Abstract
For recovering block-sparse signals with unknown block structures using compressive sensing, a block orthogonal matching pursuit- (BOMP-) like block generalized orthogonal matching pursuit (BgOMP) algorithm has been proposed recently. This paper focuses on support conditions of recovery of any -sparse block signals incorporating BgOMP under the framework of restricted isometry property (RIP). The proposed support conditions guarantee that BgOMP can achieve accurate recovery block-sparse signals within iterations.
1. Introduction
The block generalized orthogonal matching pursuit (BgOMP) [1] in compressive sensing (CS) theory has been proposed for recovering block-sparse signals, in which the signal vector has at most nonzero block elements, and satisfies [2–4]. In many practical applications, the signals usually exist in the form of block structure such as pattern recognition [5], aeronautical signal processing [6], and deoxyribonucleic acid (DNA) microarrays [7]. In other words, the nonzero coefficients of sparse signals appear in a few blocks inconsistently [8–10].
The BgOMP algorithm is mainly designated to restore nonzero indexes in the block, containing significant information, rather than restoring each entry in addition to having applications in block-objects detection, such as computer vision and pattern recognition [11]. Based on the previous literature, the main target of this paper is to explore support conditions for the high-accurate recovery of block-sparse signals through the BgOMP algorithm under the framework of restricted isometry property (RIP). The results can prove the bound of BgOMP signal recovery, which provides strong mathematical support for CS algorithm selection in sparse coding, pattern recognition, image reconstruction, and other fields.
2. Preliminaries
Consider the following linear model:where () is a sensing matrix with normalized columns, is a measurement vector, and the vector v represents a random noise. Specifically, is a concatenation of blocks ,
For any proper subset , we define that represents the submatrix of which only includes the block index of . Similarly, denotes the subvector of which only includes the block index of . A trivial summary of the BgOMP algorithm is described in Algorithm 1. For the sake of uniformity, . Signal is assumed to be block K-sparse if , where K is a positive integer. In the greedy iterations, the BgOMP algorithm takes largest correlations between the residual and columns of .
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3. Analysis of BgOMP
Signals with block property are frequently encountered in practical applications. Given that because of the above background, the purpose of this research is to explore a sufficient condition for the high-accurate recovery of the block-sparse signals by using the BgOMP algorithm in the framework of RIP.
Lemma 1. If be a proper subset of and matrix A satisfies block RIP of order , for some of and , thenwhere .
The proof of Lemma 1 has been included in Appendix A.
Theorem 1. Given any , matrix satisfies block RIP with the order . It implies that the successful recovery can be guaranteed, and the support condition of the BgOMP algorithm is satisfied. The following bound of RIP constant can be expressed asAlso, the minimum of nonzero block elements of satisfies the following condition:
Proof: . First, we assume that BgOMP chooses the right block indices in iterations, i.e., , which corresponds to . Moreover, we have . The notation indicates a set with elements belonging to but not included in . Let , and we have and . In addition, let W be the set of incorrect indexes indices. Obviously, and . Suppose denotes the j-th largest correlation between and the columns of that belong to the set of incorrect indexes. Also, let denotes the i-th largest correlation between and the columns of . Obviously, to prove the support condition of BgOMP, it is equivalent to show that . Thus, based on Lemma 1 in [7], we can define and as follows:where represents inner product operator. For the convenience of presentation, we write as . From the above, it suffices to show thatHence, we can obtain with a least square estimation as follows:Then, by step 5 of Algorithm 1 and (8), we haveWhere (a) is from the definition of and (b) follows from the fact that . Thus, on the left-hand side of (7), we obtain a lower bound asSimilarly, it is easy to obtain an upper bound with the right-hand side of (7):Integrating (7), (10), and (11), we show thatIn the following, with the aim of Lemma 1, we derive a lower bound of (12):By assumption , we haveUsing (13) and (14), it is trivial to havewhere (a) denotes the fact that from Lemma 2 in [12, 13]. Then, we reach the following equivalence thatwhere . (a) is because is a block vector and the property of norm ; (b) is reduced to (3) in [14]; and (c) follows fromWith (15) and (16), we reach thatWe rearrange (18) as follows:Because of the constant , the RIC bound of the BgOMP algorithm can be represented asThus, we complete the proof.
