1 Introduction
Over the past few years, compressed sensing [5, 8, 9, 19] has attracted considerable attention in various fields including signal processing, optics, and information theory. Basically, the goal of compressed sensing is to recover a sparse signal of interest from a small number of (noisy) linear measurements via the minimization with [4, 21, 25, 31]. It is known that the restricted isometry property (RIP) and the null space property (NSP) are two typical conditions on the measurement matrix such that the exact/stable recovery of sparse signals can be guaranteed [1, 7].
The minimization has been most widely used for sparse signal recovery in the compressed sensing literature, because of its convexity and easy computation. However, such traditional minimization does not incorporate any prior support information of the original signal, which might help improve overall performance. In many practical applications, the support information of a signal is usually available as a prior. By exploiting such partial support information, there has been a large amount of research on the recovery of a sparse signal via the weighted minimization [3, 6, 11, 14, 17, 22, 26]. Moreover, based on the weighted minimization, a weighted sparse recovery problem has been intensively studied in [2, 10, 15, 23] as well, in which a weight function is considered into the sparsity structure. Specifically, given a weight function with , a signal is called a weighted sparse signal, if
(1) 
Then, the recovery of a weighted sparse signal by using the weighted minimization is referred to as the weighted sparse recovery problem. For more details, we refer the reader to [2, 10, 15, 23].
Phase retrieval is a fundamental problem of recovering a signal only from the magnitude of its linear measurements. It can be found in several areas, such as radar signal processing [18], quantum mechanics [20], and optics [24, 28]. In these areas of application, the original signal to be recovered is often sparse. Hence, it is very natural to combine phase retrieval with compressed sensing, which is called phaseless compressed sensing [12, 16, 27, 30]. The goal of phaseless compressed sensing is to reconstruct an unknown sparse signal from the magnitude of its noisy measurements , where is the measurement matrix,
is the noise vector, and
denotes the elementwise absolute value. If the measurement matrix satisfies the strong RIP (SRIP) [13, 27] or the NSP [29], can be stably recovered by the following minimization up to a global phase:(2) 
You et al [30] considered the phaseless compressed sensing via its nonconvex relaxation minimization with , and obtained a constant such that for any , every optimal solution to the minimization solves the considered problem too.
Obviously, the above minimization (2) does not take into account the support information of the signal. Therefore, Zhou et al [33] presented that the SRIP and the weighted NSP (WNSP) are two conditions for the success of sparse signal recovery from phaseless compressed sensing measurements via the weighted minimization when partial support information is available as a prior. Zhang et al [32] used another different way to study the problem of phaseless compressed sensing using partial support information. They proposed two concepts of the partial NSP (PNSP) and the partial SRIP (PSRIP); and proved that the PNSP and the PSRIP are two exact reconstruction conditions on measurements for the problem of partially sparse phase retrieval. To the best of our knowledge, there is no study on recovery of a weighted sparse signal from the phaseless measurements via the weighted minimization. Therefore, to fill this gap, our main contribution of this paper is to build up the theoretical framework for recovery of a weighted sparse signal from the magnitude of its measurements. To be specific, the mathematical model of the weighted minimization is represented as follows:
(3) 
where the weighted norm is given by
and is the weight function with . In this paper, we first give a sufficient and necessary condition, i.e., the WNSP, which can guarantee the unique recovery of a weighted sparse signal up to a global phase. Moreover, for the noisy setting, we propose a new concept, called the strong weighted RIP (SWRIP), and it is proved to be another complementary sufficient condition for stable recovery.
The rest of this paper is organized as follows. In Section 2, we introduce the WNSP and show that it is a sufficient and necessary condition for exactly reconstructing a weighted sparse signal in phaseless compressed sensing. In Section 3, we propose a new concept, i.e., SWRIP, and prove that stable recovery of a weighted sparse signal can be guaranteed if the measurement matrix satisfies the SWRIP. Finally, we conclude this paper in Section 4.
Notations: Let , and be the measurement matrix. The null space of A is given by
For a vector , its entries are denoted as . The complement of a set is defined by . is denoted as the subvector of , whose entries only with indices in are kept. For a weight , the weighted cardinality of a set is denoted as . The best weighted term approximation error is defined as
where
2 The Weighted Null Space Property
In this section, for any weighted sparse signal , we consider the weighted minimization (3) without noise:
(4) 
Similar to phaseless compressed sensing for a sparse signal, we explore the WNSP condition for the success of the weighted minimization for weighted sparse phase retrievable.
2.1 The Real Case
We first consider the real case of the problem, i.e., the signal of interest and its measurement matrix are in the real number field. We show that the WNSP is a sufficient and necessary condition for unique recovery of a weighted sparse signal from its phaseless measurements up to a global phase.
Theorem 2.1.
Given a measurement matrix , the following two statements are equivalent:

