Optimal detection of sparse principal components in high dimension
This work addresses the challenge of efficient sparse PCA detection for high-dimensional data analysis, with incremental improvements in computational methods.
The paper tackles the problem of detecting sparse principal components in high-dimensional covariance matrices, achieving minimax optimal detection levels with a computationally efficient convex relaxation test that performs well on simulated data and reveals a statistical-computational trade-off.
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.