Theorem 2. Let a matrix satisfy the block RIP with the order and the RIC bound of BgOMP meet the condition in (20). Otherwise, the BgOMP algorithm fails to recover the block signals .
Proof. For some and , we haveHence, let and Theorem 2 is equivalent to prove the following inequivalence:The BgOMP algorithm fails to recover .
The preparatory work: for any , is a positive integer. Let us define a matrix function aswhere , with being the identity matrix. Meanwhile, we define a constant and matrix asIn the following, we divide the proof into two steps: Step 1. We plan to prove that is a symmetric positive matrix. Therefore, can be transformed into matrix by Doolittle decomposition, and then, we can ensure that satisfies the block RIP of order with . In order not to steal the attention of the main proof, we prove Step 1 in Appendix B. Step 2. Next, we show that BgOMP fails to recover with all of its entries being 1. Without loss of generality, can be extended to . Integrating (23), and .Similarly,where . By (25) and (26), since, we notice thatTherefore, in the first iteration, BgOMP will choose the position of in which is selected as the indexes; this goes against our hypothesis. By (25) and (26), to guarantee successful recovery, must satisfy ; that is, .
Theorem 3 can give sufficient condition of BOMP under the framework of RIP. Interestingly, by the established result, one can easily see that the support condition of BOMP is as restrictive as BgOMP when N equals 1; at this point, BgOMP degenerates into BOMP. But as N increases, the support condition given by (4) slacks gradually. Figure 1 compares the RIC of two algorithms. We observe that the RIC of BgOMP is significantly less restrictive than BOMP. Therefore, by suitable exploitation of RIP, a successful signal recovery scheme for BgOMP was obtained.
4. Conclusion
In this paper, the support conditions of recovery block-sparse signals by using BgOMP are proposed; the bound of RIP constant must satisfy , otherwise it may cause recovery failure. And, we have presented a sufficient condition, which is weaker than BOMP, for the exact support recovery of block K-sparse signals with K iterations of BgOMP.
Appendix
A
Proof. To prove Lemma 1, it is equivalent to the following two steps. In the first step, for some of , we denoteWe have each , in the second step, to proveBy (A.1), it suffices to showwhere (a) follows from Lemma 1 in [7]; (b) is according to the fact that ; (c) is from Cauchy–Schwarz inequality; and (d) follows from the definition of in Lemma 1. Next, we shall prove the second step, might as well set , it is not difficult to see that satisfaction . Let , and by a simple calculation, we obtainTo simplify the expression, let us defineThe above equation satisfies and . Moreover, we defineThen, it is not hard to see thatandFurthermore, we havewhere (a) follows from Lemma 3 in [14] and (b) is because . On the one hand,where steps (a) and (b) are derived from equations (36) and (37), respectively; and (c) is because of (A.4). On the other hand, we havewhere (a) follows from trigonometric inequality; (b) is from (A.11); and (c) is because of (A.4). Integrating (A.12) and (A.13), we can obtainSince (A.3) and (A.8), it is easy to see thatFor given and , we havewhich is Lemma 1. Hence, we complete the proof.
B
It is quite straightforward that is a symmetric matrix. To calculate the range of eigenvalue, let for and ; then,
Similarly, we have
Integrating (24), (B.1), and (B.2), for any ,
It is not hard to see that from (B.4). Thus, we prove is the symmetric positive matrix. Also, can transform matrix by Doolittle decomposition; that is, . By (24), we will have
Integrating (22) and (B.3), by some simple calculations, we can show that
From Definition 2 in [7], we can ensure that satisfies the block RIP of order with .
This completes the proof.
C
This mainly researches the satisfactory performance of support reconstructs of by using the BgOMP. To be specific, we compare Theorem 1 with BOMP sufficient condition (Theorem 3), which is expressed as follows.
Theorem 3. (see [7, 9]). Suppose that in (1), . Then, the BOMP algorithm can accurately reconstruct the support and when the number of iterations is , the iteration stop condition is satisfied:
The matrix A satisfies the block RIP of order , if there is a smallest constant which is defined as the restricted isometry constant (RIC). Therefore, the RIC bound of block-sparse vector of BOMP is
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors thank 2-nd Research Institute, Civil Aviation Administration of China. Furthermore, this work was supported by the Science and Technology Bureau of Sichuan Province, under Grant no. 2020YFG0414.