BLIND SEPARATION OF SPATIALLY-BLOCK-SPARSE SOURCES FROM ORTHOGONAL MIXTURES. Ofir Lindenbaum, Arie Yeredor Ran Vitek Moshe Mishali

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1 2013 IEEE INTERNATIONAL WORKSHOP ON MACHINE LEARNING FOR SIGNAL PROCESSING, SEPT , 2013, SOUTHAPMTON, UK BLIND SEPARATION OF SPATIALLY-BLOCK-SPARSE SOURCES FROM ORTHOGONAL MIXTURES Ofir Lindenbaum, Arie Yeredor Ran Vitek Moshe Mishali School of Electrical Engineering Altair EZchip Tel-Aviv University, Tel-Aviv, Israel Hod Hasharon, Israel Yokneam, Israel ABSTRACT We addresses the classical problem of blind separation of a static linear mixture, where separation is not based on statistical assumptions (such as independence) regarding the sources, but rather on their spatial (block-) sparsity, and with an additional constraint of an orthogonal mixing-matrix. An algorithm for this problem was recently proposed by Mishali and Eldar, and consists of two steps: one for recovering the support of the sources, and a subsequent one for recovering their values. That algorithm has two shortcomings: One is an assumption that the spatial sparsity level of the sources at each time-instant is constant and known; The second is the algorithm s sensitivity to the possible presence of temporal blocks of the signals with identical support. In this work we propose two pre-processing stages for improving the applicability and the performance of the algorithm. A first stage is aimed at identifying blocks of similar support, and pruning the data accordingly for the support-recovery stage. A second stage is aimed at recovering the sparsity level at each time-instant by exploiting observed structural inter-relations between the signals at different time-instants. We demonstrate the improvement over the original algorithm using both synthetic data and mixed text-images. We also show that the algorithm outperforms the recovery rate of alternative source separation methods for such contexts, including K-SVD, a leading method for dictionary learning. Index Terms Blind Source Separation, Dictionary Learning, Block-Sparsity, Orthogonal Mixtures 1. INTRODUCTION We consider the classical case of a static, square, noiseless mixture x[t] = Ψ s[t], t=1,2,...t (1) where s[t] = [s 1[t] s N [t]] T R N are N source signals, Ψ R N N is an unknown mixing matrix, and x[t] R N are the observed signals. The goal is to blindly recover the source signals (up to possible permutation and sign). However, rather than use statistical assumptions on the source signals, we employ a structural assumption, that the sources (which are not necessarily independent) are jointly sparse, in the sense that at each time instant t only K t signals are active (namely, take non-zero values) simultaneously, where typically K t N for most values of t within the observation period. We shall initially assume that all K t values are known in advance, but would later relax this assumption and consider the estimation of these values from the data. An additional, somewhat restrictive assumption employed in this work, is that the unknown mixing matrix Ψ is orthogonal, Ψ T Ψ = I (the N N identity matrix). While this might not be a realistic assumption in Blind Source Separation (BSS) applications, it can be relevant, e.g., in the context of Dictionary Learning (DL), seeking sparse representations of the observations, when the dictionary is taken to be an orthogonal basis thereof. Orthogonal mixing matrices have been considered before, e.g., in [1], [2]. We add in passing, that although, in general, it is common practice in some BSS methods to apply a (spatial) prewhitening stage, which essentially orthogonalizes the mixing-matrix - such an option is irrelevant in the context of this work, since if Ψ is not orthogonal, a whitening operation would be very likely to ruin the joint sparsity property of the (whitened) sources. The recovery of sparse sources from their observed mixtures is also known as Sparse Component Analysis (SCA). As opposed to classical Independent Component Analysis (ICA), which relies on statistical assumptions (mutual independence) regarding the sources, SCA does not require such assumptions, as it is based on structural sparsity assumptions. Previous papers on SCA, such as by Georgiev et al. [3] or Gribonval and Lesage [4], propose finding Ψ through a combinatorial search; Earlier papers, e.g., by Bofill and Zibulevsky [5] or by Li et al. [6], use an approximate maximization programming. Note that in our model the sources are not necessarily assumed to be temporally sparse - they are merely assumed to be jointly (spatially) K t-sparse (with K t N) at each time-instant t. While intuitively both kinds of sparsity should be closely related, theoretically the sources can fulfill either sparsity assumption with or without fulfilling the other. In fact, the recovery of a single temporally block-sparse signal from compressed