On the Worst-Case Approximability of Sparse PCA

TL;DR

This paper investigates the computational difficulty of approximately solving Sparse PCA, presenting an efficient approximation algorithm, hardness results, and analyzing the limitations of semidefinite programming approaches.

Contribution

It introduces a simple approximation algorithm for Sparse PCA, establishes NP-hardness and SSE-hardness of approximation, and analyzes SDP relaxation gaps.

Findings

An $n^{-1/3}$-approximation algorithm for Sparse PCA.

NP-hardness of approximation within any constant factor.

Existence of a quasi-quasi-polynomial gap for SDP relaxation.

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.