Recovery of signals by a weighted \ell_2/\ell_1 minimization under arbitrary prior support information
aa r X i v : . [ c s . I T ] J un Recovery of signals by a weighted ℓ /ℓ minimization underarbitrary prior support information Wengu Chen, Huanmin Ge ∗†‡
September 8, 2018
Abstract
In this paper, we introduce a weighted ℓ /ℓ minimization to recover block sparsesignals with arbitrary prior support information. When partial prior support informa-tion is available, a sufficient condition based on the high order block RIP is derivedto guarantee stable and robust recovery of block sparse signals via the weighted ℓ /ℓ minimization. We then show if the accuracy of arbitrary prior block support estimateis at least 50%, the sufficient recovery condition by the weighted ℓ /ℓ minimization isweaker than that by the ℓ /ℓ minimization, and the weighted ℓ /ℓ minimization pro-vides better upper bounds on the recovery error in terms of the measurement noise andthe compressibility of the signal. Moreover, we illustrate the advantages of the weighted ℓ /ℓ minimization approach in the recovery performance of block sparse signals underuniform and non-uniform prior information by extensive numerical experiments. Thesignificance of the results lies in the facts that making explicit use of block sparsity andpartial support information of block sparse signals can achieve better recovery perfor-mance than handling the signals as being in the conventional sense, thereby ignoringthe additional structure and prior support information in the problem. Keywords: Block restricted isometry property, block sparse, compressed sensing, weighted ℓ /ℓ minimization. ∗ W. Chen is with Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China,e-mail: [email protected]. † H. Ge is with Graduate School, China Academy of Engineering Physics, Beijing, 100088, China, e-mail:[email protected]. ‡ This work was supported by the NSF of China (Nos.11271050, 11371183) . Introduction
Compressed sensing, a new type of sampling theory, aims at recovering an unknown highdimensional sparse signal x ∈ R N , through the following linear measurement y = Ax + z (1.1)where A ∈ R n × N ( n ≪ N ) is a sensing matrix, y ∈ R n is a vector of measurementsand z ∈ R n is the measurement error. In last decade, compressed sensing has been a fastgrowing field of research. A multitude of different recovery algorithms including the ℓ minimization [6]-[12], [37], greedy algorithm [16, 21, 22, 27, 43, 46], [38]-[40] and iterativethreshold algorithm [3, 4, 23, 24, 25, 33] have been used to recover the sparse signal x undera variety of different conditions on the sensing matrix A .In this paper, the unknown sparse signal x of the model (1.1) has additional structure,whose nonzero coefficients occur in blocks (or clusters). Such signal is called block sparsesignal [28], [29]. The recovery of block sparse signals naturally arise in practical examplessuch as equalization of sparse communication channels [19], DNA microarrays [42], multiplemeasurement vector (MMV) problem [15], [20], [36]. A block signal x ∈ R N over I = { d , d , . . . , d M } is a concatenation of M blocks of length d i ( i = 1 , , · · · , M ), that is, x = ( x , . . . , x d | {z } x ′ [1] , x d +1 , . . . , x d + d | {z } x ′ [2] , . . . , x N − d M +1 , . . . , x N | {z } x ′ [ M ] ) ′ (1.2)where x [ i ] denotes the i th block of x and N = M P i =1 d i . Let the block index set [ M ] = { , , . . . , M } . The block signal x is referred to block k -sparse if x [ i ] has nonzero ℓ normfor at most k indices i ∈ [ M ], i.e., M P i =1 I ( k x [ i ] k > k , where I ( · ) is an indicator function.Denote k x k , = P Mi =1 I ( k x [ i ] k >
0) and T = b-supp( x ) = { i ∈ [ M ] : k x [ i ] k > } , then ablock k -sparse signal x satisfies k x k , k and | T | k . If d i = 1 for all i ∈ [ M ], the blocksparse signal x reduces to the conventional sparse signal [13], [26]. Similar to (1.2), sensingmatrix A can be expressed as a concatenation of M column blocks over I = { d , d , . . . , d M } A = [ A . . . A d | {z } A [1] A d +1 . . . A d + d | {z } A [2] . . . A N − d M +1 . . . A N | {z } A [ M ] ] , where A i is the i th column of A for i = 1 , , · · · , N .2o reconstruct the block sparse signal x in (1.2), researchers explicitly take this blockstructure into account. One of the efficient methods is the following ℓ /ℓ minimizationmin x ∈ R N k x k , subject to k y − Ax k ǫ (1.3)where k x k , = M P i =1 k x [ i ] k . To study the uniqueness and stability of the ℓ /ℓ minimizationmethod, Eldar and Mishali introduced the notion of block restricted isometry property in[28], which is a generalization of the standard RIP [14]. Definition 1.1.
Let A ∈ R n × N be a matrix. Then A has the k order block restrictedisometry property (block RIP) over I = { d , d , . . . , d M } with parameter δ I ∈ [0 , if forall block k -sparse vector x ∈ R N over I it holds that (1 − δ I ) k x k k Ax k (1 + δ I ) k x k . (1.4) The smallest constant δ I is called block restricted isometry constant (block RIC) δ I k . When k is not an integer, we define δ I k as δ I⌈ k ⌉ . For block sparse signal recovery, sufficient conditions in term of the block RIP havebeen introduced and studied in the literatures. For example, Eldar and Mishali provedthat if the sensing matrix A satisfies δ I k < √ − ℓ /ℓ minimization can recoverperfectly block sparse signals in noiseless case and can well approximate the best block k -sparse approximation in [28]. Later, Lin and Li improved the bound to δ I k < . δ I k < .
307 [34]. Recently,Chen and Li [18] have shown a sharp sufficient condition based on the high order block RIC δ I tk < q t − t for any t > .The ℓ /ℓ minimization method (1.3) is itself nonadaptive since it dose not use any priorinformation about the block sparse signal x . However, the estimate of the support of thesignal x or of its largest coefficients may be possible to be drawn in many applications (see[30]). Incorporating prior block support information of signals, we introduce a method byreplacing the ℓ /ℓ minimization (1.3) with the following weighted ℓ /ℓ minimization with L (1 L M ) weights ω , ω , . . . , ω L ∈ [0 , x ∈ R N k x w k , subject to k y − Ax k ε (1.5)3here w ∈ [0 , M and k x w k , = M P i =1 w i k x [ i ] k is the weighted ℓ /ℓ norm of x . The mainidea of the weighted ℓ /ℓ minimization approach (1.5) is to choose appropriately w suchthat in the weighted objective function, the blocks of x which are expected to be large arepenalized less. Throughout the article, given L disjoint block support estimates of the blocksignal x over I by e T j ⊆ [ M ], where j = 1 , , . . . , L and ∪ Lj =1 e T j = e T , we setw i = , i ∈ e T c ω j , i ∈ e T j (1.6)for all i ∈ [ M ]. Note that when I = { d = 1 , d = 1 , . . . , d M = 1 } , the weighted ℓ /ℓ minimization meets with the weighted ℓ minimization [5, 17, 30, 31, 32, 35, 41, 44, 45].In this paper, we establish the high order block RIP condition to ensure the stable androbust recovery of block signals x through the weighted ℓ /ℓ minimization (1.5) and derivean error bound between the unknown original block sparse signal x and the minimizer of(1.5). And we also show that when all of the accuracy of L disjoint prior block supportestimates are at least 50%, the recovery by the weighted ℓ /ℓ minimization method (1.5)is stable and robust under weaker sufficient conditions compared to the ℓ /ℓ minimization(1.3). Moreover, we analyze how many random measurements of some random measure-ments matrices A are sufficient to satisfy the block RIP condition with high probability.Last, we present an algorithm used to solve the weighted ℓ /ℓ minimization (1.5) with0 < ω i i = 1 , , . . . , L ) and illustrate the advantages of weighted ℓ /ℓ minimizationapproach in the recovery performance of block sparse signals under uniform and non-uniformprior information by extensive numerical experiments.The rest of this paper is organized as follows. In Section 2, we will introduce somenotations and some basic results that will be used. The main results and their proofs arepresented in Section 3. Section 4 discusses the measurement number of some random matri-ces satisfying the block RIP condition with high probability. In Section 5, we demonstratethe benefits of the weighted ℓ /ℓ -minimization allowing uniform and non-uniform weightsin the reconstruction of block sparse signals by numerical experiments. A conclusion isincluded in Section 6. 4 Preliminaries
Let us begin with some notations. Define a mixed ℓ /ℓ p norm with p = 1 , , ∞ as k x k ,p = ( P Mi =1 k x [ i ] k p ) p . Note that k x k , = k x k . Let Γ ⊂ [ M ] be a block index set andΓ c ⊂ [ M ] be its complement set. For arbitrary block signal x ∈ R N over I , let x k over I be its best block k -sparse approximation such that x k is block k -sparse supported on T ⊆ [ M ] with | T | k and minimizes k x − s k , over all block k -sparse vectors s over I .Then T is the block support of x k , i.e., T = b-supp( x k ). Let x [Γ] ∈ R N over I be a vectorwhich equals to x on block indices Γ and 0 otherwise. x [Γ][ i ] denotes i th block of x [Γ]. x [max( k )] over I is defined as x with all but the largest k blocks in ℓ norm set to zero,and x [ − max( k )] = x − x [max( k )]. For any i ∈ { , , · · · , L } , let e T i ⊆ [ M ] be the supportestimate of x with | e T i | = ρ i k ( ρ i >
0) and e T = ∪ Li =1 e T i , where e T i ∩ e T j = ∅ ( i = j ), | e T | = ρk and ρ = L P i =1 ρ i > T . δ k denotes the k order standard restricted isometry constant[14].The following lemma is a key technical tool for analysing the sharp restricted isometryconditions of block sparse signal recovery. It is an extension of Lemma 1.1 [9] in the blockcase, which represents block signals in a block polytope by convex combination of blocksparse signals. Lemma 2.1. ([18], Lemma 2.2) For a positive number β and a positive integer s , definethe block polytope T ( β, s ) ⊂ R N by T ( β, s ) = { v ∈ R N : k v k , ∞ β, k v k , sβ } . For any v ∈ R N , define the set of block sparse vectors U ( β, s, v ) ⊂ R N by U ( β, s, v ) = { u ∈ R N : b - supp( u ) ⊆ b - supp( v ) , k u k , s, k u k , = k v k , , k u k , ∞ β } . Then any v ∈ T ( β, s ) can be expressed as v = J X i =1 λ i u i , where u i ∈ U ( β, s, v ) and λ i , J P i =1 λ i = 1 . ℓ /ℓ , ℓ /ℓ norms in the proof ofTheorem 3.1. Lemma 2.2. ([11], Lemma 5.3) Assume m > l , a > a > · · · > a m > , l P i =1 a i > m P i = l +1 a i , then for all θ > , m X j = l +1 a θj l X i =1 a θi . More generally, assume a > a > · · · > a m > , λ > and l P i =1 a i + λ > m P i = l +1 a i , then forall θ > , m X j = l +1 a θj l (cid:16) θ s P li =1 a θi l + λl (cid:17) θ . As we mentioned in the introduction, Chen and Li have obtained a high order sufficientcondition based on the block RIP to ensure the recovery of block sparse signals in [18]. Themain result on the sufficient condition is stated as below.
