Discovering latent topology and geometry in data: a law of large dimension

Complex topological and geometric patterns often appear embedded in high-dimensional data and seem to reflect structure related to the underlying data source, with some distortion. We show that this rich data morphology can be explained by a generic and remarkably simple statistical model, demonstrating that manifold structure in data can emerge from elementary statistical ideas of correlation and latent variables. The Latent Metric Space model consists of a collection of random fields, evaluated at locations specified by latent variables and observed in noise. Driven by high dimensionality, principal component scores associated with data from this model are uniformly concentrated around a topological manifold, homeomorphic to the latent metric space. Under further assumptions this relation may be a diffeomorphism, a Riemannian metric structure appears, and the geometry of the manifold reflects that of the latent metric space. This provides statistical justification for manifold assumptions which underlie methods ranging from clustering and topological data analysis, to nonlinear dimension reduction, regression and classification, and explains the efficacy of Principal Component Analysis as a preprocessing tool for reduction from high to moderate dimension.