On the generic uniform uniqueness of the LASSO estimator

TL;DR

This paper establishes a generic sufficient condition based solely on the design matrix that guarantees the uniqueness of the LASSO estimator in linear regression, aiding understanding of its solution path.

Contribution

It introduces a new, design matrix-dependent condition ensuring LASSO uniqueness, applicable to most experimental models, advancing theoretical understanding.

Findings

Provides a generic condition for LASSO uniqueness

Condition holds with probability one for most models

Enhances understanding of LASSO solution path

Abstract

The LASSO is a variable subset selection procedure in statistical linear regression based on $\ell_1$ penalization of the least-squares operator. Uniqueness of the LASSO is an important issue, especially for the study of the LASSO path. The goal of the present paper is to provide a generic sufficient condition on the design matrix for the LASSO minimizer to be unique. Unlike previous works on the question of uniqueness, our condition only depends on the design matrix. Our study is based on a general position condition on the design matrix which holds with probability one for most experimental models.