Manifold Diffusion Geometry: Curvature, Tangent Spaces, and Dimension

Authors

Iolo Jones

Fields

Differential Geometry - Computational Geometry - Applied Mathematics

Keypoints

Diffusion Geometry: Techniques involving diffusion processes provide a robust framework for estimating geometric properties on manifolds.

Limitations of Local PCA: Traditional approaches like LPCA struggle with issues of parameter sensitivity and density variations.

Robust Tangent Space and Dimensionality Estimation: The new diffusion-based methods achieve greater stability in estimating tangent spaces and intrinsic dimensions, especially in noisy data.

Summary

The paper presents innovative methods in diffusion geometry to accurately estimate curvature, tangent spaces, and dimensionality in high-dimensional manifold data. By modeling data as a graph and simulating a Markov diffusion process, these methods evaluate the structure of data more robustly compared to local principal component analysis (LPCA). While LPCA is widely used for approximating local tangent spaces, it is often limited by its reliance on parameters such as neighborhood size, which can lead to instability in heterogeneous or noisy datasets.

In contrast, the diffusion-based method extends beyond conventional spectral decomposition by using the diffusion process to enhance signal quality over noise and density fluctuations. Specifically, this approach calculates a local covariance matrix derived from the diffusion operator, enabling more accurate tangent space recovery and dimensionality estimation. This method demonstrates resilience against noise and density changes by effectively aggregating local information over multiple scales of diffusion. As a result, it offers a more dependable approach for analyzing complex and non-uniformly sampled manifolds.

The robustness and adaptability of diffusion-based tangent space and dimensionality estimation make this technique especially valuable for applications requiring reliable geometric inference in complex, high-dimensional data environments, such as image processing, machine learning, and bioinformatics. Overall, diffusion geometry offers a promising alternative to LPCA, yielding more stable and meaningful representations of manifold structure.

Link

Arxiv