For any , we have

For all with , it holds
for all nonzero and satisfying .
Proof.
: We suppose that the statement does not hold. Then, there exists a subset with , nonzero and such that , and
Let . Then, , and
(5) 
Let be the rows of the measurement matrix . From the definitions of and , we know
and
Since , we have
Similarly, we get
Thus, we obtain
It follows from (5) that is a solution to (4), which contradicts .
: We suppose that does not hold. Then, there exists a solution to (4), i.e.,
(6) 
and
(7) 
Let be the rows of the measurement matrix . By (6), we know that there exists a subset satisfying
(8) 
and
(9) 
Let , and . Note that , then combining (8) and (9), we have , , and . From , we get
i.e.,
which contradicts (7). This completes the proof. ∎
2.2 The Complex Case
Next, we consider the same problem in for the complex case. We say that is a partition of , if
Let . The next theorem is an extension of Theorem 2.1.
Theorem 2.2.
Given a measurement matrix , the following two statements are equivalent:

For any , we have
(10) 
Assume that is any partition of , and that with
(11) for some pairwise distinct . Then, we have
(12) for all with .
Proof.
: Suppose that the statement does not hold, then there exists a solution to (4), which satisfies
(13) 
and
(14) 
Let be the rows of the measurement matrix . By (14), we have
(15) 
where . We define an equivalence relation on by , that is , when . Then, the equivalence relation leads to a partition of . Let . Obviously, are distinct and belong to . For any , we get
Let
(16) 
Then, we obtain , and
Hence, from statement (b), we know that
(17) 
Substituting (16) into (17), we obtain
i.e.,
which is a contradiction with (13). Hence, the statement (a) holds.
: Suppose that the statement does not hold, then there exists a partition of , , and some pairwise distinct satisfying (11), and
(18) 
for some distinct . Let
(19) 
and
(20) 
Then, we get
and
(21) 
Let be the rows of the measurement matrix . Since , we have
From the definitions of and (see (19) and (20)), we get
(22) 
For any , without loss of generality, we assume that . Thus, . By (11), we have
(23) 
where are distinct integers. Let
Thus,
and
Hence, and can be rewritten as
and
respectively. Since , we have
Using a similar argument, we obtain
It follows from (21) that is a solution to (3), which is a contradiction with the statement . This completes the proof. ∎
By Theorems 2.1 and 2.2, we know that if the measurement matrix satisfies the WNSP, then an unknown weighted sparse signal can be exactly recovered by solving the weighted minimization model (4) up to a global phase. However, it is very hard to check whether the measurement matrix satisfies the condition (i.e., statement ) in Theorems 2.1 and 2.2 or not. To this end, we present another property, called the SWRIP, in the following section, as an alternative way to guarantee the uniqueness of the weighted minimization model (4).
3 The Strong Weighted Restricted Isometry Property
In phaseless compressed sensing, Gao et al [13] presented that if the measurement matrix satisfies the SRIP, then the model (2) provides a stable solution. In this section, we propose to generalize the SRIP to the weighted sparsity setting, and investigate the conditions under which the weighted minimization model (4) guarantees stable recovery of a weighted sparse signal. In this section, we only focus on studying signals and matrices that are in the real number field.
Before stating the main results, we recall the definition of the weighted RIP (WRIP), and the conditions for stable recovery in the weighted sparse setting for traditional compressed sensing, where the optimization model considered is
(24) 
Definition 3.1 (Wrip).
[23, Definition 1.3] Given a weight , a matrix is said to satisfy the WRIP of order with constant , if
(25) 
holds for all weighted sparse vectors .
Proposition 3.2.
[15, Theorem 3.4] Suppose that satisfies the WRIP of order with constant
(26) 
for . Let , with , and be the solution to . Then, we have
(27) 
where
(28) 
In this paper, we derive a sufficient condition for stable recovery of a weighted sparse signal from its phaseless measurements. For any weighted sparse signal , we turn back to consider the weighted minimization in phaseless compressed sensing:
(29) 
In the following, we first propose the notion of the SWRIP, which is a combination of the WRIP and the SRIP.
Definition 3.3 (Swrip).
For a weight , a matrix is said to satisfy the SWRIP of order with bounds , if
(30) 
holds for all weighted sparse .
Based on the SWRIP, we present a reconstruction error estimation via the weighted
minimization (29) as follows.Theorem 3.4.
Proof.
For any solution to , it holds
(32) 
and
(33) 
Let be the rows of the measurement matrix . We divide the index set into two subsets:
and
Then, we know that either or . First, we assume that . It follows from (33) that
(34) 
By (34), we know that
(35) 
Combining (32) and (35), we get
(36) 
Since satisfies the SWRIP of order , then we know that satisfies the WRIP of order with
(37) 
Combining (36), (37) and Proposition 3.2, we get
where and are defined the same as in Proposition 3.2. Similarly, for the case , we obtain
This completes the proof. ∎
4 Conclusion
In this paper, we study the stable recovery of a weighted sparse signal from the magnitude of its measurements via the weighted minimization. First, we prove that the WNSP is a sufficient and necessary condition on the measurement matrix for exactly reconstructing a weighted sparse signal in phaseless compressed sensing. Moreover, we propose a new concept, called the SWRIP, and it is proved to be a sufficient condition for weighted sparse phase retrievable. In the future, we will focus on investigating some numerical algorithms for stable recovery of weighted sparse signals from its phaseless measurements.
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