I Introduction
Compressed/compressive Sensing (CS), see, e.g., [1, 2, 3], has captured a lot of attention of the researchers in a wide range of fields over the past decade. In CS, one gets the observations of signal via the following model
(1) 
where is called the measurement matrix and denotes the additive noise that satisfies a certain constraint. One of the key goals of CS is to effectively recover the original signal based on and . It has been shown that, if is sparse with and satisfies certain conditions related to , see, e.g., [4, 5, 6, 7, 8], then one can achieve this goal by solving an minimizing problem, i.e.,
(2) 
where represents for the noise level, and we take if there is no noise, i.e., .
The above minimization approach has been demonstrated to be effective in robust signal recovery. However, it does not incorporate any prior information on signal support since the norm treats the entries of variable equally. In fact in many practical applications such as the timeseries signal processing, see, e.g., [9, 10, 11], it is often possible to estimate a part of the signal support information. It thus becomes very necessary and important to use such prior information to further enhance the recovery performance of (2). This consideration directly leads to the following weighted minimization problem
(3) 
where denote the weights. For simplicity, in this paper we only consider a binary choice of , i.e.,
where is a given set, which models the Partially Known Support Information (PKSI) of . This problem has been well investigated in the past few years, see, e.g., [12, 13, 14, 15, 16, 17, 18, 19]. It was proved by Friedlander, et al. in [12] that if includes half of the accurate support of at least, then (3) will perform robustly under much weaker conditions than the analogous ones for (2). In [15], Flinth studied the optimal choice for general weights. Recently, Chen, et al. in [18] and [19] obtained some much tighter conditions for (3), and these conditions were proved to be sharp when the desired signal is exactly sparse and is also measured without noise.
In this paper, we consider the robust recovery of the signals with PKSI via the weighted Basis Pursuit DeNoising (BPDN)
(4) 
where is a positive parameter. Obviously, (4) will be reduced to the widely known BPDN if one sets (i.e., no support information is available). Although there exists a large amount of research on the BPDN, see, e.g., [20, 21, 22, 23, 24, 25, 26, 27], the theoretical analysis of (4) for sparse recovery is relatively less studied. We note that Lian, et al. recently studied (4) from both theoretical and experimental aspects in [28], where they called it weighted LASSO. However, their obtained results are established on the stochastic strategy, and they are totally different from ours that are established in a deterministic manner.
The main contribution of this paper is that a series of (tight) sufficient conditions as well as their resultant error estimates are established for (4) with the help of the Restricted Isometry Property (RIP) [1], which to some degree well complement the recent theoretical analysis of the weighted BPDN (see, [28]) that is based on the stochastic strategy.
Ii Notations and Preliminaries
In this section, we first introduce some basic notations. For any given index set , we denote
as a vector whose entries
for and 0 otherwise, and also denote the best term approximate of any signal asDefinition 1.
A matrix is said to obey the RIP of order , if there exists a constant such
(5) 
for every sparse signal . The smallest positive that satisfies (5) is denoted by ^{1}^{1}1When is not an integer, we define as . and is known as the Restricted Isometry Constant (RIC).
We also need the following two lemmas.
Lemma 1.
Proof:
Since is the optimal solution of (4), we have
which is equivalent to
(9) 
As to the lefthand side of (9), we have
(10) 
As to the righthand side of (9), we know from [12] that
(11) 
where . Since and , then and , and thus clearly
where
(12) 
This directly turns (11) to be the following inequality
(13) 
Therefore, combing (10) and (13) leads to the desired (7), and (8) follows trivially from (7). ∎
Lemma 2.
For any if satisfies the RIP of order with RIC and , then for any vector and any subset with , it holds that
(14) 
where
Remark 1.
Iii Main Results
With preparations above, we now give the main results.
Theorem 1.
Remark 2 (Recovery Condition).
The established condition (19) coincides with the one obtained recently by Chen, et al. in [18], which has been proved to be sharp for the exactly sparse signal recovery under noisefree measurements. However, their goal was to recover the signal with PKSI using the constrained model. On the other hand, our condition (19) in fact is not a simple extension of the one in [18], but is obtained in a totally different way. We refer the interested readers to [18] for more detailed discussion on (19) and its potential corollaries.
Remark 3 (Error Estimate).
It seems that the obtained error estimate (20) shows a bit complicated since it integrates and together. In what follows, we provide three special cases of (20) by selecting some simple but meaningful ’s and/or ’s.
Case 1): Suppose that , then by using (15), (16) and (17) we can deduce directly from (20) that
and
where
This directly yields
This new error estimate also coincides with the ones in [24, 25, 26, 27] in form, which are induced for the unconstrained models. However, their results do not take PKSI into consideration.
Case 2): Suppose that . Similar to the above analysis in Case 1, we can also obtain that
where
This result coincides with the ones induced for the traditional constrained models in form, see, e.g., [5, 12, 13], which to some extent indicates theoretically that the unconstrained model (4) and the constrained model (3) are equivalence in robust recovery of any (sparse) signals with PKSI.
Case 3): Suppose that , i.e., . In such case, it is also easy to deduce from (20) that
where
According to the above error estimate, it seems impossible to exactly recover any sparse signals through (4) in the absence of noise. However, if one sets the parameter to be a sufficient small positive number, the error between and will tend to depend only on the available PKSI of the original signal itself. On the other hand, from the viewpoint of nonuniform recovery [3], it has been shown that under certain conditions, one can successfully recover some (specific) sparse signals from the BPDN, see, e.g., [20]. This may brings possibility for (4) to realize the exact recovery of some sparse signals with PKSI when certain conditions are satisfied. More discussion on nonuniform recovery is beyond the scope of this paper, and we refer the interested readers to [3] and [29] for details.
Remark 4.
Due to the limited space, we can not discuss more on the obtained results in this paper. We refer the interested readers to the supplementary material for more discussion on the weight choice and its resultant performance analysis.
Proof:
We first denote , , , and . Then we have
(21) 
and also know from Lemma 2 (with ) that
(22) 
Besides, combing (8), (21) and (22) directly yields
(23) 
where we used the condition (19) and thus
(24) 
for the last inequality. Similarly, we can also deduce from (8), (21) and (22) that
On the other hand, let denote the index set of the largest entries of in magnitude. Then we can know from Lemma 2 (with ) and [24, inequality (2.3)] that
(25)  
(26) 
where . Then using (III) and (25), we have
(27) 
where we used in the first inequality.
Iv Conclusion
This paper aims to provide a deterministic (nonstochastic) analysis for the sparse recovery of signals with partially known support information from the weighted BPDN. Equipped with the powerful RIC notation, we established a series of sufficient conditions and their resultant error estimates. These theoretical results, to some degree, are well complementary for the recent ones of the weighted BPDN established in a stochastic manner.
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