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Related Experiment Video

Updated: Sep 19, 2025

Diffusion Imaging in the Rat Cervical Spinal Cord
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The G -invariant graph Laplacian part II: Diffusion maps.

Eitan Rosen1, Xiuyuan Cheng2, Yoel Shkolnisky1

  • 1Department of Applied Mathematics, Tel-Aviv University, Tel-Aviv, Israel.

Applied and Computational Harmonic Analysis
|June 18, 2025
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This study introduces novel diffusion maps that inherently handle group actions on data, improving clustering and alignment for manifold-embedded datasets. These new methods are particularly useful for analyzing complex data like rotated images.

Keywords:
Diffusion mapsGraph LaplacianGroup invariant embeddingsManifold learning

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Area of Science:

  • Data Science
  • Computational Geometry
  • Group Theory

Background:

  • Diffusion maps are effective for dimensionality reduction, clustering, and visualization of manifold data.
  • Analyzing datasets with inherent symmetries, such as those acted upon by continuous matrix groups, presents unique challenges.

Purpose of the Study:

  • To develop diffusion maps that intrinsically incorporate group actions on data.
  • To construct both equivariant and invariant embeddings for enhanced data analysis.
  • To demonstrate the application of these methods in random computerized tomography.

Main Methods:

  • Utilizing the G-invariant graph Laplacian and its eigenfunctions.
  • Employing tensor products of irreducible unitary representations and eigenvectors.
  • Deriving group-aware diffusion maps for manifold data.

Main Results:

  • Successfully constructed diffusion maps that account for group actions.
  • Developed methods for creating both equivariant and invariant embeddings.
  • Showcased the utility of these embeddings for clustering and data alignment.

Conclusions:

  • The proposed diffusion maps offer a principled way to embed data with group symmetries.
  • These methods provide powerful tools for analyzing and organizing complex, structured datasets.
  • The approach has demonstrated practical utility in applications like computerized tomography.