Computing the Riemannian curvature of image patch and single-cell RNA sequencing data manifolds using extrinsic differential geometry.

ABSTRACT: Most high-dimensional datasets are thought to be inherently low-dimensional-that is, data points are constrained to lie on a low-dimensional manifold embedded in a high-dimensional ambient space. Here, we study the viability of two approaches from differential geometry to estimate the Riemannian curvature of these low-dimensional manifolds. The intrinsic approach relates curvature to the Laplace-Beltrami operator using the heat-trace expansion and is agnostic to how a manifold is embedded in a high-dimensional space. The extrinsic approach relates the ambient coordinates of a manifold's embedding to its curvature using the Second Fundamental Form and the Gauss-Codazzi equation. We found that the intrinsic approach fails to accurately estimate the curvature of even a two-dimensional constan

SUBMITTER: Sritharan D 

PROVIDER: S-EPMC8307776 | biostudies-literature | 2021 Jul

REPOSITORIES: biostudies-literature