sensing thereof has been treated extensively in recent years, see, e.g., [7], [8], [9], [10]. However, temporal block-sparsity of the individual sources do not necessarily imply spatial block-sparsity, and vice-versa. Our current work is an extension of the work by Mishali and Eldar [1], where an efficient two-stage approach was proposed for SCA with an orthogonal Ψ and with a known constant sparsity-level K t = K t of the signals. In the first stage the support-patterns of the signals are recovered, namely the K t active sources in each s[t] are identified. This is accomplished by exploiting the orthogonality of Ψ to derive a set of mutual activity rules based on the T T temporal sample-covariance matrix of the observed mixtures. Solving these rules reveals the unknown support. In the second stage the values are recovered by solving the remaining set of equations using an alternating minimization procedure. The original algorithm in [1] encounters difficulties in recovering the full support-patterns when handling natural signals: Some typical sparse signals (e.g., text images) contain segments ( blocks ) of energy in between segments of silence (zero energy). Our contribution in this work are two major modifications to the original algorithm. First, we add a preprocessing stage to reveal these block segments, thereby improv /13/$31.00 c 2013 IEEE

2 ing the ability of the algorithm to recover the support-patterns. Next, we relax the restrictive assumption of a constant, known sparsitylevel (K t = K t) and allow different, unknown sparsity levels which are estimated at each time-instant t from the correlation patterns of the data. The paper is structured as follows: Section II presents the problem formulation. In Section III the support recovery algorithm of [1] is briefly described. In section IV we describe the proposed preprocessing stage for addressing natural signals which contain blocks. Section V describes the estimation of the sparsity-level at each time frame. Section VI describes the values recovery algorithm. Experimental results are presented and analyzed in section VII, followed by conclusions in Section VIII Model 2. PROBLEM FORMULATION Expressing (1) in matrix notations, we have X = ΨS, (2) where S = [s[1] s[2] s[t ]] R N T, with a similar structure for X R N T. Each column s[t] of S is assumed to have exactly (not more and not less) K t non-zero-values, and we shall assume for now that all K t are known (an assumption that will be relaxed later on). We denote that property of s[t] as K t-sparsity, expressed mathematically as s[t] 0 = K t, t=1,...,t (3) where the l 0 pseudo-norm 0 counts the number of non-zero values of its vector argument. The separation problem can thus be recast as the following optimization program: ( Ψ, Ŝ) = arg min X ΨS F Ψ,S s.t. Ψ T Ψ = I and s[t] 0 = K t, t = 1,..., T. (4) (here F denotes the Frobenius norm). The pair of matrices Ψ, Ŝ are the primal solution of (4). Next, we outline the conditions for uniqueness of this solution Uniqueness The BSS problem as presented in (2) has a well-known inherent permutation and scale indeterminacy (which, when considering the orthogonality constraint on Ψ, reduces to sign indeterminacy). Therefore, unique recovery of Ψ and S is not possible. Nevertheless, uniqueness of the solution to (4) up to these permutation and sign ambiguities can be addressed. Here we quote the following result by Aharon et al. [11], which guarantees such uniqueness. Theorem 1. The factorization B = AS, where A R M N has normalized columns, and where all columns of S R N T are at most K-sparse, is unique up to signed permutations under the following conditions: Support: Every 2K columns of A are linearly independent. Richness: Upon clustering the columns of S according to their support, each cluster consists of at least K + 1 columns, sharing the same support. In addition, for every 1 n N, there exist k, l such that the intersection supp(s :,k ) supp(s :,l ) = {n}, where supp(s :,k ) denotes the support of the k-th column of S. Non-degeneracy: Every column subset S :,J of cardinality J = D, with R rows that are non-identically zero, has Rank(S :,J) = min{d, R}. In our case we assume that the mixing matrix A = Ψ is orthogonal, which leads to the following corollary (see [12] for details): Corollary 1. Consider the setting of Theorem 1 with an orthogonal A. Then, uniqueness holds if N 2K, S is non-degenerate and rich, possibly up to the following richness exceptions: The intersection property may not hold for n = N, and for each cluster of columns (sharing the same support), it may also not hold for one entry of the support. A rich matrix S can be constructed with T = 2N(K + 1) columns [11], whereas in our case of an orthogonal mixture, the exceptions of Corollary 1 allows to reach a similar construction with T = 2(N N 0)(K + 1) columns, where N 0 = 1 + (N 1)/K (here denotes the lower integer part of its argument). Both the Theorem and Corollary are proved by explicitly constructing Ψ and S through a procedure with combinatorial complexity - which however is irrelevant for the retrieval of these matrices. 