ABSTRACT
We investigate the difference between using an ℓ1 penalty versus an ℓ1 constraint in generalized eigenvalue problems arising in multivariate analysis. Our main finding is that the ℓ1 penalty may fail to provide very sparse solutions; a severe disadvantage for variable selection that can be remedied by using an ℓ1 constraint. Our claims are supported both by empirical evidence and theoretical analysis. Finally, we illustrate the advantages of the ℓ1 constraint in the context of discriminant analysis and principal component analysis. Supplementary materials for this article are available online.
Supplementary Materials
Supplementary file: Proofs for all Propositions and derivation of Algorithm 1 (pdf file).
Computer code: All the R script files and C extensions with R wrappers that were used for simulations and real data calculations performed in the article (zip file).
Acknowledgments
The authors thank Michael Todd for a useful discussion of duality theory and Daniela Witten for the helpful comments on the earlier drafts of this manuscript. The authors also thank the associate editor and two anonymous referees for valuable comments and suggestions that helped to improve this manuscript. This research was partially supported by NSF grants DMS-1208488 and DMS-0808864.