Full Length ArticleSignal separation under coherent dictionaries and -bounded noise☆
Section snippets
Background
Recovering information, in which one cannot afford to collect or transmit a lot of measurements, has drawn a lot of interest in the last few decades [4], [9], [17], [35]. The phrase most often used in this context is compressed sensing, which deals with signals that have sparse (or approximately sparse) representations in some orthonormal basis. The core problem in compressed sensing is to recover a sparse signal (i.e. ) from , given a set of linear measurements where
General noise assumption
In certain applications, it is of interest to consider different kinds of noises. In [22], instead of the -norm constraint, Fuchs chose the -norm constraint when the noise is Laplacian or in the presence of outliers. Candès et al. in [10] mentioned that the -norm constraint should be replaced by norm to handle the quantization distortion of measurements. Therefore, it is more suitable to model the constraints corresponding to different types of noises. In order to deal with the
Notations
For , denote as . For a vector , is the support index set of . is the number of nonzero entries of . For any , denote and . denotes the vector consisting of the largest entries of in magnitude. Besides, is the unit sphere in , and .
The discrete Fourier matrix is taken as , for . For a matrix , write and as the transpose and the
Main results
First of all, we establish the motivation on introducing the modified -RNSP. In order to simplify the notations, denote Since the columns of are highly correlated, might be much smaller than . For example,take with , and choose , . Assume that . Then we can find that for any . Therefore, we can replace the left-hand side of (13) by , and (13)
Technical lemmas
Before the proof of the main theorems, we introduce some auxiliary lemmas as below.
Lemma 5.1 Let and be two tight frames with mutual coherence defined in (21). Denote and as those in (16). Then for any with , and , we have Here with , and and are defined as those in (15).
Proof The proof follows the techniques in [29]. Since
CRediT authorship contribution statement
Yu Xia: Formal analysis, Methodology, Writing - original draft, Funding acquisition. Song Li: Conceptualization, Writing - review & editing, Supervision, Funding acquisition.
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This work is supported by the key project of NSFC, China grant (11531013), NSFC, China grant (11901143, 12071426, 11971427), the key research and development project of Zhejiang Province, China grant (2015C01028), the Zhejiang Provincial Natural Science Foundation, China grant (LQ19A010008), Education Department of Zhejiang Province Science Foundation, China (Y201840082), the fundamental research funds for the central universities, China (2020XZZX002-03).