3. SUPPORT RECOVERY In this section we provide a brief description of the support recovery algorithm proposed by Mishali and Eldar in [1]. The recovery is based on the sample-correlation matrix C = X T X R T T, which, thanks to the orthogonality of Ψ, provides direct information on the correlation between the sources, since C = X T X = S T Ψ T ΨS = S T S. (5) Using C, the recovery algorithm relies on the following set of properties, ensuing from (5) and from our model assumptions, in addition to mild statistical assumptions on the distributions of values of the sources (see below): (P1) C k,l = 0 implies that supp(s :,k ) and supp(s :,l ) are disjoint; (P2) C k,l 0 implies that supp(s :,k ) supp(s :,l ) is not empty; (P3) A column S :,t = s[t] of S contains exactly K t non-zeros; (P4) The order of rows of S is immaterial for successful recovery of the support. Property (P1) holds (deterministically) provided that the nonzero values of all sources are all positive; It also holds probabilistically with probability 1 (w.p.1) if these values are drawn independently from any continuous distribution. The support recovery algorithm is based on defining rules (see below) and on exploiting these rules, in accordance with the four properties above. The algorithm iteratively constructs an N T matrix denoted Z, which contains activity indicators regarding the respective elements in S. The elements of Z can take three optional values: Z n,t = 1 indicating an active signal (namely that s n[t] is nonzero); Z n,t = 0 indicating inactivity (s n[t] = 0); and Z n,t = φ indicating that s n[t] is unresolved. The matrix Z is initialized such that all of its elements are φ, and these elements are updated (inferred) as 1-s or 0-s in an iterative process as described in Algorithm 1 below. To explain the algorithm, let us first define the term A Rule for Column t as a set of row-indices

3 I, indicating that column t should have at least one non-zero element with a row-index in that set. The algorithm maintains a list of rules for each column, dynamically removing old rules or creating new rules at each iteration, as described in detail below. A special rule is a rule which is set (for each column) upon initialization, defining I = {1,..., N} (the entire set of all possible row-indices) and, unlike a regular rule (a rule created by the algorithm), a special rule for column t is not removed until K t 1-s are identified for that column. We say that symmetry holds with respect to a given rule if all elements of the respective set I either are all included in, or are all excluded from, all other rules for the same column. Using this terminology, the algorithm takes the following form: Algorithm 1 Recover support Input: C = X T X, sparsity levels {K t} T t=1 Output: Support pattern Z Initialization: Z n,t = φ n {1,..., N}, t {1,..., T } A special rule I = {1,..., N} for each column. for t = 1 to T do for all rules r in the list L t of rules for column t do Let I r denote the set of row-indices pointed by rule r L t if Z n,t = 1 for some n I r and r is a regular rule then Remove r form L t end if if Symmetry holds with respect to I r then Choose n I r and set Z n,t = 1 end if for all {n, t C t,t = 0, Z n,t = 1} do Z n,t = 0 if Z :,t indicates exactly K t non-zero values then supp(s :,t) = {n Z n,t = 1} Z m,t = 0 for every m / supp(s :,t) Remove the special rule from L t for all {C t,t 0} do Add a regular rule for column t with I = supp(s :,t) end if The algorithm iteratively utilizes the properties (P1)-(P4), until there are no longer any changes to the matrix Z. Although convergence (in the sense of meeting the stopping condition in a finite number of iterations) is guaranteed, perfect recovery of the support is not guaranteed, since Z might still contain a large number of unresolved elements. This usually happens when the sources are characterized by temporal blocks of the same spatial sparsity pattern. In the next section we propose a modification to the algorithm, which addresses such cases. 