On the Worst-Case Approximability of Sparse PCA

Siu On Chan; Dimitris Papailiopoulos; Aviad Rubinstein

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arXiv · 1507.05950

On the Worst-Case Approximability of Sparse PCA

Siu On Chan·Dimitris Papailiopoulos·Aviad Rubinstein

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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 $(1-\varepsilon)$, for some small constant $\varepsilon > 0$; 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.

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BibTeX RIS

Siu On Chan, Dimitris Papailiopoulos, Aviad Rubinstein. 2015-07-21. On the Worst-Case Approximability of Sparse PCA. https://arxiv.org/abs/1507.05950

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