# The Sparse Principal Component of a Constant-rank Matrix

arXiv: https://arxiv.org/abs/1312.5891 · HTML: https://mlsanity.com/paper/1312.5891/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Dec 2013.

## Built on by (most cited first, 2 of 2)

- [Sparse PCA on fixed-rank matrices](https://mlsanity.com/paper/2201.02487/) (2022)
- [Principled and Scalable Diversity-Aware Retrieval via Cardinality-Constrained Binary Quadratic Programming](https://mlsanity.com/paper/2604.02554/) (2026)

## Abstract (arXiv)

The computation of the sparse principal component of a matrix is equivalent to the identification of its principal submatrix with the largest maximum eigenvalue. Finding this optimal submatrix is what renders the problem NP-hard. In this work, we prove that, if the matrix is positive semidefinite and its rank is constant, then its sparse principal component is polynomially computable. Our proof utilizes the auxiliary unit vector technique that has been recently developed to identify problems that are polynomially solvable. Moreover, we use this technique to design an algorithm which, for any sparsity value, computes the sparse principal component with complexity mathcal Oleft(ND+1right), where N and D are the matrix size and rank, respectively. Our algorithm is fully parallelizable and memory efficient.