Network exploration via the adaptive LASSO and SCAD penalties

TL;DR

This paper introduces adaptive LASSO and SCAD penalties for network exploration via graphical models, addressing optimization challenges and bias in precision matrix estimation, with applications to real data and theoretical validation.

Contribution

It proposes a novel approach using nonconcave and adaptive LASSO penalties with local linear approximation for improved precision matrix estimation in graphical models.

Findings

Effective network exploration with reduced bias.

Successful application to real datasets.

Theoretical justification through simulations and asymptotic analysis.

Abstract

Graphical models are frequently used to explore networks, such as genetic networks, among a set of variables. This is usually carried out via exploring the sparsity of the precision matrix of the variables under consideration. Penalized likelihood methods are often used in such explorations. Yet, positive-definiteness constraints of precision matrices make the optimization problem challenging. We introduce nonconcave penalties and the adaptive LASSO penalty to attenuate the bias problem in the network estimation. Through the local linear approximation to the nonconcave penalty functions, the problem of precision matrix estimation is recast as a sequence of penalized likelihood problems with a weighted $L_1$ penalty and solved using the efficient algorithm of Friedman et al. [Biostatistics 9 (2008) 432--441]. Our estimation schemes are applied to two real datasets. Simulation experiments and asymptotic theory are used to justify our proposed methods.