Theorem 2.3. ([18], Theorem 3.1) Let x ∈ R N be an arbitrary vector consistent with (1.1) and k z k ε . If the measurement matrix A satisfies the block RIP with δ I tk < r t − t (2.1) for t > , the solution ˆ x to (1.3) obeys k ˆ x − x k C ε + C k x [ T c ] k , √ k , (2.2) where C = q t ( t − δ I tk ) t ( p ( t − /t − δ I tk ) ,C = √ δ I tk + q t ( p ( t − /t − δ I tk ) δ I tk t ( p ( t − /t − δ I tk ) + 1 . (2.3)Note that if t > /
3, the condition (2.1) is sharp in Theorem 3.2 of [18].It is clear that the weighted ℓ /ℓ minimization problem (1.5) is equivalent to theweighted ℓ minimization problem when I = { d = 1 , d = 1 , · · · , d M = 1 } , i.e, M = N . Inthe case, Theorem 2.4 below states the main result of [41] for the weighted ℓ minimizationwith L (1 L N ) weights. 6 heorem 2.4. ([41],Theorem 2) Let x ∈ R N , x k denote its best k -sparse approximation,and denote the support of x k by T ⊆ { , , . . . , N } . Let e T i ⊆ { , , · · · , N } for i = 1 , . . . , L ,where L N , be arbitrary disjoint sets and denote e T = ∪ Li =1 e T i . Without loss ofgenerality, assume that the weights in (1.6) are ordered so that > ω > ω > · · · > ω L > .For each i , define the relative size ρ i and α i via | e T i | = ρ i k and | e T i ∩ T || e T i | = α i . Suppose thatthere exists a > , a ∈ k Z with P Li =1 ρ i (1 − α i ) a , and that the measurement matrix A has the standard RIP with δ ak + aK L δ ( a +1) k < aK L − , (2.4) where K L = ω L + (1 − ω ) s L P i =1 ( ρ i − α i ρ i ) + L P j =2 (cid:18) ( ω j − − ω j ) s L P i = j ( ρ i − α i ρ i ) (cid:19) .Then the minimizer ˆ x to (1.5) obeys k ˆ x − x k εC ′ + 2 C ′ k − (cid:18) k x − x k k L X i =1 ω i + (1 − L X i =1 ω i ) k x e T c ∩ T c k − L X i =1 L X j =1 ,j = i ω j k x e T i ∩ T c k (cid:19) where the constants C ′ = 1 + K L √ a p − δ ( a +1) k − K L √ a √ δ ak , C ′ = a − / ( p − δ ( a +1) k + √ δ ak ) p − δ ( a +1) k − K L √ a √ δ ak . (2.5) Remark 2.5.
Since δ ak δ ( a +1) k , the sufficient condition for (2.4) to hold is δ ( a +1) k < a − K L a + K L , δ ( a + 1 , K L ) . (2.6)From now on, let h = ˆ x − x , where ˆ x over I is the minimizer of the weighted ℓ /ℓ minimization problem (1.5) and x be the original block signal over I . For any block indexset, one establishes a cone constraint to prove our results (in Section 3) as following. Lemma 2.6. (Block cone constraint) For any block index set Γ ⊆ [ M ] , it holds that k h [Γ c ] k , ω L k h [Γ] k , + (1 − ω ) k h [Γ ∪ ∪ Li =1 e T i \ ∪ Li =1 ( e T i ∩ Γ)] k , + L X i =2 ( ω i − − ω i ) k h [Γ ∪ ∪ Lj = i e T j \ ∪ Lj = i ( e T j ∩ Γ)] k , + 2 L X i =1 ω i k x [Γ c ] k , +(1 − L X i =1 ω i ) k x [ e T c ∩ Γ c ] k , − L X i =1 ( L X j =1 ω j − ω i ) k x [ e T i ∩ Γ c ] k , . (2.7)7 emark 2.7. When ω i = 1 for all i ∈ { , , . . . , L } , the result of Lemma 2.6 is identical tothat of Lemma 2.3 in [18]. Suppose d i = 1 for all i ∈ [ M ] , the result of Lemma 2.6 meets withthat of Lemma 1 in [41]. Under the above assumption, if ω = ω = · · · = ω L = ω < ,the inequality (2.7) is k h [Γ c ] k , ω k h [Γ] k , + (1 − ω ) k h [Γ ∪ e T \ e T ∩ Γ] k , + 2( ω k x [Γ c ] k , + (1 − ω ) k x [ e T c ∩ Γ c ] k , ) , which is (21) of [30].Proof. Using the fact that ˆ x = x + h is a minimizer of the weighted ℓ /ℓ minimizationproblem (1.5), we have k ˆ x w k , = k ( x + h ) w k , k x w k , . We then obtain that L X i =1 ω i k x [ e T i ] + h [ e T i ] k , + k x [ e T c ] + h [ e T c ] k , L X i =1 ω i k x [ e T i ] k , + k x [ e T c ] k , , since e T i ∩ e T j = ∅ ( i = j ). Therefore, L X i =1 ω i k x [ e T i ∩ Γ] + h [ e T i ∩ Γ] k , + L X i =1 ω i k x [ e T i ∩ Γ c ] + h [ e T i ∩ Γ c ] k , + k x [ e T c ∩ Γ] + h [ e T c ∩ Γ] k , + k x [ e T c ∩ Γ c ] + h [ e T c ∩ Γ c ] k , L X i =1 ω i k x [ e T i ∩ Γ] k , + L X i =1 ω i k x [ e T i ∩ Γ c ] k , + k x [ e T c ∩ Γ] k , + k x [ e T c ∩ Γ c ] k , . From the triangle inequality, it follows that L X i =1 ω i k x [ e T i ∩ Γ] k , − L X i =1 ω i k h [ e T i ∩ Γ] k , + L X i =1 ω i k h [ e T i ∩ Γ c ] k , − L X i =1 ω i k x [ e T i ∩ Γ c ] k , + k x [ e T c ∩ Γ] k , − k h [ e T c ∩ Γ] k , + k h [ e T c ∩ Γ c ] k , − k x [ e T c ∩ Γ c ] k , L X i =1 ω i k x [ e T i ∩ Γ] k , + L X i =1 ω i k x [ e T i ∩ Γ c ] k , + k x [ e T c ∩ Γ] k , + k x [ e T c ∩ Γ c ] k , , i.e., L X i =1 ω i k h [ e T i ∩ Γ c ] k , + k h [ e T c ∩ Γ c ] k , L X i =1 ω i k h [ e T i ∩ Γ] k , + k h [ e T c ∩ Γ] k , (cid:18) k x [ e T c ∩ Γ c ] k , + L X i =1 ω i k x [ e T i ∩ Γ c ] k , (cid:19) . Adding and subtracting L P i =1 ω i k h [ e T ci ∩ Γ c ] k , on the left hand side, and L P i =1 ω i k h [ e T ci ∩ Γ] k , , 2 L P i =1 ω i k x [ e T ci ∩ Γ c ] k , on the right hand side respectively, we obtain L X i =1 ω i k h [Γ c ] k , + k h [ e T c ∩ Γ c ] k , − L X i =1 ω i k h [ ˜ T ci ∩ Γ c ] k , L X i =1 ω i k h [Γ] k , + k h [ e T c ∩ Γ] k , − L X i =1 ω i k h [ e T ci ∩ Γ] k , +2 L X i =1 ω i k x [Γ c ] k , + x [ e T c ∩ Γ c ] k , − L X i =1 ω i k x [ e T ci ∩ Γ c ] k , ! . (2.8)Note that k h [ e T c ∩ Γ c ] k , and L P i =1 ω i k h [ ˜ T ci ∩ Γ c ] k , are written as k h [ e T c ∩ Γ c ] k , = (1 − L X i =1 ω i ) k h [ e T c ∩ Γ c ] k , + L X i =1 ω i k h [ e T c ∩ Γ c ] k , = (1 − L X i =1 ω i ) k h [ e T c ∩ Γ c ] k , + L X i =1 ω i (cid:18) k h [Γ c ] k , − k h [ e T ∩ Γ c ] k , (cid:19) = (1 − L X i =1 ω i ) k h [ e T c ∩ Γ c ] k , + L X i =1 ω i (cid:18) k h [Γ c ] k , − L X j =1 k h [ e T j ∩ Γ c ] k , (cid:19) and L X i =1 ω i k h [ e T ci ∩ Γ c ] k , = L X i =1 ω i (cid:18) k h [Γ c ] k , − k h [ e T i ∩ Γ c ] k , (cid:19) . Then it is clear that k h [ e T c ∩ Γ c ] k , − L X i =1 ω i k h [ e T ci ∩ Γ c ] k , = (1 − L X i =1 ω i ) k h [ e T c ∩ Γ c ] k , − L X i =1 ( L X j =1 ω j − ω i ) k h [ e T i ∩ Γ c ] k , . Similarly, there