Orthonormalized partial least squares (OPLS) is a popular multivariate analysis method to perform supervised feature extraction. Usually, in machine learning papers OPLS projections are obtained by solving a generalized eigenvalue problem. However, in statistical papers the method is typically formulated in terms of a reduced-rank regression problem, leading to a formulation based on a standard eigenvalue decomposition. A first contribution of this paper is to derive explicit expressions for matching the OPLS solutions derived under both approaches and discuss that the standard eigenvalue formulation is also normally more convenient for feature extraction in machine learning. More importantly, since optimization with respect to the projection vectors is carried out without constraints via a minimization problem, inclusion of penalty terms that favor sparsity is straightforward. In the paper, we exploit this fact to propose modified versions of OPLS. In particular, relying on the ℓ1 norm, we propose a sparse version of linear OPLS, as well as a non-linear kernel OPLS with pattern selection. We also incorporate a group-lasso penalty to derive an OPLS method with true feature selection. The discriminative power of the proposed methods is analyzed on a benchmark of classification problems. Furthermore, we compare the degree of sparsity achieved by our methods and compare them with other state-of-the-art methods for sparse feature extraction.This work was partly supported by MINECO projects TEC2011-22480 and PRIPIBIN-2011-1266.Publicad
