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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 acoherence-based performance guarantee for a regularized $\ell_{1}$-normminimization 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 thenextend these theoretical results to guarantee the robust recovery of thesignals corrupted with the Dantzig Selector (DS) type noise, i.e.,$\|A^{T}\bm{z}\|_{\infty}\leq\epsilon$, and the structured block-sparse signalrecovery in the presence of the bounded noise. To the best of our knowledge, wefirst 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 unconstrainedregularized $\ell_{1}$-norm minimization models to guarantee the robustrecovery 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 arealso established for the first time to our knowledge, for the robustblock-sparse signal recovery using a regularized mixed $\ell_{2}/\ell_{1}$-normminimization model, and these results well complement the existing theoreticalinvestigation on this model which focuses on the non-uniform recoveryconditions and/or the robust signal recovery in presence of the random noise.

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