are k h [ e T c ∩ Γ] k , − L X i =1 ω i k h [ e T ci ∩ Γ] k , = (1 − L X i =1 ω i ) k h [ e T c ∩ Γ] k , − L X i =1 ( L X j =1 ω j − ω i ) k h [ e T i ∩ Γ] k , k x [ e T c ∩ Γ c ] k , − L X i =1 ω i k x [ e T ci ∩ Γ c ] k , = (1 − L X i =1 ω i ) k x [ e T c ∩ Γ c ] k , − L X i =1 ( L X j =1 ω j − ω i ) k x [ e T i ∩ Γ c ] k , . Combining (2.8) with the above equalities, one easily deduces that L X i =1 ω i k h [Γ c ] k , L X i =1 ω i k h [Γ] k , + (1 − L X i =1 ω i ) (cid:18) k h [ e T c ∩ Γ] k , − k h [ e T c ∩ Γ c ] k , (cid:19) − L X i =1 ( L X j =1 ω j − ω i ) (cid:18) k h [ e T i ∩ Γ] k , − k h [ e T i ∩ Γ c ] k , (cid:19) + 2 (cid:18) L X i =1 ω i k x [Γ c ] k , + (1 − L X i =1 ω i ) k x [ e T c ∩ Γ c ] k , − L X i =1 ( L X j =1 ω j − ω i ) k x [ e T i ∩ Γ c ] k , (cid:19) . In the remainder of the proof, denote Z = L X i =1 ω i k x [Γ c ] k , + (1 − L X i =1 ω i ) k x [ e T c ∩ Γ c ] k , − L X i =1 ( L X j =1 ω j − ω i ) k x [ e T i ∩ Γ c ] k , . Then the above inequality can be expressed as L X i =1 ω i k h [Γ c ] k , L X i =1 ω i k h [Γ] k , + (1 − L X i =1 ω i ) (cid:18) k h [ e T c ∩ Γ] k , − k h [ e T c ∩ Γ c ] k , (cid:19) − L X i =1 ( L X j =1 ω j − ω i ) (cid:18) k h [ e T i ∩ Γ] k , − k h [ e T i ∩ Γ c ] k , (cid:19) + 2 Z. Since k h [Γ c ] k , = L P i =1 ω i k h [Γ c ] k , + (1 − L P i =1 ω i ) (cid:18) k h [ e T c ∪ Γ c ] k , + k h [ e T ∪ Γ c ] k , (cid:19) and( e T c ∩ Γ) ∪ ( e T ∩ Γ c ) = ( e T ∪ Γ) \ ( e T ∩ Γ) = Γ ∪ ∪ Li =1 e T i \ ∪ Li =1 ( e T i ∩ Γ), we deduce that k h [Γ c ] k , L X i =1 ω i k h [Γ] k , + (1 − L X i =1 ω i ) (cid:18) k h [ e T c ∩ Γ] k , + k h [ e T ∪ Γ c ] k , (cid:19) − L X i =1 ( L X j =1 ω j − ω i ) (cid:18) k h [ e T i ∩ Γ] k , − k h [ e T i ∩ Γ c ] k , (cid:19) + 2 Z = ω L k h [Γ] k , + (1 − ω ) k h [Γ ∪ ∪ Li =1 e T i \ ∪ Li =1 ( e T i ∩ Γ)] k , + L − X i =1 ω i k h [Γ] k , +( ω − L X i =1 ω i ) k h [ e T c ∩ Γ] k , − L X i =1 ( L X j =1 ω j − ω i ) k h [ e T i ∩ Γ] k , ω − L X i =1 ω i ) k h [ e T ∪ Γ c ] k , + L X i =1 ( L X j =1 ω j − ω i ) k h [ e T i ∩ Γ c ] k , + 2 Z = ω L k h [Γ] k , + (1 − ω ) k h [Γ ∪ ∪ Li =1 e T i \ ∪ Li =1 ( e T i ∩ Γ)] k , + L − X i =1 ω i k h [Γ] k , +( ω − L X i =1 ω i ) (cid:18) k h [Γ] k , − L X i =1 k h [ e T i ∩ Γ] k , (cid:19) − L X i =1 ( L X j =1 ω j − ω i ) k h [ e T i ∩ Γ] k , + L X i =1 ( ω − L X j =1 ω j ) k h [ e T i ∪ Γ c ] k , + L X i =1 ( L X j =1 ω j − ω i ) k h [ e T i ∩ Γ c ] k , + 2 Z = ω L k h [Γ] k , + (1 − ω ) k h [Γ ∪ ∪ Li =1 e T i \ ∪ Li =1 ( e T i ∩ Γ)] k , +( ω − ω L ) k h [Γ] k , + L X i =2 ( ω − ω i ) (cid:18) k h [ e T ci ∩ Γ] k , − k h [Γ] k , (cid:19) + L X i =2 ( ω − ω i ) k h [ e T i ∪ Γ c ] k , + 2 Z = ω L k h [Γ] k , + (1 − ω ) k h [Γ ∪ ∪ Li =1 e T i \ ∪ Li =1 ( e T i ∩ Γ)] k , + L − X i =2 ( ω i − ω ) k h [Γ] k , + L X i =2 ( ω − ω i ) (cid:18) k h [ e T ci ∩ Γ] k , + k h [ e T i ∪ Γ c ] k , (cid:19) + 2 Z. (2.9)Using the facts that k h [ e T cj ∩ Γ] k , = k h [Γ] k , − k h [ e T j ∩ Γ] k , = L X i =1 ,i = j k h [ e T j ∩ Γ] k , + k h [Γ ∩ ∩ Li =1 e T ci ] k , and L X j = i k h [ e T j ∩ Γ c ] k , + k h [Γ ∩ ∩ Lj = i e T cj ] k , = k h [Γ ∪ ∪ Lj = i e T j \ ∪ Lj = i ( e T j ∩ Γ)] k , , we obtain L X i =2 ( ω − ω i ) (cid:18) k h [ e T ci ∩ Γ] k , + k h [ e T i ∪ Γ c ] k , (cid:19) = L X i =2 ( ω i − − ω i ) L X j = i (cid:18) k h [ e T cj ∩ Γ] k , + k h [ e T j ∪ Γ c ] k , (cid:19) = L X i =2 ( ω i − − ω i ) (cid:18) L X j = i ( k h [Γ] k , − k h [ e T j ∩ Γ] k , ) − k h [Γ ∩ ∩ Lj = i e T cj ] k , k h [Γ ∩ ∩ Lj = i e T cj ] k , + L X j = i k h [ e T j ∩ Γ c ] k , (cid:19) = L X i =2 ( ω i − − ω i )( L − i ) k h [Γ] k , + L X i =2 ( ω i − − ω i ) k h [Γ ∪ ∪ Lj = i e T j \ ∪ Lj = i ( e T j ∩ Γ)] k , = − L − X i =2 ( ω i − ω ) k h [Γ] k , + L X i =2 ( ω i − − ω i ) k h [Γ ∪ ∪ Lj = i e T j \ ∪ Lj = i ( e T j ∩ Γ)] k , . Substituting the above equality into (2.9), we get the result (2.7).
In this section, we present the main results. First, we consider the signal recovery model(1.1) in the setting where the error vector z = 0 and the block signal x is not exactly block k -sparse and establish the sufficient condition based on the high order block RIP. The resultimplies that the condition guarantees the exact recovery in the noiseless setting and stablerecovery in noisy setting when the block signal x is block k -sparse. Theorem 3.1.
Consider the signal recovery model (1.1) with k z k ε , where x ∈ R N over I is an arbitrary block signal. Suppose that x k over I is the best block k -sparse approximationand ˆ x is the minimizer of (1.5) . Let e T i ⊆ [ M ] with i = 1 , , . . . , L be disjoint block index setsand denote e T = ∪ Li =1 e T i where L is a positive integer, such that | e T i | = ρ i k and | e T i ∩ T | = α i ρ i k where T is the support of x k , ρ i > and α i . Without loss of generality, assumethat the weights in (1.6) are ordered so that ω L ω L − · · · ω . If A satisfiesthe block RIP with δ I tk < s t − dt − d + Υ L , δ I ( t, Υ L ) (3.1) for t > d , where Υ L = ω L + (1 − ω ) vuut L X i =1 ρ i − L X i =1 α i ρ i + L X i =2 ( ω i − − ω i ) vuut L X j = i ρ j − L X j = i α j ρ j and d = , L Q i =1 ω i = 1max i ∈{ , ,...,L } { b i (1 − L P j = i α j ρ j + a i ) } , L Q i =1 ω i < ith a i = max { L P j = i α j ρ j , L P j = i (1 − α j ) ρ j } and b i = , i = 1 sgn ( ω i − − ω i ) , i = 2 , . . . , L. Then k ˆ x − x k D ε + 2 D √ k (cid:16) L X i =1 ω i k x − x k k , + (1 − L X i =1 ω i ) k x [ e T c ∩ T c ] k , − L X i =1 ( L X j =1 ω j − ω i ) k x [ e T i ∩ T c ] k , (cid:17) , (3.2) where D = q t − d )( t − d + Υ L )(1 + δ I tk )( t − d + Υ L )( q t − dt − d +Υ L − δ I tk ) ,D = √ δ I tk Υ L + r ( t − d + Υ L )( q t − dt − d +Υ L − δ I tk ) δ I tk ( t − d + Υ L )( q t − dt − d +Υ L − δ I tk ) + 1 √ d . (3.3) Proof.
