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The Manifold Scattering Transform for High-Dimensional Point Cloud Data

Joyce Chew, Holly R. Steach, Siddharth Viswanath, Hau-Tieng Wu, Matthew Hirn, Deanna Needell, Smita Krishnaswamy, Michael Perlmutter

TL;DR

The work addresses high-dimensional manifold-structured data by extending the scattering transform to point clouds using diffusion-map–based approximations of the Laplace–Beltrami operator. It introduces two practical schemes: (i) a spectrum-based approach leveraging a data-driven Laplacian eigenbasis and (ii) an eigenvector-free approach using adaptive kernels to propagate the heat semigroup, enabling computation of multiscale wavelets $W_j$ and moments $Sf$. The authors demonstrate strong empirical performance on mesh-free two-dimensional surfaces (e.g., spherical MNIST, FAUST) and on high-dimensional single-cell datasets, achieving notable gains over clustering baselines and matching mesh-based results in several tasks. This provides a scalable blueprint for spectral manifold networks on arbitrary-dimensional manifolds, with potential impact in biomedical data analysis and beyond, by delivering robust, geometry-aware feature representations without requiring meshes or explicit manifold models.

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.

The Manifold Scattering Transform for High-Dimensional Point Cloud Data

TL;DR

The work addresses high-dimensional manifold-structured data by extending the scattering transform to point clouds using diffusion-map–based approximations of the Laplace–Beltrami operator. It introduces two practical schemes: (i) a spectrum-based approach leveraging a data-driven Laplacian eigenbasis and (ii) an eigenvector-free approach using adaptive kernels to propagate the heat semigroup, enabling computation of multiscale wavelets and moments . The authors demonstrate strong empirical performance on mesh-free two-dimensional surfaces (e.g., spherical MNIST, FAUST) and on high-dimensional single-cell datasets, achieving notable gains over clustering baselines and matching mesh-based results in several tasks. This provides a scalable blueprint for spectral manifold networks on arbitrary-dimensional manifolds, with potential impact in biomedical data analysis and beyond, by delivering robust, geometry-aware feature representations without requiring meshes or explicit manifold models.

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.
Paper Structure (12 sections, 20 equations, 2 figures, 2 tables)

This paper contains 12 sections, 20 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Manifold Scattering Transform
  3. Implementing the Manifold Scattering Transform from Pointclouds
  4. Approximating $H^t$ via the Spectrum of the Data-Driven Laplacian
  5. Approximating $H$ without Eigenvectors
  6. Related work
  7. Graph and Manifold Scattering
  8. Manifold Neural Networks
  9. Results
  10. Two-dimensional surfaces without a mesh
  11. Single-cell datasets
  12. Conclusions

Figures (2)

  • Figure 1: The MNIST dataset projected onto the sphere.
  • Figure 2: Wavelets on the FAUST dataset, $j = 1, 3, 5$ from left to right. Positive values are red, while negative values are blue.