The Manifold Scattering Transform for High-Dimensional Point Cloud Data

TL;DR

This paper introduces practical methods for implementing the manifold scattering transform on high-dimensional point cloud data, enabling effective signal and manifold classification in naturalistic systems.

Contribution

It extends the manifold scattering transform to high-dimensional point clouds using diffusion maps, providing a practical implementation for real-world data.

Findings

Effective for signal classification

Effective for manifold classification

Applicable to high-dimensional biological data

Abstract

The manifold scattering transform is a deep feature extractor for data defined on a Riemannian manifold. It is one of the first examples of extending convolutional neural network-like operators to general manifolds. The initial work on this model focused primarily on its theoretical stability and invariance properties but did not provide methods for its numerical implementation except in the case of two-dimensional surfaces with predefined meshes. In this work, we present practical schemes, based on the theory of diffusion maps, for implementing the manifold scattering transform to datasets arising in naturalistic systems, such as single cell genetics, where the data is a high-dimensional point cloud modeled as lying on a low-dimensional manifold. We show that our methods are effective for signal classification and manifold classification tasks.