A New Analysis for Support Performance with Block Generalized Orthogonal Matching Pursuit

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.*is paper focuses on support conditions of recovery of any K-sparse block signals incorporating BgOMP under the framework of restricted isometry property (RIP). *e proposed support conditions guarantee that BgOMP can achieve accurate recovery block-sparse signals within k iterations.


Introduction
e block generalized orthogonal matching pursuit (BgOMP) [1] in compressive sensing (CS) theory has been proposed for recovering block-sparse signalsx, in which the signal vector x has at most K nonzero block elements, and satisfies x 2,0 ≤ K [2][3][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][9][10].
e 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 blocksparse signals through the BgOMP algorithm under the framework of restricted isometry property (RIP). e 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.

Preliminaries
Consider the following linear model: where A ∈ R M 1 ×M 2 (M 1 ≪ M 2 ) is a sensing matrix with normalized columns, Y ∈ R M 1 ×1 is a measurement vector, and the vector v represents a random noise. Specifically, represents the submatrix of A which only includes the block index of S. Similarly, x[S] denotes the subvector of x which only includes the block index of S. A trivial summary of the BgOMP algorithm is described in Algorithm 1. For the sake of uniformity, Ω � supp(x) � i|xt[i]n ≠ q0 . Signal x is assumed to be block K-sparse if |Ω| � K, where K is a positive integer. In the k th greedy iterations, the BgOMP algorithm takes N largest correlations between the residual r k− 1 and columns of A.

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 S be a proper subset of |Ω| ≠ 0 and matrix A satisfies block RIP of order Nk + N + |Ω| − |ℓ|, for some of υ ≥ 1 and N ≥ 1, then where e proof of Lemma 1 has been included in Appendix A.
Also, the minimum of nonzero block elements of x satisfies the following condition: Proof: . First, we assume that BgOMP chooses the right block indices in k iterations, i.e., S k , which corresponds to |A T [i]r k− 1 |. Moreover, we have |S k | � kN. e notation Ω − S k indicates a set with elements belonging to Ω but not included in S k . Let Ω ∩ S k � ℓ, and we have |Ω ∩ S k | � |Ω| − |Ω − S k | ≥ k and 0 < k ≤ |ℓ| ≤ |Ω|. In addition, let W be the set of N incorrect indexes indices. Obviously, W ⊂ Ω ∪ S k and |W| � N. Suppose α k j denotes the j-th largest correlation between r k and the columns of A that belong to the set of incorrect indexes. Also, let β k i denotes the i-th largest correlation between r k and the columns of A Ω− S k . Obviously, to prove the support condition of BgOMP, it is equivalent to show that β k 1 > α k N . us, based on Lemma 1 in [7], we can define β k 1 and α k N as follows: where 〈·〉 represents inner product operator. For the convenience of presentation, we write A[i] as A i . From the above, it suffices to show that Hence, we can obtain with a least square estimation as follows: en, by step 5 of Algorithm 1 and (8), we have us, on the left-hand side of (7), we obtain a lower bound as Similarly, it is easy to obtain an upper bound with the right-hand side of (7): Integrating (7), (10), and (11), we show that Input: sensing matrix A, measurement vector Y, select N indexes in each iteration, number of blocks L, the number of iterations t, and the accuracy of recovery ε. Initialization: r 0 � Y, S 0 � ∅. Iterate the following steps until stopping criterion t > k + 1 or r t ≤ ε is met.

Mathematical Problems in Engineering
In the following, with the aim of Lemma 1, we derive a lower bound of (12): By assumption |Ω ∩ S k | � |ℓ|, we have Using (13) and (14), it is trivial to have where (a) denotes the fact that 0 < k ≤ |ℓ| from Lemma 2 in [12,13]. en, we reach the following equivalence that [14]; and (c) follows from With (15) and (16), we reach that We rearrange (18) as follows: Because of the constant δ Nk+N+|Ω|− |ℓ| ∈ [0, 1), the RIC bound of the BgOMP algorithm can be represented as us, we complete the proof. Proof. For some N ≥ 1 and 0 < k ≤ |Ω|, we have Hence, let δ Nk+N+|Ω|− k � δ and eorem 2 is equivalent to prove the following inequivalence: e BgOMP algorithm fails to recover x. e preparatory work: for any λ � Nk + N + |Ω| − k − 1 ≥ 1, λ is a positive integer. Let us define a matrix function as where E (dλ)×d � (I d , · · · I d ) ∈ R (dλ)×d , with I d being the d × d identity matrix. Meanwhile, we define a constant s and matrix C(d) as In the following, we divide the proof into two steps: Step 1. We plan to prove that C(d) is a symmetric positive matrix. erefore, C(d) can be transformed into matrix A T (d)A(d) by Doolittle decomposition, and then, we can ensure that A(d) satisfies the block RIP of order Nk + N + |Ω| − k with δ Nk+N+|Ω|− k � δ. 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 x � (1 T d , 1 T d · · · 1 T d ) T ∈ R dλ with all of its entries being 1. Without loss of generality,x can be extended to Similarly, where 1 ≤ i ≤ λ. By (25) and (26), sinces ≥ 0, we notice that erefore, in the first iteration, BgOMP will choose the position of 0 T d in which x(d) is selected as the indexes; this goes against our hypothesis. By (25) and (26), to guarantee successful recovery, xmust satisfy s < 0; that is, δ < 1/ �������������� � Nk + N + |Ω| − k ≤ 1/ ���������� (|Ω|/N) + 1. eorem 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. erefore, by suitable exploitation of RIP, a successful signal recovery scheme for BgOMP was obtained.

Conclusion
In this paper, the support conditions of recovery blocksparse signals by using BgOMP are proposed; the bound of RIP constant must satisfy 1/ ���������� (|Ω|/N) + 1, 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 v ≥ 1, we denote We have each j ∈ W, in the second step, to prove where (a) follows from Lemma 1 in [7]

B
It is quite straightforward that B(d) is a symmetric matrix. To calculate the range of eigenvalue, let u i � v i � x[i] 2 for 1 ≤ i ≤ λ and u λ+1 � − v λ+1 � x[λ + 1] 2 ; then, Similarly, we have Integrating (24), (B.1), and (B.2), for any x, It is not hard to see that x T C(d)x > 0 from (B.4). us, we prove C(d) is the symmetric positive matrix. Also, C(d) (B.6) From Definition 2 in [7], we can ensure that A(d) satisfies the block RIP of order Nk + N + |Ω| − k with δ Nk+N+|Ω|− k � δ.
is completes the proof.

C
is mainly researches the satisfactory performance of support reconstructs of x by using the BgOMP. To be specific, we compare eorem 1 with BOMP sufficient condition ( eorem 3), which is expressed as follows.
Theorem 3 (see [7,9]). Suppose that in (1), v 2 ≤ ε. en, the BOMP algorithm can accurately reconstruct the support Ω and when the number of iterations is k, the iteration stop condition is satisfied: e matrix A satisfies the block RIP of order |Ω| + 1, if there is a smallest constant δ k+1 ∈ [0, 1) which is defined as the restricted isometry constant (RIC). erefore, the RIC bound of block-sparse vector x of BOMP is Data Availability e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest. 6 Mathematical Problems in Engineering