The Manifold Scattering Transform for High-Dimensional Point Cloud Data
This work provides a practical method for applying manifold-based deep learning to real-world high-dimensional data, addressing a gap in numerical implementation for domains like biology.
The authors tackled the problem of implementing the manifold scattering transform for high-dimensional point cloud data, such as in single-cell genetics, by developing practical schemes based on diffusion maps, and showed effectiveness in signal and manifold classification tasks.
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.