The Continuous Relaxation of Sparse PCA is NP-hard

Linbin Li; Yong Xia

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arXiv · 2607.09429

The Continuous Relaxation of Sparse PCA is NP-hard

Linbin Li·Yong Xia

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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.

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Linbin Li, Yong Xia. 2026-07-10. The Continuous Relaxation of Sparse PCA is NP-hard. https://arxiv.org/abs/2607.09429

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