We will prove the associated recovery guarantees (3.2) of the weighted ℓ /ℓ mini-mization. To this end, assume that tk is an integer and ˆ x = x + h , where x is the originalblock signal over I and ˆ x over I is a solution of the weighted ℓ /ℓ minimization problem(1.5). From Lemma 2.6 and the block support T ⊆ [ M ], it follows that k h [ T c ] k , ω L k h [ T ] k , + (1 − ω ) k h [ T ∪ ∪ Li =1 e T i \ ∪ Li =1 ( e T i ∩ T )] k , + L X i =2 ( ω i − − ω i ) k h [ T ∪ ∪ Lj = i e T j \ ∪ Lj = i ( e T j ∩ T )] k , + 2 (cid:18) L X i =1 ω i k x [ T c ] k , + (1 − L X i =1 ω i ) k x [ e T c ∩ T c ] k , − L X i =1 ( L X j =1 ω j − ω i ) k x [ e T i ∩ T c ] k , (cid:19) . (3.4)Based on d = , L Q i =1 ω i = 1max i ∈{ , ,...,L } { b i (1 − L P j = i α j ρ j + a i ) } , L Q i =1 ω i < a i = max { L P j = i α j ρ j , L P j = i (1 − α j ) ρ j } and b i = , i = 1 sgn ( ω i − − ω i ) , i = 2 , . . . , L,
13t is clear that d is an integer and d >
1. Recall h [max( dk )] as the block dk -sparse vector h over I with all but the largest dk blocks in ℓ norm set to zero. From (3.4) and d >
1, wehave k h [ − max( dk )] k , k h [ T c ] k , ω L k h [ T ] k , + (1 − ω ) k h [ T ∪ ∪ Li =1 e T i \ ∪ Li =1 ( e T i ∩ T )] k , + L X i =2 ( ω i − − ω i ) k h [ T ∪ ∪ Lj = i e T j \ ∪ Lj = i ( e T j ∩ T )] k , + 2 (cid:18) L X i =1 ω i k x [ T c ] k , + (1 − L X i =1 ω i ) k x [ e T c ∩ T c ] k , − L X i =1 ( L X j =1 ω j − ω i ) k x [ e T i ∩ T c ] k , (cid:19) . (3.5)Let r = 1 k h ω L k h [ T ] k , + (1 − ω ) k h [ T ∪ ∪ Li =1 e T i \ ∪ Li =1 ( e T i ∩ T )] k , + L X i =2 ( ω i − − ω i ) k h [ T ∪ ∪ Lj = i e T j \ ∪ Lj = i ( e T j ∩ T )] k , + 2 (cid:18) L X i =1 ω i k x [ T c ] k , + (1 − L X i =1 ω i ) k x [ e T c ∩ T c ] k , − L X i =1 ( L X j =1 ω j − ω i ) k x [ e T i ∩ T c ] k , (cid:19)(cid:21) , then r >
0. If r = 0, then k h [ T c ] k , = 0 and h is a block k -sparse vector. From the definitionof the block RIP and t > d >
1, it follows that (1 − δ I tk ) k h k k Ah k = k A ˆ x − Ax k ( k y − A ˆ x k + k Ax − y k ) ε , that is, k h k ε √ − δ I tk . By means of a series of calculation,we obtain D > √ − δ I tk . Therefore, we get the associated recovery guarantees (3.2) of theweighted ℓ /ℓ minimization under r = 0.From now on, we only consider r >
0. Decompose h [ − max( dk )] over I into two parts, h [ − max( dk )] = h (1) + h (2) , where for all i ∈ [ M ], the i th block of the block vectors h (1) and h (2) satisfies respectively h (1) [ i ] = h [ − max( dk )][ i ] , k h [ − max( dk )][ i ] k > rt − d ∈ R d i , elseand h (2) [ i ] = h [ − max( dk )][ i ] , k h [ − max( dk )][ i ] k rt − d ∈ R d i , else.In view of the definition of the block vector h (1) and (3.5), we obtain k h (1) k , k h [ − max( dk )] k , kr. k h (1) k , = m. Because the ℓ norm of every non-zero blocks of h (1) is larger than rt − d ( t > d, r > kr > k h (1) k , = X i ∈ b-supp( h (1) ) k h (1) [ i ] k > X i ∈ b-supp( h (1) ) rt − d = mrt − d . Namely m k ( t − d ). In addition, we have k h [max( dk )] + h (1) k , = dk + m dk + k ( t − d ) = tk, (3.6) k h (2) k , = k h [ − max( dk )] k , − k h (1) k , kr − mrt − d = ( k ( t − d ) − m ) · rt − d and k h (2) k , ∞ rt − d , where the last inequality follows from all non-zero blocks of h (2) having ℓ norm smallerthan rt − d .Now, using Lemma 2.1 with s = k ( t − d ) − m and β = rt − d , then h (2) can be expressedas a convex combination of block-sparse vectors, i.e., h (2) = J P i =1 λ i u i , where J P i =1 λ i = 1 and u i ∈ U ( rt − d , k ( t − d ) − m, h (2) ). In the remainder of the proof, one considers the followingtwo case.Case 1: Υ L = 0In the case, denote X = k h [max( dk )] + h (1) k and P = 2 (cid:18) L P i =1 ω i k x [ T c ] k , + (1 − L P i =1 ω i ) k x [ e T c ∩ T c ] k , − L P i =1 ( L P j =1 ω j − ω i ) k x [ e T i ∩ T c ] k , (cid:19) √ k Υ L . Thus, we have the upper bound k u i k = k u i k , q k u i k , k u i k , ∞ p k ( t − d ) − m k u i k , ∞ p k ( t − d ) · rt − d r kt − d r = r kt − d · k h ω L k h [ T ] k , + (1 − ω ) k h [ T ∪ ∪ Li =1 e T i \ ∪ Li =1 ( e T i ∩ T )] k , + L X i =2 ( ω i − − ω i ) k h [ T ∪ ∪ Lj = i e T j \ ∪ Lj = i ( e T j ∩ T )] k , + 2 (cid:16) L X i =1 ω i k x [ T c ] k , + (1 − L X i =1 ω i ) k x [ e T c ∩ T c ] k , − L X i =1 ( L X j =1 ω j − ω i ) k x [ e T i ∩ T c ] k , (cid:17)i
15 1 p k ( t − d ) h ω L k h [ T ] k , + (1 − ω ) k h [ T ∪ ∪ Li =1 e T i \ ∪ Li =1 ( e T i ∩ T )] k , + L X i =2 ( ω i − − ω i ) k h [ T ∪ ∪ Lj = i e T j \ ∪ Lj = i ( e T j ∩ T )] k , i + Υ L P √ t − d √ t − d h ω L k h [ T ] k + (1 − ω ) vuut L X i =1 ρ i − L X i =1 α i ρ i k h [ T ∪ ∪ Li =1 e T i \ ∪ Li =1 ( e T i ∩ T )] k + L X i =2 ( ω i − − ω i ) vuut L X j = i ρ j − L X j = i α j ρ j k h [ T ∪ ∪ Lj = i e T j \ ∪ Lj = i ( e T j ∩ T )] k i + Υ L P √ t − d k h [max( dk )] k √ t − d h ω L + (1 − ω ) vuut L X i =1 ρ i − L X i =1 α i ρ i + L X i =2 ( ω i − − ω i ) vuut L X j = i ρ i − L X j = i α i ρ i i + Υ L P √ t − d Υ L k h [max( dk )] + h (1) k √ t − d + Υ L P √ t − d = Υ L √ t − d ( X + P ) , (3.7)where we use the facts | T ∪ ∪ Li =1 e T i \ ∪ Li =1 ( e T i ∩ T ) | = (1 + L P i =1 ρ i − L P i =1 α i ρ i ) k dk and for i =2 , . . . , L , | T ∪ ∪ Lj = i e T j \∪ Lj = i ( e T j ∩ T ) | = (1+ L P j = i ρ i − L P j = i α i ρ i ) k dk when sgn ( ω i − − ω i ) = 1.Let β i = h [max( dk )] + h (1) + µu i where 0 µ
1, then we get J X j =1 λ j β j − β i = h [max( dk )] + h (1) + µh (2) − β i =( 12 − µ )( h [max( dk )] + h (1) ) − µu i + µh, (3.8)where J P i =1 λ i = 1 and h (1) + h (2) = h − h [max( dk )]. Since h [max( dk )], h (1) , u i are block dk -, m -, (( t − d ) k − m )-sparse vectors respectively, β i and N P j =1 λ j β j − β i − µh = ( − µ )( h [max( dk )] + h (1) ) − µu i are block tk -sparse vectors.Next, we compute an upper bound of X = k h [max( dk )] + h (1) k . We shall use the factsthat k Ah k = k A ˆ x − Ax k k y − A ˆ x k + k Ax − y k ε (3.9)16nd the following identity (see (25) in [9]) J X i =1 λ i (cid:13)(cid:13)(cid:13) A (cid:16) J X j =1 λ j β j − β i (cid:17)(cid:13)(cid:13)(cid:13) = J X i =1 λ i k Aβ i k . (3.10)Besides, h A ( h [max( dk )] + h (1) ) , Ah i k A ( h [max( dk )] + h (1) ) k k Ah k q δ I tk k h [max( dk )] + h (1) k · (2 ε ) , (3.11)where the last inequality uses the definition of block RIP with δ I tk , (3.6) and (3.9). Com-bining (3.11) and (3.8), we estimate the left hand side of (3.10) J X i =1 λ i (cid:13)(cid:13)(cid:13) A (cid:16) J X j =1 λ j β j − β i (cid:17)(cid:13)(cid:13)(cid:13) = J X i =1 λ i (cid:13)(cid:13)(cid:13) A (cid:16) ( 12 − µ )( h [max( dk )] + h (1) ) − µu i + µh (cid:17)(cid:13)(cid:13)(cid:13) = J X i =1 λ i (cid:13)(cid:13)(cid:13) A (cid:16) ( 12 − µ )( h [max( dk )] + h (1) ) − µu i (cid:17)(cid:13)(cid:13)(cid:13) + µ k Ah k + 2 J X i =1 λ i h A ( 12 − µ )( h [max( dk )] + h (1) ) − µu i , µh i = J X i =1 λ i (cid:13)(cid:13)(cid:13) A (cid:16) ( 12 − µ )( h [max( dk )] + h (1) ) − µu i (cid:17)(cid:13)(cid:13)(cid:13) + µ (1 − µ ) h A ( h [max( dk )] + h (1) ) , Ah i (1 + δ I tk ) J X i =1 λ i (cid:13)(cid:13)(cid:13) ( 12 − µ )( h [max( dk )] + h (1) ) − µu i (cid:13)(cid:13)(cid:13) + 2 εµ (1 − µ ) q δ I tk k h [max( dk )] + h (1) k = (1 + δ I tk ) h ( 12 − µ ) k h [max( dk )] + h (1) k + µ J X i =1 λ i k u i k i + 2 εµ (1 − µ ) q δ I tk k h [max( dk )] + h (1) k = (1 + δ I tk )( 12 − µ ) X + µ (1 + δ I tk )4 J X i =1 λ i k u i k + 2 εµ (1 − µ ) q δ I tk X, where the last but one equality applies J P i =1 λ i = 1 and h λ i u i , h [max( dk )] + h (1) i = 0 . β i and the definition of theblock RIP with δ I tk we have J X i =1 λ i k Aβ i k = J X i =1 λ i k A ( h [max( dk )] + h (1) + µu i ) k > J X i =1 λ i − δ I tk ) k h [max( dk )] + h (1) + µu i k = (1 − δ I tk ) J X i =1 λ i (cid:16) k h [max( dk )] + h (1) k + µ k u i k (cid:17) = 1 − δ I tk X + µ (1 − δ I tk )4 J X i =1 λ i k u i k . In consideration of the above two inequalities and (3.10) we have0 = J X i =1 λ i (cid:13)(cid:13)(cid:13) A (cid:16) J X j =1 λ j β j − β i (cid:17)(cid:13)(cid:13)(cid:13) − J X i =1 λ i k Aβ i k (cid:18) (1 + δ I tk )( 12 − µ ) −
14 (1 − δ I tk ) (cid:19) X + 12 δ I tk µ J X i =1 λ i k u i k + 2 εµ (1 − µ ) q δ I tk X (cid:20) (1 + δ I tk )( 12 − µ ) −
14 (1 − δ I tk ) + δ I tk µ Υ L t − d ) (cid:21) X + (cid:20) µ (1 − µ ) q δ I tk · (2 ε ) + δ I tk µ Υ L Pt − d (cid:21) X + δ I tk µ Υ L P t − d )= h ( µ − µ ) + (cid:16) − µ + (1 + Υ L t − d ) ) µ (cid:17) δ I tk i X + h εµ (1 − µ ) q δ I tk + δ I tk µ Υ L Pt − d i X + δ I tk µ Υ L P t − d ) , (3.12)where we apply the estimate of k u i k in (3.7). Substituting µ = √ ( t − d )( t − d +Υ L ) − ( t − d )Υ L ∈ (0 ,
1) into (3.12) yields − t − d + Υ L t − d µ (cid:18)s t − dt − d + Υ L − δ I tk (cid:19) X + (cid:18) εµ t − d + Υ L t − d s (1 + δ I tk )( t − d ) t − d + Υ L + δ I tk µ Υ L Pt − d (cid:19) X + δ I tk µ Υ L P t − d ) > , i.e., µ t − d h − ( t − d + Υ L ) (cid:16)s t − dt − d + Υ L − δ I tk (cid:17) X (cid:16) ε q ( t − d )( t − d + Υ L )(1 + δ I tk ) + δ I tk Υ L P (cid:17) X + δ I tk Υ L P i > , which is a second-order inequality for X . Hence, under the conditions δ I tk < q t − dt − d +Υ L and t > d we have X (cid:26)(cid:16) ε q ( t − d )( t − d + Υ L )(1 + δ I tk ) + δ I tk Υ L P (cid:17) + h(cid:16) ε q ( t − d )( t − d + Υ L )(1 + δ I tk ) + δ I tk Υ L P (cid:17) + 2( t − d + Υ L ) (cid:16)s t − dt − d + Υ L − δ I tk (cid:17) δ I tk Υ L P i / (cid:27) · (cid:16) t − d + Υ L )( s t − dt − d + Υ L − δ I tk ) (cid:17) − ε q ( t − d )( t − d + Υ L )(1 + δ I tk )( t − d + Υ L )( q t − dt − d +Υ L − δ I tk )+ 2 δ I tk Υ L + r t − d + Υ L )( q t − dt − d +Υ L − δ I tk ) δ I tk Υ L t − d + Υ L )( q t − dt − d +Υ L − δ I tk ) P, which is an upper bound of X = k h [max( dk )] + h (1) k .Last, it remains to develop an upper bound on k h k . To this end, we express k h k = k h [max( dk )] k + k h [ − max( dk )] k .Considering the inequality (3.5) and the definition of P , we have k h [ − max( dk )] k , (cid:0) ω L + (1 − ω ) + L X i =2 ( ω i − − ω i ) (cid:1) k h [max( dk )] k , + P √ k Υ L = k h [max( dk )] k , + P √ k Υ L , where we use that | T | dk ( d > | T ∪∪ Li =1 e T i \∪ Li =1 ( e T i ∩ T ) | = (1+ L P i =1 ρ i − L P i =1 α i ρ i ) k dk and for all i ∈ { , . . . , L } as sgn ( ω i − − ω i ) = 1 | T ∪ ∪ Lj = i e T j \ ∪ Lj = i ( e T j ∩ T ) | = (1 + L X j = i ρ j − L X j = i α j ρ j ) k dk. Thanks to Lemma 2.2 with θ = 2 , l = dk , and λ = P √ k Υ L , we have k h [ − max( dk )] k = k h [ − max( dk )] k , k h [max( dk )] k , + P Υ L √ d = k h [max( dk )] k + P Υ L √ d . k h k s k h [max( dk )] k + (cid:18) k h [max( dk )] k + P Υ L √ d (cid:19) √ k h [max( dk )] k + P Υ L √ d √ k h [max( dk )] + h (1) k + P Υ L √ d = √ X + P Υ L √ d ε q t − d )( t − d + Υ L )(1 + δ I tk )( t − d + Υ L )( q t − dt − d +Υ L − δ I tk )+ √ δ tk Υ L + r ( t − d + Υ L )( q t − dt − d +Υ L − δ I tk ) δ I tk Υ L ( t − d + Υ L )( q t − dt − d +Υ L − δ I tk ) + Υ L √ d ! P = 2 ε q t − d )( t − d + Υ L )(1 + δ I tk )( t − d + Υ L )( q t − dt − d +Υ L − δ I tk )+ √ δ I tk Υ L + r ( t − d + Υ L )( q t − dt − d +Υ L − δ I tk ) δ I tk ( t − d + Υ L )( q t − dt − d +Υ L − δ I tk ) + 1 √ d ! L P i =1 ω i k x [ T c ] k , + (1 − L P i =1 ω i ) k x [ e T c ∩ T c ] k , − L P i =1 ( L P j =1 ω j − ω i ) k x [ e T i ∩ T c ] k , ! √ k . Case 2: Υ L = 0Similarly, let X = k h [max( dk )] + h (1) k and P ′ = 2 L P i =1 ω i k x [ T c ] k , + (1 − L P i =1 ω i ) k x [ e T c ∩ T c ] k , − L P i =1 ( L P j =1 ω j − ω i ) k x [ e T i ∩ T c ] k , ! √ k . In the same way, we have that k u i k P ′ √ t − d , k h [ − max( dk )] k , k h [max( dk )] k , + P ′ √ k and k h [ − max( dk )] k k h [max( dk )] k + P ′ √ d . (3.13)Then 0 = J X i =1 λ i (cid:13)(cid:13)(cid:13) A (cid:16) J X j =1 λ j β j − β i (cid:17)(cid:13)(cid:13)(cid:13) − J X i =1 λ i k Aβ i k (cid:18) (1 + δ I tk )( 12 − µ ) −
14 (1 − δ I tk ) (cid:19) X + 12 δ I tk µ J X i =1 λ i k u i k + 2 εµ (1 − µ ) q δ I tk X (cid:20) (1 + δ I tk )( 12 − µ ) −
14 (1 − δ I tk ) (cid:21) X + 2 εµ (1 − µ ) q δ I tk X + δ I tk µ ( P ′ ) t − d ) . Taking µ = , we have − (1 − δ I tk ) X + 2 ε q δ I tk X + δ I tk ( P ′ ) t − d ) > . So, under the conditions δ I tk < q t − dt − d +Υ L , i.e., δ I tk < t > d we obtain X ε q δ I tk − δ I tk + s δ I tk t − d )(1 − δ I tk ) P ′ . From the above inequality and (3.13), it follows that k h k s k h [max( dk )] k + (cid:18) k h [max( dk )] k + P ′ √ d (cid:19) √ X + P ′ √ d ε q δ I tk )1 − δ I tk + (cid:18)s δ I tk t − d )(1 − δ I tk ) + 1 √ d (cid:19) P ′ = 2 ε q t − d )( t − d + Υ L )(1 + δ I tk )( t − d + Υ L )( q t − dt − d +Υ L − δ I tk )+ √ δ I tk Υ L + r ( t − d + Υ L )( q t − dt − d +Υ L − δ I tk ) δ I tk ( t − d + Υ L )( q t − dt − d +Υ L − δ I tk ) + 1 √ d ! L P i =1 ω i k x [ T c ] k , + (1 − L P i =1 ω i ) k x [ e T c ∩ T c ] k , − L P i =1 ( L P j =1 ω j − ω i ) k x [ e T i ∩ T c ] k , ! √ k where in last equality Υ L = 0.When tk is not an integer, take t ′ = ⌈ tk ⌉ /k , then t ′ k is an integer, t < t ′ and δ I t ′ k = δ I tk < s t − dt − d + γ < s t ′ − dt ′ − d + γ which implies that the case can be deduced to the former case ( tk is an integer). To sumup, we complete the proof of Theorem 3.1. Remark 3.2.
