# Precise Performance Analysis of the Box-Elastic Net under Matrix Uncertainties

## TL;DR

This paper provides a precise analysis of the Box-Elastic Net method for sparse signal recovery under matrix uncertainties, demonstrating its improved performance over standard Elastic-Net through theoretical and numerical validation.

## Contribution

The paper introduces a modified Elastic-Net with box constraints and characterizes its mean squared error and support recovery probability under matrix uncertainties.

## Key findings

- Box-Elastic Net outperforms standard Elastic-Net in support recovery.
- Theoretical predictions match numerical simulations.
- Enhanced robustness to matrix uncertainties.

## Abstract

In this letter, we consider the problem of recovering an unknown sparse signal from noisy linear measurements, using an enhanced version of the popular Elastic-Net (EN) method. We modify the EN by adding a box-constraint, and we call it the Box-Elastic Net (Box-EN). We assume independent identically distributed (iid) real Gaussian measurement matrix with additive Gaussian noise. In many practical situations, the measurement matrix is not perfectly known, and so we only have a noisy estimate of it. In this work, we precisely characterize the mean squared error and the probability of support recovery of the Box-Elastic Net in the high-dimensional asymptotic regime. Numerical simulations validate the theoretical predictions derived in the paper and also show that the boxed variant outperforms the standard EN.