Sparse PCA on fixed-rank matrices

TL;DR

This paper investigates the computational complexity of sparse PCA on fixed-rank matrices, demonstrating polynomial-time algorithms for certain cases and clarifying the problem's complexity landscape.

Contribution

It establishes polynomial-time solvability of sparse PCA for fixed-rank matrices and disjoint support variants, resolving key complexity questions.

Findings

Polynomial-time algorithm for fixed-rank sparse PCA

Complexity results depend on the rank of the covariance matrix

Efficient solutions for disjoint support sparse PCA

Abstract

Sparse PCA is the optimization problem obtained from PCA by adding a sparsity constraint on the principal components. Sparse PCA is NP-hard and hard to approximate even in the single-component case. In this paper we settle the computational complexity of sparse PCA with respect to the rank of the covariance matrix. We show that, if the rank of the covariance matrix is a fixed value, then there is an algorithm that solves sparse PCA to global optimality, whose running time is polynomial in the number of features. We also prove a similar result for the version of sparse PCA which requires the principal components to have disjoint supports.