Variable selection in saturated and supersaturated designs via lp-lq minimization

Abstract

In many real world problems it is of interest to ascertain which factors are most relevant for determining a given outcome. This is the so-called variable selection problem. The present paper proposes a new regression model for its solution. We show that the proposed model satisfies continuity, sparsity, and unbiasedness properties. A generalized Krylov subspace method for the practical solution of the minimization problem involved is described. This method can be used for the solution of both small-scale and large-scale problems. Several computed examples illustrate the good performance of the proposed model. We place special focus on screening studies using saturated and supersaturated experimental designs.

Keywords:

• ${\mathcal{l}}_{\mathit{p}}$-${\mathcal{l}}_{\mathit{q}}$ minimization
• Nonconvex minimization
• Saturated design
• Screening experiments
• Supersaturated design
• Variable selection

Acknowledgments

The authors thank the referees for comments that lead to an improved presentation. A.B. is a member of the GNCS-INdAM group that partially supported this work with the Young Researchers Project (Progetto Giovani Ricercatori) “Variational methods for the approximation of sparse data”. Moreover, A.B. research is partially supported by the Regione Autonoma della Sardegna research project “Algorithms and Models for Imaging Science [AMIS]” (RASSR57257, intervento finanziato con risorse FSC 2014-2020 - Patto per lo Sviluppo della Regione Sardegna). The work of O.D.C and L.R. is supported in part by NSF grant DMS-1720259.