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...as been devoted to the sparse representation model, which stems from the observation that many signals can be approximated using a small number of elements, or “atoms,” chosen from a large dictionary =-=[1]-=-, [2]. Thus, we may write , where the signal is a linear combination of a small number of columns of the dictionary matrix , corrupted by noise . Since only a small number of elements of are required ...

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...in assessing the quality of an estimator is to check its proximity to the best possible performance in the given setting. To this end, it is common practice to compute the CRB for unbiased estimators =-=[31]-=-, i.e., those techniques for which the bias equals zero. The CRB is a lower bound on the mean-squared error for any unbiased estimator . 1 The remaining terms in (23) are always no worse than the corr...

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...en devoted to the sparse representation model, which stems from the observation that many signals can be approximated using a small number of elements, or “atoms,” chosen from a large dictionary [1], =-=[2]-=-. Thus, we may write , where the signal is a linear combination of a small number of columns of the dictionary matrix , corrupted by noise . Since only a small number of elements of are required for t...

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...model. Apart from ordinary sparsity, unions of subspaces have been applied to estimate signals as diverse as pulse streams [9]–[12], multi-band communications [13]–[15], and block sparse vectors [8], =-=[16]-=-–[19], the latter being the focus of this paper. The common thread running through these applications is the ability to exploit the union of subspaces structure in order to achieve accurate reconstruc...

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...re relatively weak, ensuring only that the error in is on the order of [2]–[4]. By contrast, when the noise is random, estimation performance is considerably improved for most noise realizations [3], =-=[21]-=-–[23]. Even better performance can be obtained in the Bayesian case, in which itself is random with a known distribution [24]–[26]. This paper contributes to the ongoing effort to extend such performa...

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...given by (16)BEN-HAIM AND ELDAR: NEAR-ORACLE PERFORMANCE OF GREEDY BLOCK-SPARSE ESTIMATION TECHNIQUES 1035 b) Block Orthogonal Matching Pursuit (BOMP): The BOMP algorithm, based on the OMP algorithm =-=[30]-=-, was independently proposed in [20] and [18]. Given a measurement vector , perform the following steps: 1) Define . 2) For each , do the following: a) Set b) Set (17) (18) c) Set . 3) The estimate is...

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...CA 94305 USA (e-mail: yonina@ee.technion.ac.il). Digital Object Identifier 10.1109/JSTSP.2011.2160250 in many fundamental fields of signal processing, including compressed sensing [1], [2], denoising =-=[3]-=-, deblurring [5], and interpolation [6]. The assumption of sparsity is an example of a much more general class of signal models which can be described as a union of subspaces [7], [8]. Indeed, each su...

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...rsion of the CoSaMP approach under adversarial noise [19]. Some research has focused on the related problem in which the same dictionary is used to obtain measurements of a series of distinct signals =-=[29]-=-. The performance of BOMP under Gaussian noise was first addressed by Lozano et al. [20], who provided a bound on the estimation error, i.e., the maximum componentwise error, in the presence of Gaussi...

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...ons of subspaces are proving to be a powerful generalization of the sparsity model. Apart from ordinary sparsity, unions of subspaces have been applied to estimate signals as diverse as pulse streams =-=[9]-=-–[12], multi-band communications [13]–[15], and block sparse vectors [8], [16]–[19], the latter being the focus of this paper. The common thread running through these applications is the ability to ex...

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...ng effort to extend such performance guarantees to the block sparse setting. Such guarantees are already available in the context of Group Lasso in the random [27] and adversarial noise settings [8], =-=[28]-=-. Performance assurances were also derived for a block version of the CoSaMP approach under adversarial noise [19]. Some research has focused on the related problem in which the same dictionary is use...

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..., [2], denoising [3], deblurring [5], and interpolation [6]. The assumption of sparsity is an example of a much more general class of signal models which can be described as a union of subspaces [7], =-=[8]-=-. Indeed, each support pattern defines a subspace of the space of possible parameter vectors. Saying that the parameter contains no more than nonzero entries is equivalent to stating that belongs to t...

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...It turns out that the sparsity assumption can be used to accurately estimate from , even when the number of possible atoms (and thus, the length of ) is greater than the number of measurements in [2]–=-=[4]-=-. This model has been used to great advantage Manuscript received September 05, 2010; revised February 14, 2011; accepted June 09, 2011. Date of publication June 23, 2011; date of current version Augu...

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...estimators designed for the ordinary sparsity model can be readily adapted to the block sparse setting. Previous work has described techniques such as Group Lasso [16], also known as L-OPT [8], [17], =-=[18]-=-, block orthogonal matching pursuit (BOMP) [18], [20], and a block version of the CoSaMP algorithm [19]. In this paper, we also consider a block-sparse version of the thresholding algorithm, which we ...

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... powerful generalization of the sparsity model. Apart from ordinary sparsity, unions of subspaces have been applied to estimate signals as diverse as pulse streams [9]–[12], multi-band communications =-=[13]-=-–[15], and block sparse vectors [8], [16]–[19], the latter being the focus of this paper. The common thread running through these applications is the ability to exploit the union of subspaces structur...

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...tely, estimators designed for the ordinary sparsity model can be readily adapted to the block sparse setting. Previous work has described techniques such as Group Lasso [16], also known as L-OPT [8], =-=[17]-=-, [18], block orthogonal matching pursuit (BOMP) [18], [20], and a block version of the CoSaMP algorithm [19]. In this paper, we also consider a block-sparse version of the thresholding algorithm, whi...

