On the Worst-Case Approximability of Sparse PCA

Abstract

It is well known that Sparse PCA (Sparse Principal Component Analysis) is NP-hard to solve exactly on worst-case instances. What is the complexity of solving Sparse PCA approximately? Our contributions include: 1) a simple and efficient algorithm that achieves an $n^{-1/3}$-approximation; 2) NP-hardness of approximation to within $(1-\epsilon )$, for some small constant $\epsilon >0$; 3) SSE-hardness of approximation to within any constant factor; and 4) an $\exp \exp \left(\Omega \left(\sqrt{\log \log n}\right)\right)$ ("quasi-quasi-polynomial") gap for the standard semidefinite program.

20 pages

References in corpus (3)

• Generalized power method for sparse principal component analysis
• Sparse PCA via Bipartite Matchings
• Computational Complexity of Certifying Restricted Isometry Property

Cited by in corpus (1)

• Using L1-relaxation and integer programming to obtain dual bounds for sparse PCA