On Consistency and Sparsity for Principal Components Analysis in High Dimensions

• 成果类型: Article
• 署名作者: Johnstone, Iain M.; Lu, Arthur Yu
• 署名单位: Stanford University
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1198/jasa.2009.0121
• 发表日期: 2009
• 页码: 682-693
• 关键词: feature-selection; PCA

摘要

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 it 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.