4. BLOCKS PRUNING Some typical natural sparse signals may contain segments ( blocks ) of energy in between segments of silence (zero energy). As mentioned above (and demonstrated in simulation in the sequel), Algorithm 1 encounters difficulties in fully recovering the support of such signals, due to the compromised diversity in their structure. Let us define a Block as a subset of columns of S sharing the exact same support, {B} = {t 1, t 2,..., t L supp(s[t 1]) = supp(s[t 2]) = = supp(s[t L])}, (6) where L = B is the cardinality of the block. The basic reason for the performance degradation of the original algorithm in the support-recovery in the presence of blocks, is that since all columns in a block share the exact same support, they also share the same set of rules, and fully resolving one column in the block still does not imply resolution of the other columns. We propose to add a preprocessing stage, which helps solving this problem. Let us first define the correlation indicators matrix C = sign(c) R T T, which only contains 1-s and 0-s, such that for nonzero elements in C the respective element in C is 1, and for zero elements in C the respective element in C is 0. We make the following observations regarding the matrix C: Observation 1. Columns in C corresponding to columns in S with the same support are identical. Observation 2. The probability that two columns in C corresponding to columns in S with different support are be identical decreases with T and will vanish for T >> N. To prove the Observation 1 (by contradiction), let supp(s[t 1]) = supp(s[t 2]) for some columns t 1 and t 2, and assume that t C t,t1 C t,t2. This implies that sign(s T [t]s[t 1]) sign(s T [t]s[t 2]). Without loss of generality assume that sign(s T [t]s[t 1]) = 0 and sign(s T [t]s[t 2]) = 1, implying (respectively) that supp(s[t]) supp(s[t 1]) = φ, whereas supp(s[t]) supp(s[t 2]) φ, which in turn implies that supp(s[t 1]) supp(s[t 2]), contradicting our assumption. Observation 2 can be explained by noting that if T is large enough such that all possible supports exist in S, then every two columns in S with different supports must lead to different columns in C. Based on these two observation we seek the blocks in the available correlation indicators matrix C in order to estimate the blocks in the unknowns matrix S. Once these blocks are identified, the matrix X is pruned accordingly, leaving one column from each block as the representative of the block. The original algorithm is then applied to the pruned version of X, and the resulting estimated supports are then re-expanded by blocks for retrieval of the full support pattern of S. Algorithm 2 Block Pruning 1. Estimate the blocks using the correlation indicators matrix C. 2. Prune the matrix X, leaving only one column representing each block. 3. Recover the support of the trimmed version of X using Algorithm Re-expand to recover the support of S using the estimated blocks. In order to get empirical quantification of the percentage of identification errors inflicted by Observation 2 for moderate values of T, we ran the following experiment. We built random block source matrices S R N T with constant sparsity levels K t = K, where the indices of the K active signals in each column were randomly

4 drawn, uniformly and independently. The values for these active signals were drawn independently from a standard Normal distribution. The blocking effect was created by temporally convolving each signal with a rectangular window of width B, thereby obtaining blocks of typical width B. We ran simulations for several values of N = 10, 12, 14 and K = 2, 3, 4, with B = 2. Figure 1 shows the ratio of falsely identified blocks vs. the observation length T, averaged over 1000 independent trials. Note that there can be no mis-detected blocks, Since Observation 1 holds deterministically. it can be shown, using some statistical model on the support distribution, that if the number of available columns is sufficiently large and S is sufficiently rich, then these lower-bounds on the sparsity levels would become tight and the estimates would converge to the true K t. The main difference between this estimation approach and a naïve energy-based approach is that in the proposed approach the estimation of a column s sparsity level makes use of the inter-relations between that column and all the other columns (as well as of the internal relations between these other columns), as opposed to basing the estimate on the column data alone. We propose a computationally efficient algorithm for computation of the proposed estimator: Algorithm 3 Sparsity estimation Input: C t - a square sub-matrix of C, containing only the rows and columns indices taken from A t. while C t I do Delete from C t the row and column with the largest number of non-zero elements. end while Set ˆK t to the dimension of C t. Fig. 1. The ratio of columns with the same correlation indicators and different support vs. the number of columns. 