From Theorem 3.1, it is clear that the block signal x can be recovered exactlyand stably from y and A in the noiseless and noisy cases as x is a block k -sparse over I . Proposition 3.3. (1) If ω = ω = · · · = ω L = ω ∈ [0 , , then Υ L = ω + (1 − ω ) vuut L X i =1 ρ i − L X i =1 α i ρ i , d = , ω = 11 − L P i =1 α i ρ i + a , ω < and k ˆ x − x k D ε + D √ k ( ω k x [ T c ] k , +(1 − ω ) k x [ e T c ∩ T c ] k , ) , which can be regardedas an extension of Theorem 3.1 [17] to block signals. In this case, denote Υ L by Υ ωL .For d = d = · · · = d M = 1 , the above result of Theorem 3.1 is identical to that ofTheorem 3.1 in [17] with ρ = L P i =1 ρ i and α = L P i =1 α i ρ i / L P i =1 ρ i . (2) If ω i = 1 for all i ∈ { , , . . . , L } , then Υ L = 1 and d = 1 . The result reduces to that ofTheorem 2.3. That is, D = C , D = C and the sufficient condition for Theorem3.1 given in (3.1) is identical to (2.1) in Theorem 2.3. (3) If α i = for all i ∈ { , , . . . , L } , then Υ L = 1 , d = 1 , D = C , D = C and thesufficient condition for Theorem 3.1 given in (3.1) is identical to (2.1) in Theorem2.3. (4) Suppose that Q Li =1 ω i < and α i > for all i = 1 , . . . , L , then d = 1 , Υ L < , D < C , D < C and the sufficient condition (3.1) is weaker than (2.1) in Theorem2.3. For d = · · · = d M = 1 then D < C ′ , D < C ′ and the sufficient condition (3.1) is weaker than the sufficient condition (2.6) in Remark 2.5. Furthermore, we compare the sufficient condition (3.1) used the single weight with thatused the combination of weights when all accuracies α i are greater than . Proposition 3.4.
Let ω > ω > · · · > ω L , L P i =1 ρ i = ρ and α = α = · · · = α L = α . Then δ I ( t, Υ ω L ) δ I ( t, Υ L ) δ I ( t, Υ ω L L ) if and only if α > .Proof. Needell et.al have shown Υ ω L L Υ L Υ ω L if and only if α > in the proof ofProposition 1 in [41]. From the definition of δ I ( t, Υ L ) in (3.1), it is clear that δ I ( t, Υ ω L ) δ I ( t, Υ L ) δ I ( t, Υ ω L L ) if and only if α > . 22 ω ) B l o c k R I C δ I ( t , Υ L ) α =0.1 α =0.3 α =0.5 α =0.7 α =0.9 (a) δ I ( t, Υ ωL ) versus ω ρ B l o c k R I C δ I ( t , Υ ω ) a nd δ I ( t , Υ ) α =0.1 α =0.3 α =0.5 α =0.7 α =0.9 (b) δ I ( t, Υ ω ) and δ I ( t, Υ ) versus ρ (c) δ I ( t, Υ ) versus ρ and ρ ρ E rr o r bound no i s e c on s t an t D \sqrt{2} α =0.3 α =0.5 α =0.7 α =0.9 (d) D versus ρ ρ E rr o r bound no i s e c on s t an t D α =0.1 α =0.3 α =0.5 α =0.7 α =0.9 (d ′ ) D versus ρ Figure 1: Comparison of the sufficient conditions for recovery and stability constants forthe weighted ℓ /ℓ reconstruction. In all figures, we set t = 5. In (b), (d) and (d ′ ), thered dotted lines and the blue dotted lines indicate respectively the cases of ω = 0 . ω = 0 .
25 while the two weights case uses the solid lines. In (d) and (d ′ ), we fix δ I tk = 0 . ρ B l o c k R I C δ I ( t , Υ ) a nd s t a nd a r d R I C δ ( a + , K ) α =0.1 α =0.3 α =0.5 α =0.7 α =0.9 (a) δ I ( t, Υ ) and δ ( a + 1 , K ) versus ρ ρ E rr o r bound no i s e c on s t an t D and C ’ α =0.1 α =0.3 α =0.5 α =0.7 α =0.9 (b) D and C ′ versus ρ ρ E rr o r bound no i s e c on s t an t D and C ’ α =0.1 α =0.3 α =0.5 α =0.7 α =0.9 (c) D and C ′ versus ρ Figure 2: Comparison of δ I ( t, Υ ) (solid lines) and δ ( a + 1 , K ) (dotted lines) and stabilityconstants. In all the figures, we set t = 5. In (b) and (c), comparing stability constants D (solid lines) and C ′ (dotted lines) as well as D (solid lines) and C ′ (dotted lines) we fix δ I tk = 0 . δ ak = 0 .
15. 24ig.1 illustrates how the bound δ I ( t, Υ L ) of the block RIP constant δ I tk given in (3.1)and the stability constants given in (3.3) change with weights and the prior block supportestimate sizes for the different accuracy of prior block support estimate in the case ofweighted ℓ /ℓ when t = 5. And we also compare the bound δ I ( t, Υ L ) and the stabilityconstants when the single weight is used with that using two or three distinct weights asa function of the block support estimate sizes for the different accuracy of prior supportestimate.In Fig.1(a), we set ω = ω = · · · = ω L = ω ∈ [0 , α = α = · · · = α L = α and ρ + ρ + · · · + ρ L = ρ = 1. That is, we only consider the weighted ℓ /ℓ minimization withthe single weight ω , the block support estimate size ρ and the accuracy α . We plot thebound δ I ( t, Υ ωL ) versus ω with different values of α . We observe that the bound δ I ( t, Υ ωL )gets larger as α increases, which implies the sufficient condition on the block RIP constantbecomes weaker as α increases in the case of the weighted ℓ /ℓ minimization with thesingle weight. And when ω = 1 or α = , δ I ( t, Υ ωL ) is a constant (see Proposition 3.3). Inaddition, δ I ( t, Υ ωL ) decreases as ω increases with α > , which means the condition (3.1) isweaker for smaller weight ω .In Fig.1(b), we compare the bound δ I ( t, Υ L ) ( L = 2) when using either two disjointprior block support estimates e T and e T or a single prior block support estimate e T = e T ∪ e T ,which implies ρ = ρ + ρ . Let ω = 0 . e T ), ω = 0 .
25 (applied on e T ), thesingle weight ω = 0 . .
25 (applied on e T ), ρ = 1 and α = α = α . The figure displaysthe bounds δ I ( t, Υ L ) and δ I ( t, Υ ωL ) as a function of the size ρ for different α . As expected,the bounds δ I ( t, Υ L ) and δ I ( t, Υ ωL ) get larger as α increases both in the single and twoweights cases. The bound δ I ( t, Υ L ) lies between the bound δ I ( t, Υ ωL ) in the single weight ω = 0 . δ I ( t, Υ ωL ) applied the single weight ω = 0 .
25 when ρ ∈ [0 , α = 0 .
5. In addition, when α > , the figure demonstrates the result of Proposition(3.4), i.e., δ I ( t, Υ . L ) δ I ( t, Υ L ) δ I ( t, Υ . L ).Fig.1(c) displays the transition of δ I ( t, Υ L ) as ρ and ρ vary with L = 3, ρ + ρ + ρ = 1, α = α = α = 0 . ω = 0 . , ω = 0 . ω = 0 . ′ ), set L = 2, δ I tk = 0 . α = α = α . And we set ω = 0 . ω = 0 .
25 and the single weight ω = 0 . .
25. One can easily see that D and D in(3.3) decreases as α increases for the case of two distinct weights and the cases of the single25eight. We observe that constants D and D with two distinct weights lie between thosewith a single weight for the accuracy α . For α > , the smallest weight results in thebest (smallest) constants D and D and the largest weight results in the worst (largest)constants D and D .Fig.2 compares the bound δ I ( t, Υ L ) in (3.1) for I = { d = 1 , d = 1 , · · · , d M = 1 } withthe bound of the standard RIC δ ( a + 1 , K L ) in (2.6) as well as stability constants in (3.3)and (2.5) for various accuracy α = α = α . Set L = 2, ρ + ρ = 1, t = 5 , a = 4, ω = 0 . ω = 0 .
25. Here we depict the bounds δ I ( t, Υ L ) in (3.1), δ ( a + 1 , K L ) in (2.6) and theconstants in (3.3) and (2.5) versus ρ with various α .Fig.2(a) illustrates δ I ( t, Υ L ) is larger than δ ( a +1 , K L ) under the same support estimate.Moreover, Fig.2(b) and (c) describe that constants D and D are always smaller than C ′ and C ′ , respectively. Therefore, the sufficient condition (3.1) is weaker than (2.6), and errorbound constants (3.3) in Theorem 3.1 are better than those (2.5) in Theorem 2.4. Theorem 3.1 established that the block k -sparse signal x can be exactly recovered under asufficient condition δ I tk < δ I ( t, Υ L ) = q t − dt − d +Υ L . In this section, we prove that how manyrandom measurements are needed for δ I tk < δ I ( t, Υ L ) to be satisfied with high probability.Firstly, we recall Lemma 5.1 of [2], which plays an important role in the proof of Theorem4.2. Lemma 4.1. ([2] Lemma 5.1) Let Φ( ω ) , ω ∈ Ω nN , be a random matrix of size n × N drawnaccording to any distribution that satisfies the concentration inequality P ( |k Φ( ω ) x k − k x k | > ε k x k ) e − nc ( ε ) , < ε < , (4.1) where c ( ε ) is a constant depending on ε . Then, for any set T with | T | = k < n and any < δ < , we have that (1 − δ ) k x k k Φ( ω ) x k (1 + δ ) k x k , for all x ∈ X T (4.2) with probability > − (cid:18) δ (cid:19) k exp( − c ( δ/ n ) , (4.3) where X T denotes the set of all signals in R N that are zero outside of T .
26n the section, we consider special random measurement matrices A = ( A ij ) n × N , where A ij ∼ N (0 , /n ) , A ij = / √ n, w.p. 1 / − / √ n, w.p. 1 / A ij = p /n, w.p. 1 / , w.p. 2 / − p /n, w.p. 1 /
6. (4.4)Achlioptas [1] showed that the above random measurement matrices (4.4) satisfy (4.1)with c ( ε ) = ε / − ε /
6. Therefore, for each of the k -dimensional spaces X T , randommeasurement matrices (4.4) will fail to satisfy (4.2) with probability (cid:18) δ (cid:19) k exp (cid:18) − n (cid:18) δ − δ (cid:19)(cid:19) (4.5)by Lemma 4.1. Theorem 4.2.
