# Coherence-Based Performance Guarantee of Regularized $\ell_{1}$-Norm Minimization and Beyond

## TL;DR

This paper establishes coherence-based guarantees for regularized -norm minimization in signal recovery, extending existing conditions to noisy, block-sparse, and structured signals, with new uniform recovery conditions and error bounds.

## Contribution

It extends the sharp uniform recovery condition based on coherence to unconstrained  models for robust signal recovery under various noise models, including structured block-sparse signals.

## Key findings

- Established coherence-based performance guarantees for  minimization with noise.
- Extended recovery conditions to Dantzig Selector and block-sparse signals.
- Provided new error estimates and uniform recovery conditions for structured signals.

## Abstract

In this paper, we consider recovering the signal $\bm{x}\in\mathbb{R}^{n}$ from its few noisy measurements $\bm{b}=A\bm{x}+\bm{z}$, where $A\in\mathbb{R}^{m\times n}$ with $m\ll n$ is the measurement matrix, and $\bm{z}\in\mathbb{R}^{m}$ is the measurement noise/error. We first establish a coherence-based performance guarantee for a regularized $\ell_{1}$-norm minimization model to recover such signals $\bm{x}$ in the presence of the $\ell_{2}$-norm bounded noise, i.e., $\|\bm{z}\|_{2}\leq\epsilon$, and then extend these theoretical results to guarantee the robust recovery of the signals corrupted with the Dantzig Selector (DS) type noise, i.e., $\|A^{T}\bm{z}\|_{\infty}\leq\epsilon$, and the structured block-sparse signal recovery in the presence of the bounded noise. To the best of our knowledge, we first extend nontrivially the sharp uniform recovery condition derived by Cai, Wang and Xu (2010) for the constrained $\ell_{1}$-norm minimization model, which takes the form of \begin{align*} \mu<\frac{1}{2k-1}, \end{align*} where $\mu$ is defined as the (mutual) coherence of $A$, to two unconstrained regularized $\ell_{1}$-norm minimization models to guarantee the robust recovery of any signals (not necessary to be $k$-sparse) under the $\ell_{2}$-norm bounded noise and the DS type noise settings, respectively. Besides, a uniform recovery condition and its two resulting error estimates are also established for the first time to our knowledge, for the robust block-sparse signal recovery using a regularized mixed $\ell_{2}/\ell_{1}$-norm minimization model, and these results well complement the existing theoretical investigation on this model which focuses on the non-uniform recovery conditions and/or the robust signal recovery in presence of the random noise.