5. SPARSITY LEVEL ESTIMATION As explained earlier, the original algorithm in [1] is based on the often unrealistic assumption that the exact sparsity level K t at each time frame is at least known (if not constant, as assumed in [1]). To relax this restrictive assumption, we tried using statistically-based estimators in order to estimate the sparsity levels, using the energy of the columns (which is preserved by the orthogonal mixing matrix). However such estimators yielded poor results in the supportrecovery stage, which in turn resulted in poor separation. Here we present an alternative estimator of sparsity levels, based on the correlation indicators matrix C, exploiting the mutual structure information of S, rather than the statistical properties of the sources. The basic idea is to use the correlation indicators values to find a lower bound on the number of active signal in each column. Let us define for each column t a set A t - the set of indices of all columns which are correlated with that column (according to the correlation indicators matrix): A t = {s C s,t = 1} (7) Our proposed estimation scheme is based on the following observation: Observation 3. K t is larger than or equal to the size of any subset of A t containing columns that are uncorrelated with each other. Therefore, it is larger than or equal to the size of the largest of these subsets. Consequently, we should search for the largest uncorrelated subset of A t, and set K t to be the cardinality of that subset. Observation 3 can be corroborated by a simple example: If s[t] and s[t 1] are correlated, and s[t] and s[t 2] are correlated, but s[t 1] and s[t 2] are uncorrelated, then s[t] has at least two active values. Moreover, To evaluate the accuracy of the sparsity levels estimation we ran an experiment with N = 15 sources and varying sparsity levels drawn uniformly and independently between 1 and 5 (inclusive). The mean percentage of errors (averaged over 200 independent trials) is shown in Figure 2 vs. the number of observations T. As expected, the errors vanish when the observation length is sufficiently long. Fig. 2. Sparsity estimation error percentage vs. the number of columns. 6. SOURCE SEPARATION In this Section we provide a brief description of the algorithm for recovering the sources values (following the recovery of their support patterns), as proposed in [1]. With the support information available in Z, the estimation of the sources (and of the mixing matrix) translates into the following optimization problem: ( Ψ, Ŝ) = arg min X ΨS F Ψ,S s.t. Ψ T Ψ = I and S n,t = 0, {(n, t) Z n,t = 0}. (8)

5 Note that we consider the unresolved elements of Z (if any) as indicators of active signals (namely, all Z n,t = φ are considered Z n,t = 1) - the values recovery can still assign very small (or zero) values to mis-identified active locations. Any solution of (8) will minimize the Lagrangian: L(Ψ, S; Γ, Π) = X ΨS 2 F + Tr[Γ(Ψ T Ψ I)] + Tr(Π T S) (9) where Γ R N N and Π R N T are matrices of Lagrangian multipliers, with Π n,t = 0 {(n, t) Z n,t 0}. Calculating the gradients with respect to each variable yields: ΠL = 0 S n,t = 0, {(n, t) Z n,t = 0} ΓL = 0 Ψ T Ψ = I ΨL = 0 (X ΨS)S T = Ψ T S = 0 2Ψ T X + 2Ψ T ΨS + Π = 0 (10a) (10b) (10c) (10d) We use an alternating minimization procedure to find Ψ and S satisfying (10a)-(10d). When keeping S fixed, the solution to (10b)- (10d) in terms of Ψ is given by Ψ = UV T, where U R N N and V R N N are orthogonal matrices taken from the singular value decomposition (SVD) of the matrix XS T = UΣV T [1]. When keeping the Ψ fixed, the solution to (10a), (10c) and (10d) in terms of S is given (as could be expected) by S = (Ψ T X) Z, where denotes Hadamard s (element-wise) product, and where in case Z contains unresolved elements (φ), these elements are taken as 1. Namely, given Ψ, the recovered sources are the result of unmixing X, limited to the recovered supports. The problem as formulated in (8) is not convex, and thus the system (10a)-(10d) only guarantees a stationary point. Moreover, the alternating approach does not guarantee convergence to a true minimum. Nonetheless, empirically (according to our experience), algorithm 4 always converged to an optimal point of (8). In many cases S is recovered correctly, even when its support is only partially recovered in the initial stage. Algorithm 4 Recover values Input: X, support pattern Z Output: Ψ, Ŝ Initialization: Ψ as any N N orthogonal matrix repeat 1. Ŝ = ( Ψ T X) Z 2. Decompose XŜT = UΣV T using SVD 3. Ψ = UV T until No change of K-SVD to an orthogonal solution. The sparsity-level input to the original algorithm was K = 2 in both experiments: In the first experiment it is the true constant sparsity level, and in the second experiment it was observed to be the most common sparsity level in the mixtures and yielded best results for that algorithm. For the second experiment we also compare the results tho those of JADE [13], a classical ICA-based algorithm, which exploits the sources statistical independence. Although the independence assumption happens to be quite justified in this experiment, the JADE results are seen to be inferior to those of our structural sparsity based approach. Fig. 4. Experiment 1: Recovery percentage of the original algorithm, of the proposed algorithm and of K-SVD vsṫhe number of columns. B indicates the block-lengths: one (no blocks) or two. In the first experiment a K-sparse matrix S R N T is constructed by selecting the locations of the non zeros