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...g [1], [2], denoising [3], deblurring [5], and interpolation [6]. The assumption of sparsity is an example of a much more general class of signal models which can be described as a union of subspaces =-=[7]-=-, [8]. Indeed, each support pattern defines a subspace of the space of possible parameter vectors. Saying that the parameter contains no more than nonzero entries is equivalent to stating that belongs...

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...n . Observe that, for a deterministic value , the pdf defines a Gaussian random vector whose mean depends linearly on , but whose covariance is constant in . Therefore, using a result due to Anderson =-=[39]-=-, it follows that is a non-increasing function of . Next, denoting by the marginal pdf of ,wehave (83) Of the two bounds provided in (88), the first is somewhat tighter, but obviously more cumbersome....

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...d estimator . 1 The remaining terms in (23) are always no worse than the corresponding terms in (25). To utilize the information inherent in the block sparsity structure, we apply the constrained CRB =-=[32]-=-–[35] to the present setting. In the constrained estimation scenario, one often seeks estimators which are unbiased for all parameter values in the constraint set [32], [33]. However, as we will see b...

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...rful generalization of the sparsity model. Apart from ordinary sparsity, unions of subspaces have been applied to estimate signals as diverse as pulse streams [9]–[12], multi-band communications [13]–=-=[15]-=-, and block sparse vectors [8], [16]–[19], the latter being the focus of this paper. The common thread running through these applications is the ability to exploit the union of subspaces structure in ...

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... apply the constrained CRB [32]–[35] to the present setting. In the constrained estimation scenario, one often seeks estimators which are unbiased for all parameter values in the constraint set [32], =-=[33]-=-. However, as we will see below, this requirement is too strict in the block sparse setting. Indeed, in Theorem 3 we show that it is not possible to construct any method which is unbiased for all feas...

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...n.ac.il). Digital Object Identifier 10.1109/JSTSP.2011.2160250 in many fundamental fields of signal processing, including compressed sensing [1], [2], denoising [3], deblurring [5], and interpolation =-=[6]-=-. The assumption of sparsity is an example of a much more general class of signal models which can be described as a union of subspaces [7], [8]. Indeed, each support pattern defines a subspace of the...

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...of subspaces are proving to be a powerful generalization of the sparsity model. Apart from ordinary sparsity, unions of subspaces have been applied to estimate signals as diverse as pulse streams [9]–=-=[12]-=-, multi-band communications [13]–[15], and block sparse vectors [8], [16]–[19], the latter being the focus of this paper. The common thread running through these applications is the ability to exploit...

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...t factor of the CRB at high SNR, for dictionaries satisfying suitable requirements. Once again, when each block contains one element, we can recover previously known guarantees for non-block sparsity =-=[22]-=- from our results. Comparing our results with those available for ordinary or “scalar” sparsity reveals the conditions under which the block sparse approach is beneficial. Our results are a function o...

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...[26]. This paper contributes to the ongoing effort to extend such performance guarantees to the block sparse setting. Such guarantees are already available in the context of Group Lasso in the random =-=[27]-=- and adversarial noise settings [8], [28]. Performance assurances were also derived for a block version of the CoSaMP approach under adversarial noise [19]. Some research has focused on the related pr...

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...imator . 1 The remaining terms in (23) are always no worse than the corresponding terms in (25). To utilize the information inherent in the block sparsity structure, we apply the constrained CRB [32]–=-=[35]-=- to the present setting. In the constrained estimation scenario, one often seeks estimators which are unbiased for all parameter values in the constraint set [32], [33]. However, as we will see below,...

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...tion performance is considerably improved for most noise realizations [3], [21]–[23]. Even better performance can be obtained in the Bayesian case, in which itself is random with a known distribution =-=[24]-=-–[26]. This paper contributes to the ongoing effort to extend such performance guarantees to the block sparse setting. Such guarantees are already available in the context of Group Lasso in the random...

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...an be readily adapted to the block sparse setting. Previous work has described techniques such as Group Lasso [16], also known as L-OPT [8], [17], [18], block orthogonal matching pursuit (BOMP) [18], =-=[20]-=-, and a block version of the CoSaMP algorithm [19]. In this paper, we also consider a block-sparse version of the thresholding algorithm, which we refer to as block-thresholding (BTH). The BOMP and BT...

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...latively weak, ensuring only that the error in is on the order of [2]–[4]. By contrast, when the noise is random, estimation performance is considerably improved for most noise realizations [3], [21]–=-=[23]-=-. Even better performance can be obtained in the Bayesian case, in which itself is random with a known distribution [24]–[26]. This paper contributes to the ongoing effort to extend such performance g...

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...heorem. APPENDIX C PROOFS FOR GAUSSIAN NOISE We begin with two lemmas which prove some useful properties of the Gaussian distribution. The first of these is a generalization of a result due to ˇSidák =-=[38]-=-. Lemma 3: Let be a set of jointly Gaussian random vectors. Suppose that for all , but that the covariances of the vectors are unspecified and that the vectors are not necessarily independent. We then...

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...performance is considerably improved for most noise realizations [3], [21]–[23]. Even better performance can be obtained in the Bayesian case, in which itself is random with a known distribution [24]–=-=[26]-=-. This paper contributes to the ongoing effort to extend such performance guarantees to the block sparse setting. Such guarantees are already available in the context of Group Lasso in the random [27]...