.

Conclusions:

• The unified geometric scattering model provides a flexible and powerful tool for analyzing non-Euclidean data.
• The framework advances the understanding of neural network architectures in geometric deep learning.
• The proposed methods demonstrate practical utility on diverse datasets, including spherical images and single-cell data.