Statistical exploration of the manifold hypothesis

Abstract The manifold hypothesis is a widely accepted tenet of machine learning which asserts that nominally high-dimensional data are in fact concentrated near a low-dimensional manifold, embedded in high-dimensional space. This phenomenon is observed empirically in many real-world situations, has led to development of a wide range of statistical methods in the last few decades, and has been suggested as a key factor in the success of modern AI technologies. We show that rich and sometimes intricate manifold structure in data can emerge from a generic and remarkably simple statistical model—the latent metric space (LMS) model—via elementary concepts such as latent variables, correlation, and stationarity.…