NP-Hardness and Inapproximability of Sparse PCA

TL;DR

This paper proves that solving sparse PCA optimally is computationally hard (NP-hard) and cannot be approximated efficiently within certain factors, based on reductions from clique problems and complexity assumptions.

Contribution

It establishes NP-hardness and inapproximability results for sparse PCA using reductions from clique, highlighting fundamental computational limitations.

Findings

Sparse PCA is NP-hard to solve exactly.

No polynomial-time approximation scheme exists for sparse PCA unless P=NP.

Constant-factor approximation algorithms are also unlikely under weaker assumptions.

Abstract

We give a reduction from {\sc clique} to establish that sparse PCA is NP-hard. The reduction has a gap which we use to exclude an FPTAS for sparse PCA (unless P=NP). Under weaker complexity assumptions, we also exclude polynomial constant-factor approximation algorithms.