OPTIMAL DETECTION OF SPARSE PRINCIPAL COMPONENTS IN HIGH DIMENSION

• 成果类型: Article
• 署名作者: Berthet, Quentin; Rigollet, Philippe
• 署名单位: Princeton University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/13-AOS1127
• 发表日期: 2013
• 页码: 1780-1815
• 关键词: large hidden clique; LARGEST EIGENVALUE; NORM; size; PCA; submatrices; relaxations; Consistency; matrices; noisy

摘要

We perform a finite sample analysis of the detection levels for sparse principal components of a high-dimensional covariance matrix. Our minimax optimal test is based on a sparse eigenvalue statistic. Alas, computing this test is known to be NP-complete in general, and we describe a computationally efficient alternative test using convex relaxations. Our relaxation is also proved to detect sparse principal components at near optimal detection levels, and it performs well on simulated datasets. Moreover, using polynomial time reductions from theoretical computer science, we bring significant evidence that our results cannot be improved, thus revealing an inherent trade off between statistical and computational performance.