Signal separation under coherent dictionaries and [formula omitted]-bounded noise

Abstract

In this paper, we discuss the compressed data separation problem. In order to reconstruct the distinct subcomponents, which are sparse in morphologically different dictionaries D1∈Rn×d1 and D2∈Rn×d2, we present a general class of convex optimization decoder. It can deal with signal separation under the corruption of different kinds of noises, including Gaussian noise (p=2), Laplacian noise (p=1), and uniformly bounded noise (p=∞).Although the restricted isometry property adapted to frames is a commonly used tool, the measurement number is suboptimal when p>2. Furthermore, the ℓp robust nullspace property adapted to Ψ, which is constructed by D1 and D2, may fail to work on data separation problem. Here we introduce the modified ℓp robust nullspace property adapted to Ψ (abbreviated as the modified (ℓp, Ψ)-RNSP). First of all, we show the robust recovery of signals based on the modified (ℓp, Ψ)-RNSP and the mutual coherence between D1 and D2. Besides, we find that Gaussian measurements meet the modified (ℓp, Ψ)-RNSP for any 1≤p≤∞, provided with the optimal number of measurements O(slog(d∕s)), where s is the sparsity level and d=d1+d2.Furthermore, we introduce another properly constrained ℓ1-analysis optimization model, called the Split Dantzig Selector. It can recover signals which are approximately sparse in terms of different frame representations, when the measurement matrix satisfies the modified (ℓp, Ψ)-RNSP. As a special case, when considering Gaussian white noise, the recovery error by the Split Dantzig Selector is Oslogdm. It outperforms the ℓ2-constrained model, whose recovery error is O(logm), if the sparsity level is small.