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Optimal detection of sparse principal components in high dimension

Quentin Berthet, Philippe Rigollet
2013 Annals of Statistics  
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.
more » ... imulated 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.
doi:10.1214/13-aos1127 fatcat:vij6wvcynjaolgym2cowi3bgx4