The Continuous Relaxation of Sparse PCA is NP-hard

Linbin Li; Yong Xia

The Continuous Relaxation of Sparse PCA is NP-hard

Optimization and Control 2026-07-10 v1

Authors: Linbin Li , Yong Xia

Abstract

Maximizing a symmetric quadratic form under simultaneous L1 norm inequality and L2 norm equality constraints is a standard and widely used continuous relaxation for Sparse Principal Component Analysis (SPCA). This paper settles the computational complexity of this continuous formulation by proving it is NP-hard. Furthermore, the variant with both L1 and L2 norm inequalities is also shown to be NP-hard.

Keywords

sparse PCA continuous relaxation NP-hard computational complexity quadratic form

Cite

@article{arxiv.2607.09429,
  title  = {The Continuous Relaxation of Sparse PCA is NP-hard},
  author = {Linbin Li and Yong Xia},
  journal= {arXiv preprint arXiv:2607.09429},
  year   = {2026}
}

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5 pages