rs-Sparse principal component analysis: A mixed integer nonlinear programming approach with VNS

Principal component analysis is a popular data analysis dimensionality reduction technique, aiming to project with minimum error for a given dataset into a subspace of smaller number of dimensions.In order to improve interpretability, different variants of the method have been proposed in the literature, in which, besides error minimization, sparsity is sought. In this paper we formulate as a mixed integer nonlinear program the problem of finding a subspace with a sparse basis minimizing the sum of squares of distances between the points and their projections. Contrary to other attempts in the literature, with our model the user can fix the level of sparseness of the resulting basis vectors. Variable neighborhood search is proposed to solve the problem obtained this way.Our numerical experience on test sets shows that our procedure outperforms benchmark methods in the literature, both in terms of sparsity and errors.