Precise Performance Analysis of the LASSO under Matrix Uncertainties

TL;DR

This paper provides a precise analysis of the LASSO's performance in recovering sparse signals when the measurement matrix is noisy, characterizing mean squared error and support recovery probability in high-dimensional settings.

Contribution

It offers the first detailed theoretical characterization of LASSO performance under matrix uncertainties in high-dimensional regimes.

Findings

Accurate predictions of mean squared error in noisy matrix scenarios

Explicit support recovery probability formulas

Validation through numerical simulations

Abstract

In this paper, we consider the problem of recovering an unknown sparse signal $\xv_0 \in \mathbb{R}^n$ from noisy linear measurements $\yv = \Hm \xv_0+ \zv \in \mathbb{R}^m$. A popular approach is to solve the $\ell_1$-norm regularized least squares problem which is known as the LASSO. In many practical situations, the measurement matrix $\Hm$ is not perfectely known and we only have a noisy version of it. We assume that the entries of the measurement matrix $\Hm$ and of the noise vector $\zv$ are iid Gaussian with zero mean and variances $1/n$ and $\sigma_{\zv}^2$. In this work, an imperfect measurement matrix is considered under which we precisely characterize the limiting behavior of the mean squared error and the probability of support recovery of the LASSO. The analysis is performed when the problem dimensions grow simultaneously to infinity at fixed rates. Numerical simulations validate the theoretical predictions derived in this paper.