Sparse PCA on fixed-rank matrices
This resolves a long-standing open problem in computational theory for machine learning, providing efficient solutions for sparse PCA under fixed-rank constraints.
The paper tackles the computational complexity of sparse PCA by showing that when the covariance matrix has a fixed rank, there exists a polynomial-time algorithm that solves it to global optimality, and extends this result to a version with disjoint supports.
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.