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Coherence-Based Performance Guarantee of Regularized ℓ_1-Norm Minimization and Beyond

by   Wendong Wang, et al.
Southwest University

In this paper, we consider recovering the signal x∈R^n from its few noisy measurements b=Ax+z, where A∈R^m× n with m≪ n is the measurement matrix, and z∈R^m is the measurement noise/error. We first establish a coherence-based performance guarantee for a regularized ℓ_1-norm minimization model to recover such signals x in the presence of the ℓ_2-norm bounded noise, i.e., z_2≤ϵ, 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^Tz_∞≤ϵ, 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 ℓ_1-norm minimization model, which takes the form of μ<1/2k-1, where μ is defined as the (mutual) coherence of A, to two unconstrained regularized ℓ_1-norm minimization models to guarantee the robust recovery of any signals (not necessary to be k-sparse) under the ℓ_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 ℓ_2/ℓ_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.


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