randomly for each column, then drawing the non zero values from a standard Normal distribution. Another N N matrix is drawn from a standard Normal distribution, and we use the left orthogonal matrix in its SVD as the orthogonal mixing matrix Ψ, yielding the mixture X = ΨS R N T. The four-stages process described above (block-pruning, estimation of sparsity-levels, support-recovery and sources-recovery) is applied to the measurements matrix X, and the recovered matrices Ψ, Ŝ are compared to the original pair. For experiment 1, success is counted if the Frobenius norms of all of the following matrixdifferences (after resolving sign ambiguities and row permutations) are smaller than 10 3 : Ŝ S, Ψ Ψ and X ΨŜ. The successrates of Experiment 1 (averaged over 200 independent trials) are shown in Figure 4. Separation results of Experiment 2 are shown in Figure CONCLUSION 7. EXPERIMENTAL RESULTS In order to examine and compare the overall performance of the proposed algorithm, we ran two experiments: In the first experiment we examine the performance of the algorithm with N = 12 sources with a fixed sparsity level K t = K = 2, with and without blocks. In the second experiment we show the recovery rate for a mixture of N = 12 natural sparse text images. In both experiments we compare our method to the original algorithm in [1], as well as to K-SVD [11], a leading dictionary learning algorithm. K-SVD was slightly modified by adding an orthogonalization stage between every 5 consecutive iterations, this choice was found to provide fast convergence We extended the method proposed by Mishali and Eldar in [1] for sparse source separation with an orthogonal mixing matrix. We exploit the knowledge from the correlation indicators matrix in order to recover the sparsity level, blocks dimensions and the support of each column of the mixed matrix. This knowledge enables us to derive a set of linear equations (at the size of the sparsity level of the original data matrix), without relying on statistical properties of the sources (such as mutual independence, as common in ICA), but only on their spatial sparsity. The improvement over the original algorithm of [1], as well as over other competing methods, was demonstrated in simulation. The algorithm can be applicable not only in the context of BSS, but also for dictionary learning (DL) problems.

6 Fig. 3. Experiment 2: From left to right: One of the mixed images (first), recovered image using the proposed algorithm (second), recovered image using JADE (third), recovered image using K-SVD (fourth), recovered image using original algorithm (fifth). For each algorithm we chose to show the one recovered image (among the 12) that best reflects the common quality of recovered images by that algorithm. 9. REFERENCES [1] M. Mishali and Y. C. Eldar, Sparse source separation from orthogonal mixtures, in IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP 2009), 2009, pp [2] T. E. Abrudan, J. Eriksson, and V. Koivunen, Steepest descent algorithms for optimization under unitary matrix constraint, IEEE Trans. Signal Process, vol. 56, no. 3, pp , [3] P. Georgiev, F. Theis, and A. Cichocki, Blind source separation and sparse component analysis of overcomplete mixtures, in ICASSP, vol. 5, [4] R. Gribonval and S. Lesage, A survey of sparse component analysis for blind source separation: Principles, perspectives, and new challenges,, in Proceedings of ESANN, vol. 6, 2006, pp [5] P. Bofill and M. Zibulevsky, Underdetermined blind source separation using sparse representations, Signal Processing, vol. 1, no. 1, pp , [6] Y. Li, A. Cichocki, and S. Amari, Analysis of sparse representation and blind source separation, Neural Computation, vol. 16, no. 6, pp , [7] M. Stojnic, F. Parvaresh, and B. Hassibi, On the reconstruction of block-sparse signals with an optimal number of measurements, Signal Processing, IEEE Transactions on, vol. 57, no. 8, pp , [8] Z. Ben-Haim and Y. Eldar, Near-oracle performance of greedy block-sparse estimation techniques from noisy measurements, Selected Topics in Signal Processing, IEEE Journal of, vol. 5, no. 5, pp , [9] E. Elhamifar and R. Vidal, Block-sparse recovery via convex optimization, Signal Processing, IEEE Transactions on, vol. 60, no. 8, pp , [10] Z. Zhang and B. Rao, Extension of sbl algorithms for the recovery of block sparse signals with intra-block correlation, Signal Processing, IEEE Transactions on, vol. 61, no. 8, pp , [11] M. Aharon, M. Elad, and A. M. Bruckstein, On the uniqueness of overcomplete dictionaries, and a practical way to retrieve them, Linear Algebra and Its Applications, vol. 416, no. 1, pp , [12] M. Mishali and Y. C. Eldar, Sparse source separation from orthogonal mixtures, CCIT Report no.704, EE Dept., Techion - Israel Institute of Technology, Sept [13] J.-F. Cardoso and A. Souloumiac, Blind beamforming for non Gaussian signals, IEE Proceedings-F, vol. 140, no. 6, pp , Dec