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, datapoints 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. Keeping in mind practical constraints of real-world datasets, like small sample sizes and measurement noise, we found that estimating curvature is only feasible for even simple, low-dimensional toy manifolds, when the extrinsic approach is used. 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 approach to study single-cell transcriptomic sequencing (scRNAseq) datasets of blood, gastrulation, and brain cells, revealing for the first time the intrinsic curvature of scRNAseq manifolds.