For random measurement matrices (4.4) , suppose n > tk log Mkt − d t − d +Υ L ) − (cid:16) √ ( t − d ) / ( t − d +Υ L ) (cid:17) . Then δ I tk < δ I ( t, Υ L ) = q ( t − d ) / ( t − d + Υ L ) ( t > d ) holds in high probability.Proof. Without loss of generality, let tk is a positive integer. By (4.5), a n × tk ˆ d submatrixof random measurement matrices A (4.4) fails to fulfil (4.2) with probability (cid:18) δ I ( t, Υ L ) (cid:19) tk ˆ d exp (cid:18) − n (cid:18) ( δ I ( t, Υ L )) − ( δ I ( t, Υ L )) (cid:19)(cid:19) . As discussed in [28], we know that a block sparse signal lies in a structured union ofsubspaces. Then random measurement matrices (4.4) fail to satisfy (1.4) with probability (cid:0) Mtk (cid:1) (cid:18) δ I ( t, Υ L ) (cid:19) tk ˆ d exp (cid:18) − n (cid:18) ( δ I ( t, Υ L )) − ( δ I ( t, Υ L )) (cid:19)(cid:19) . Note that (cid:0)
Mtk (cid:1) ( eMtk ) tk . Then for t > d and δ I ( t, Υ L ) = q ( t − d ) / ( t − d + Υ L ), wehave P ( δ I tk > δ I ( t, Υ L )) (cid:0) Mtk (cid:1) (cid:18) δ I ( t, Υ L ) (cid:19) tk ˆ d exp (cid:18) − n (cid:18) ( δ I ( t, Υ L )) − ( δ I ( t, Υ L )) (cid:19)(cid:19) (cid:18) eMtk (cid:19) tk (cid:18) q ( t − d ) / ( t − d + Υ L ) (cid:19) tk ˆ d × exp (cid:18) − n (cid:18) t − d t − d + Υ L ) − (cid:16)q ( t − d ) / ( t − d + Υ L ) (cid:17) (cid:19)(cid:19) = 2 exp (cid:18) − n (cid:18) t − d t − d + Υ L ) − (cid:16)q ( t − d ) / ( t − d + Υ L ) (cid:17) (cid:19) + tk (cid:18) log eMtk + ˆ d log 12 q ( t − d ) / ( t − d + Υ L ) (cid:19)(cid:19) Hence, P ( δ I tk < q ( t − d ) / ( t − d + Υ L )) > − (cid:18) − n (cid:18) t − d t − d + Υ L ) − (cid:16)q ( t − d ) / ( t − d + Υ L ) (cid:17) (cid:19) + tk (cid:18) log eMtk + ˆ d log 12 q ( t − d ) / ( t − d + Υ L ) (cid:19)(cid:19) It is easy to see that the random measurements n > tk log( M/k )( t − d ) / (16( t − d +Υ L )) − (( t − d ) / ( t − d +Υ L )) / / when M/k → ∞ to sure δ I tk < q ( t − d ) / ( t − d + Υ L ) ( t > d ) to hold in high probability.We have completed the proof of the theorem. In this section, we present several numerical experiments to compare the weighted ℓ /ℓ minimization method with the ℓ /ℓ minimization method in the context of block signalrecovery. By numerical experiments, we illustrate the benefits of the weighted ℓ /ℓ mini-mization to recover block sparse signals in both noiseless and noisy cases. In addition, wealso demonstrate that non-uniform block support information can be preferable to uniformblock support information.For the solution of the ℓ /ℓ minimization problem, Wang et.al adopt an efficient iter-atively reweighted least squares (IRLS) algorithm [47], [48]. Inspired by the ideas of [47],we present a generalized algorithm of the IRLS to solve the weighted ℓ /ℓ minimizationproblem (1.5) with (1.6). First, we rewrite the problem (1.5) as the following regularized28nconstrained smoothed weighted ℓ /ℓ minimizationmin x k x w k ε , + 12 τ k y − Ax k , (5.1)where k x w k ε , = M P i =1 w i ( k x [ i ] k + ε ) / and w i ∈ (0 , f ( x, ε, τ ) = M X i =1 w i ( k x [ i ] k + ε ) / + 12 τ k y − Ax k be the objective function associated with the minimization problem (5.1). For the solutionof x , it is known that the first-order necessary condition is (cid:20) w i x [ i ]( k x [ i ] k + ε ) / (cid:21) i M + 1 τ A ′ ( Ax − y ) = 0 . Let the block vector e x ∈ R N over I = { d , d , . . . , d M } satisfy e x [ i ] = ( √ w i ( k x [ i ] k + ε ) − / , . . . , √ w i ( k x [ i ] k + ε ) − / ) ′ ∈ R d i for all i ∈ [ M ]. Define the diagonal weighting matrix W = diag( e x ), Therefore, we obtainthe necessary optimality condition ( τ W + A ′ A ) x = A ′ y . Due to the nonlinearity of theabove system, we apply an iterative method to solve the above equations. That is, if we fix W = W ( t ) to be that determined already in the t -th iteration step, we set the solution ofthe above equations x ( t +1) = ( W ( t ) ) − (( A ( W ( t ) ) − ) ′ ( A ( W ( t ) ) − ) + τ I ) − ( A ( W ( t ) ) − ) ′ y asthe ( t + 1)-th iterate.By the above analysis, we extend naturally the IRLS algorithm to the above problem(5.1) denoting by Algorithm 1 as following: Input : measurements y ∈ R n , sensing matrix A ∈ R n × N , estimated block-sparsity ˆ k ,weighted vector w ∈ R M . Step 1 : choose appropriate parameter τ >
0, set iteration count t = 0 and ε = 1,initialize x (0) = arg min k y − Ax k . Step 2 : “stopping criterion is not met” do W ( t ) = diag( √ w i ( ε t + k x ( t ) [ i ] k ) − / ), i = 1 , . . . , M ;2: B ( t ) = A ( W ( t ) ) − ;3: x ( t +1) = ( W ( t ) ) − (( B ( t ) ) ′ B ( t ) + τ I ) − ( B ( t ) ) ′ y ;4: ε t +1 = min { ε t , νr ( x ( t +1) ) ˆ k +1 /N } ;5: t=t+1. 29 ndOutput x ( t +1) is an approximation solution.In the algorithm 1, r ( x ( t +1) ) ˆ k +1 is the (ˆ k + 1)-th largest ℓ norm value of the block of x ( t +1) in the decreasing order, ν ∈ (0 ,
1) satisfies νr ( x (1) ) ˆ k +1 /N < τ is an appropri-ately chosen parameter, which controls the tolerance of noise term. Note that the algorithm1 is the IRLS when w i = 1 for all i = [ M ], i.e., Π Li =1 ω i = 1. In this paper, we don’t makea detailed analysis including convergence, local convergence rate and error bound of thealgorithm leaving to the interested reader.In all of our experiments, we apply the algorithm 1 to solve the weighted ℓ /ℓ mini-mization problem with 0 < Π Li =1 ω i
1. For the algorithm 1, we set the estimated ˆ k = k and ν = 0 .
7. If ε t +1 < − or k x ( t +1) − x ( t ) k < − , the iteration terminates andoutputs x ( t +1) ; otherwise, the maximum number of iterations is 1000. The measurementmatrix A ∈ R n × N was generated randomly with i.i.d draws from a standard Gaussiandistribution and the measurement vector y was observed from y = Ax + z , where z waszero-mean Gaussian noise with standard deviation σ or zero vector. In the noise-free case( σ = 0), τ = 10 − and the average exact recovery frequency over 50 experiments is plot-ted by the following figures. If k x ( t +1) − x k / k x k − , the recovery is regarded exact.For the presence of noise ( σ = 0 . τ = 10 − max | A ′ y | and we draw up the averagereconstruction signal to noise ratio (SNR) over 50 experiments. The SNR is given bySNR( x ( t +1) , x ) = 20 log k x k / k x ( t +1) − x k , where the measure of the SNR is dB. We first consider the uniform weight ω ∈ (0 , ω = · · · = ω L = ω , applied on e T = ∪ Li =1 e T i for the block k -sparse signal x over I with length N = 256 and k = 10,generated by choosing k blocks uniformly at random, where I = { d = ˆ d, . . . , d M = ˆ d } . Forthese k blocks, we choose the nonzero values from a standard Gaussian distribution. Let α = α = · · · = α L = α .In Fig.3(a)-5(a), the average exact recovery frequency is plotted versus measurementlevel n for the accuracy of the prior block support estimate: α = 0 . , α = 0 . , α = 0 . , which illustrates the reconstruction performance of the block k -sparse signal x with threedifferent block sizes ˆ d = 2 , ˆ d = 4 and ˆ d = 8 in the noiseless case. Fig.3(b)-5(b) depict the30 = 0 .
20 30 40 50 60 70 80 90 100 11000.10.20.30.40.50.60.70.80.91 Number of measurements n E x a c t r e c o v e r y f r equen cy ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 α = 0 .
20 30 40 50 60 70 80 90 100 11000.10.20.30.40.50.60.70.80.91 Number of measurements n E x a c t r e c o v e r y f r equen cy ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 α = 0 .
20 30 40 50 60 70 80 90 100 11000.10.20.30.40.50.60.70.80.91 Number of measurements n E x a c t r e c o v e r y f r equen cy ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 (a) Noise Free α = 0 .
20 30 40 50 60 70 80 90 100 110−10−50510152025303540 Number of measurements n S NR ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 α = 0 .
20 30 40 50 60 70 80 90 100 110−10−505101520253035 Number of measurements n S NR ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 α = 0 .
