On Asymptotics of Local Principal Component Analysis. Hitotsubashi Journal of Commerce and Management 1996 Vol.31 pp.1-11.
Assume data are randomly distributed on or near a smooth submanifold of IRd When applied to a subset of data in a neighborhood, principal component analysis (PCA) gives an estimate of the tangent and normal spaces of the underlying manifold. Asymptotic properties of the estimate is surveyed in connection with variations of data and curvatures of the manifold. A dimension estimate based on the work of Waternaux (1975, 1976) is also considered.