Open AccessComputing the Riemannian curvature of image patch and single-cell RNA sequencing data manifolds using extrinsic differential geometry
Author(s) -
Duluxan Sritharan,
Shu Wang,
Sahand Hormoz
Publication year - 2021
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 5.011
H-Index - 771
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.2100473118
Subject(s) - curvature , differential geometry , curvature of riemannian manifolds , riemannian geometry , ansatz , mathematics , computer science , sectional curvature , scalar curvature , topology (electrical circuits) , geometry , combinatorics , mathematical physics
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).