Precise Error Analysis of the LASSO under Correlated Designs

TL;DR

This paper provides a detailed high-dimensional asymptotic analysis of the LASSO algorithm's performance in recovering sparse signals from noisy, correlated Gaussian measurements, using the CGMT framework.

Contribution

It offers the first precise asymptotic performance characterization of LASSO under correlated Gaussian designs with additive noise.

Findings

Derived explicit formulas for MSE, support recovery probability, EER, and cosine similarity.

Validated theoretical results with numerical simulations.

Enhanced understanding of LASSO behavior in correlated design scenarios.

Abstract

In this paper, we consider the problem of recovering a sparse signal from noisy linear measurements using the so called LASSO formulation. We assume a correlated Gaussian design matrix with additive Gaussian noise. We precisely analyze the high dimensional asymptotic performance of the LASSO under correlated design matrices using the Convex Gaussian Min-max Theorem (CGMT). We define appropriate performance measures such as the mean-square error (MSE), probability of support recovery, element error rate (EER) and cosine similarity. Numerical simulations are presented to validate the derived theoretical results.