Statistical exploration of the manifold hypothesis

• 成果类型: Article
• 署名作者: Whiteley, Nick; Gray, Annie; Rubin-Delanchy, Patrick
• 署名单位: University of Bristol; Alan Turing Institute; University of Edinburgh; University of Edinburgh; Heriot Watt University
• 刊物名称: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
• ISSN/ISSBN: 1369-7412; 1467-9868
• DOI: 10.1093/jrsssb/qkag055
• 发表日期: 2026-04
• 页码: 353-385
• 关键词: nonlinear dimensionality reduction; PRINCIPAL COMPONENT SCORES; gaussian-processes; pca consistency; regression; CLASSIFICATION; asymptotics; confidence; eigenmaps; HOMOLOGY

摘要

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. This establishes a general statistical explanation for why the manifold hypothesis seems to hold in so many situations. Informed by the LMS model we derive procedures to discover and interpret the geometry of high-dimensional data, and explore hypotheses about the data-generating mechanism. These procedures operate under minimal assumptions and make use of well-known dimension reduction methods and graph-analytic algorithms.