SPARSISTENCY AND AGNOSTIC INFERENCE IN SPARSE PCA

• 成果类型: Article
• 署名作者: Lei, Jing; Vu, Vincent Q.
• 署名单位: Carnegie Mellon University; University System of Ohio; Ohio State University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/14-AOS1273
• 发表日期: 2015
• 页码: 299-322
• 关键词: Principal component analysis; high-dimensional analysis; variable selection; power method; Consistency; likelihood; rates

摘要

The presence of a sparse truth has been a constant assumption in the theoretical analysis of sparse PCA and is often implicit in its methodological development. This naturally raises questions about the properties of sparse PCA methods and how they depend on the assumption of sparsity. Under what conditions can the relevant variables be selected consistently if the truth is assumed to be sparse? What can be said about the results of sparse PCA without assuming a sparse and unique truth? We answer these questions by investigating the properties of the recently proposed Fantope projection and selection (FPS) method in the high-dimensional setting. Our results provide general sufficient conditions for sparsistency of the FPS estimator. These conditions are weak and can hold in situations where other estimators are known to fail. On the other hand, without assuming sparsity or identifiability, we show that FPS provides a sparse, linear dimension-reducing transformation that is close to the best possible in terms of maximizing the predictive covariance.