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

Significance

High-dimensional datasets are becoming increasingly prevalent in many scientific fields. A universal theme connecting these high-dimensional datasets is the ansatz that data points are constrained to lie on nonlinear low-dimensional manifolds, whose structure is dictated by the natural laws governing the data. While tools have been developed for estimating global properties of these data manifolds, estimating the Riemannian curvature, a local property, has not been considered. Computing curvature of data manifolds offers both detailed criteria with which to evaluate models of these complex data (e.g., a Klein bottle model of image patches) and a way to explore detailed geometric features that cannot simply be visualized by the naked eye (e.g., in single-cell RNA-sequencing data).

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 constant-curvature manifold, whereas the extrinsic approach was able to handle more complex toy models, even when confounded by practical constraints like small sample sizes and measurement noise. To test the applicability of the extrinsic approach to real-world data, we computed the curvature of a well-studied manifold of image patches and recapitulated its topological classification as a Klein bottle. Lastly, we applied the extrinsic approach to study single-cell transcriptomic sequencing (scRNAseq) datasets of blood, gastrulation, and brain cells to quantify the Riemannian curvature of scRNAseq manifolds.