Penalized Versus Constrained Generalized Eigenvalue Problems Irina Gaynanova James G. Booth Martin T. Wells 10.6084/m9.figshare.3159700 https://tandf.figshare.com/articles/dataset/Penalized_versus_constrained_generalized_eigenvalue_problems/3159700 <p>We investigate the difference between using an ℓ<sub>1</sub> penalty versus an ℓ<sub>1</sub> constraint in generalized eigenvalue problems arising in multivariate analysis. Our main finding is that the ℓ<sub>1</sub> penalty may fail to provide very sparse solutions; a severe disadvantage for variable selection that can be remedied by using an ℓ<sub>1</sub> constraint. Our claims are supported both by empirical evidence and theoretical analysis. Finally, we illustrate the advantages of the ℓ<sub>1</sub> constraint in the context of discriminant analysis and principal component analysis. Supplementary materials for this article are available online.</p> 2017-04-25 05:40:45 Discriminant analysis Duality gap Nonconvex optimization Variable selection