Lasso and elastic nets by orthants

TL;DR

This paper introduces a novel geometric approach to compute the lasso and elastic net paths using orthant-based moves, providing explicit formulas and interpretability, with practical R implementation.

Contribution

It presents a new orthant-based algorithm for the lasso and elastic net paths, offering explicit formulas and improved interpretability over existing methods.

Findings

The method computes the lasso path with orthant moves.

Explicit formulas for elastic net path are derived.

The approach is validated with computational examples and R code.

Abstract

We propose a new method for computing the lasso path, using the fact that the Manhattan norm of the coefficient vector is linear over every orthant of the parameter space. We use simple calculus and present an algorithm in which the lasso path is series of orthant moves. Our proposal gives the same results as standard literature, with the advantage of neat interpretation of results and explicit lasso formul{\ae}. We extend this proposal to elastic nets and obtain explicit, exact formul{\ae} for the elastic net path, and with a simple change, our lasso algorithm can be used for elastic nets. We present computational examples and provide simple R prototype code.