title: On Consistency and Sparsity for Principal Components Analysis in High Dimensions.
abstract: Principal components analysis (PCA) is a classic method for the reduction of dimensionality of data in the form of n observations (or cases) of a vector with p variables. Contemporary datasets often have p comparable with or even much larger than n. Our main assertions, in such settings, are (a) that some initial reduction in dimensionality is desirable before applying any PCA-type search for principal modes, and (b) the initial reduction in dimensionality is best achieved by working in a basis in which the signals have a sparse representation. We describe a simple asymptotic model in which the estimate of the leading principal component vector via standard PCA is consistent if and only if p(n)/n→0. We provide a simple algorithm for selecting a subset of coordinates with largest sample variances, and show that if PCA is done on the selected subset, then consistency is recovered, even if p(n) ⪢ n.
authors.affiliation: Iain M. Johnstone is Professor of Statistics and Biostatistics, Stanford University, Department of Statistics, 390 Serra Mall, Stanford, CA 94305 ( imj@stanford.edu ). Arthur Yu Lu is Principal, Renaissance Technologies LLC, 600 Route 25A, East Setauket, NY 11733.
authors.forename: Iain M
authors.initials: IM
authors.lastname: Johnstone
authors.forename: Arthur Yu
authors.initials: AY
authors.lastname: Lu