20 30 40 50 60 70 80 90 100 110−10−505101520253035 Number of measurements n S NR ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 (b) Noisy Case Figure 3: The recovery performance of the weighted ℓ /ℓ minimization is in terms of theexact recovery frequency in noiseless case and the SNR in noisy case. The block sparsesignal x with ˆ d = 2 has k = 10 nonzero blocks.31 = 0 .
40 50 60 70 80 90 100 110 120 13000.10.20.30.40.50.60.70.80.91 Number of measurements n E x a c t r e c o v e r y f r equen cy ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 α = 0 .
40 50 60 70 80 90 100 110 120 13000.10.20.30.40.50.60.70.80.91 Number of measurements n E x a c t r e c o v e r y f r equen cy ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 α = 0 .
40 50 60 70 80 90 100 110 120 13000.10.20.30.40.50.60.70.80.91 Number of measurements n S NR ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 (a) Noise Free α = 0 .
20 30 40 50 60 70 80 90 100 110−10−50510152025303540 Number of measurements n S NR ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 α = 0 .
40 50 60 70 80 90 100 110 120 130−14−12−10−8−6−4−20246 Number of measurements n S NR ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 α = 0 .
40 50 60 70 80 90 100 110 120 130−10−50510152025 Number of measurements n S NR ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 (b) Noisy Case Figure 4: The recovery performance of the weighted ℓ /ℓ minimization is in terms of theexact recovery frequency in noiseless case and the SNR in noisy case. The block sparsesignal x with ˆ d = 4 has k = 10 nonzero blocks.32 = 0 .
80 90 100 110 120 130 140 150 160 17000.10.20.30.40.50.60.70.80.91 Number of measurements n E x a c t r e c o v e r y f r equen cy ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 α = 0 .
80 90 100 110 120 130 140 150 160 17000.10.20.30.40.50.60.70.80.91 Number of measurements n E x a c t r e c o v e r y f r equen cy ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 α = 0 .
80 90 100 110 120 130 140 150 160 17000.10.20.30.40.50.60.70.80.91 Number of measurements n E x a c t r e c o v e r y f r equen cy ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 (a) Noise Free α = 0 .
80 90 100 110 120 130 140 150 160 17005101520253035 Number of measurements n S NR ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 α = 0 .
80 90 100 110 120 130 140 150 160 170−50510152025 Number of measurements n S NR ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 α = 0 .
80 90 100 110 120 130 140 150 160 170−50510152025 Number of measurements n S NR ω =0.1 ω =0.3 ω =0.5 ω =0.7 ω =1 (b) Noisy Case Figure 5: The recovery performance of the weighted ℓ /ℓ minimization is in terms of theexact recovery frequency in noiseless case and the SNR in noisy case. The block sparsesignal x with ˆ d = 8 has k = 10 nonzero blocks.33ase of recovering the block k -sparse signal x with three different block sizes ˆ d = 2 , ˆ d = 4and ˆ d = 8 in the presence of noise by the SNR. When the accuracy of the prior block supportestimate α > .
5, one can easily see that the best recovered performance is achieved forweight ω = 0 . ω = 1 results in the worst exact recovery frequency in thenoiseless case and the worst SNR in the noisy case. In addition, reducing the uniform weight ω below 1 reduces the number of measurements required for the recovery of x . On the otherhand, the exact recovery frequency and the SNR are shifted towards larger weights ω forsmall n as α < .
5, which means that the performance of the reconstruction algorithm isshifted. In a word, applying a larger prior block support estimate favors better recoveryand the experimental results are consistent with our theoretical results in Theorem 3.1.And it is also shown that the curves are very close for different weights ω when α = 0 . In this subsection, we demonstrate that multiple weights can be preferable to a singleweight by designing serval numerical experiments. In these experiments, we set N = 256, k = | T | = 20 and ˆ d = 2. We compare the exact recovery frequency and SNR when applyingeither a single prior block support e T or two disjoint prior block supports e T and e T satisfying e T = e T + e T , which means ρ = ρ + ρ .In Fig.6, we set α = α = α = 0 . ρ + ρ = ρ = 1 and vary the size of ρ and ρ .For the two prior block supports e T and e T , e T applies the larger weight ω = 0 . e T applies the smaller weight ω = 0 .
25. In the single prior block support case, the weight ω = 0 . ω = 0 .
25 is used on e T . Fig.6 (a) and (b) plot the exact recovery frequency andthe SNR versus the number of measurements n , respectively. One can see that using thesmaller weight ω = 0 .
25 prefers the best, using the larger weight ω = 0 . ω = 0 . ω = 0 .
25 as ρ and ρ are varied producesintermediate performance.Fig.7 (a) and (b) depict the exact recovery frequency and the SNR versus the numberof measurements n for some different α and α maintaining ρ α + ρ α = ρα , where α = 0 . , ρ = 1 and ρ = ρ = 0 .
5, which imply α + α = 1. The weights ω = 0 . ω = 0 .
25 are applied on e T and e T , respectively. Note that e T ⊆ T c and e T ⊆ T when34 E x a c t r e c o v e r y f r equen cy ρ =0.0, ρ =1.0 ρ =0.2, ρ =0.8 ρ =0.5, ρ =0.5 ρ =0.8, ρ =0.2 ρ =1.0, ρ =0.0 ω =0.5 ω =0.25 (a) Noise Free
40 50 60 70 80 90 100 110 120 13051015202530354045 Number of measurements n S NR ρ =0.0, ρ =1.0 ρ =0.2, ρ =0.8 ρ =0.5, ρ =0.5 ρ =0.8, ρ =0.2 ρ =1.0, ρ =0.0 ω =0.5 ω =0.25 (b) Noisy CaseFigure 6: Comparison of the exact recovery frequency and the SNR over 50 trials versusthe number of measurements n while using the weighted ℓ /ℓ -minimization with a singleweight ω (blue dotted lines) and two distinct weights ω and ω (black solid lines). Let ρ + ρ = ρ = 1 and α = α = α = 0 .
40 50 60 70 80 90 100 110 120 13000.10.20.30.40.50.60.70.80.91 Number of measurements n E x a c t r e c o v e r y f r equen cy α =0.0, α =1.0 α =0.2, α =0.8 α =0.5, α =0.5 α =0.8, α =0.2 α =1.0, α =0.0 ω =0.5 ω =0.25 (a) Noise Free
40 50 60 70 80 90 100 110 120 130−50510152025 Number of measurements n S NR α =0.0, α =1.0 α =0.2, α =0.8 α =0.5, α =0.5 α =0.8, α =0.2 α =1.0, α =0.0 ω =0.5 ω =0.25 (b) Noisy CaseFigure 7: Under α + α = 1, α = 0 . ρ = ρ = 0 .
5, we compare the exact recoveryfrequency in the noiseless case and the SNR in the noisy case over 50 trials versus the numberof measurements n while using the weighted ℓ /ℓ -minimization with a single weight ω (bluedotted lines) and two distinct weights ω and ω (black solid lines).35 = 0 . α = 1 .
0. As expected, we observe that the exact recovery frequency andSNR are largest when α = 0 . α = 1 . ℓ /ℓ -minimization in this case is best from Fig.7 (a) and (b). As α increasesfrom 0 to 1 and α decreases from 1 to 0, fewer correctly identified block indexes in T receive the smaller weight ω = 0 .
25, but rather the larger weight ω = 0 .
5. Moreover,we also see that the values of the exact recovery frequency and the SNR are very close, asusing a single weight ω = 0 . ω = 0 .
25. In fact, the recovery is slightly better applyingthe single weight ω = 0 .
25 than that using ω = 0 .
5, and the recovery falls in between the ω = 0 .
25 and ω = 0 . ω = 0 . ω = 0 .
25 are used with α = α = 0 .
5. Here, we turn to make the fact clear in the theory. Based on Theorem 3.1and Proposition 3.3 (1) and (3), for the case of ω = 0 . ω = 0 . ω = 0 .
25 we need tocompare the terms ω k x [ T c ] k , + (1 − ω ) k x [ e T c ∩ T c ] k , = 0 . k x [ T c ] k , + 0 . k x [ e T c ∩ T c ] k , and( ω + ω ) k x [ T c ] k , + (1 − ( ω + ω )) k x [ e T c ∩ T c ] k , − ω k x [ e T ∩ T c ] k , − ω k x [ e T ∩ T c ] k , = 0 . k x [ T c ] k , + 0 . k x [ e T c ∩ T c ] k , − . k x [ e T ∩ T c ] k , − . k x [ e T ∩ T c ] k , . k x [ T c ] k , + 0 . k x [ e T c ∩ T c ] k , − . k x [ e T ∩ T c ] k , where we use k x [ e T ∩ T c ] k , + k x [ e T ∩ T c ] k , = k x [ e T ∩ T c ] k , = k x [ T c ] k , − k x [ e T c ∩ T c ] k , .It is obvious that( ω + ω ) k x [ T c ] k , + (1 − ( ω + ω )) k x [ e T c ∩ T c ] k , − ω k x [ e T ∩ T c ] k , − ω k x [ e T ∩ T c ] k , ω k x [ T c ] k , + (1 − ω ) k x [ e T c ∩ T c ] k , . Similarly, for ω = 0 . ω = 0 . ω = 0 .
25 there is( ω + ω ) k x [ T c ] k , + (1 − ( ω + ω )) k x [ e T c ∩ T c ] k , − ω k x [ e T ∩ T c ] k , − ω k x [ e T ∩ T c ] k , > ω k x [ T c ] k , + (1 − ω ) k x [ e T c ∩ T c ] k , . In this paper, the problem of reconstructing unknown block sparse signals under arbitraryprior block support information is studied from incomplete linear measurements. Firstly,36e introduce the weighted ℓ /ℓ minimization and obtain a high order block RIP conditionto guarantee stable and robust recovery of block signals in bounded ℓ noise setting. Thecondition is weaker than that of block sparse signals by the standard ℓ /ℓ minimizationwhen all of the accuracy of L disjoint prior block support estimates are at least 50%.Secondly, we determine how many random measurements are needed to fulfill the high orderblock RIP condition with high probability for some random matrices. 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