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38
Robust Recovery of Signals From a Structured Union of Subspaces
, 2008
"... Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structu ..."
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Cited by 213 (49 self)
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Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which x lies in a union of subspaces. In this paper we develop a general framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which x lies in a sum of k subspaces, chosen from a larger set of m possibilities. The samples are modelled as inner products with an arbitrary set of sampling functions. To derive an efficient and robust recovery algorithm, we show that our problem can be formulated as that of recovering a blocksparse vector whose nonzero elements appear in fixed blocks. We then propose a mixed ℓ2/ℓ1 program for block sparse recovery. Our main result is an equivalence condition under which the proposed convex algorithm is guaranteed to recover the original signal. This result relies on the notion of block restricted isometry property (RIP), which is a generalization of the standard RIP used extensively in the context of compressed sensing. Based on RIP we also prove stability of our approach in the presence of noise and modeling errors. A special case of our framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Adapting our results to this context leads to new MMV recovery methods as well as equivalence conditions under which the entire set can be determined efficiently.
Average Case Analysis of Multichannel Sparse Recovery Using Convex Relaxation
"... In this paper, we consider recovery of jointly sparse multichannel signals from incomplete measurements. Several approaches have been developed to recover the unknown sparse vectors from the given observations, including thresholding, simultaneous orthogonal matching pursuit (SOMP), and convex relax ..."
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Cited by 98 (23 self)
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In this paper, we consider recovery of jointly sparse multichannel signals from incomplete measurements. Several approaches have been developed to recover the unknown sparse vectors from the given observations, including thresholding, simultaneous orthogonal matching pursuit (SOMP), and convex relaxation based on a mixed matrix norm. Typically, worstcase analysis is carried out in order to analyze conditions under which the algorithms are able to recover any jointly sparse set of vectors. However, such an approach is not able to provide insights into why joint sparse recovery is superior to applying standard sparse reconstruction methods to each channel individually. Previous work considered an average case analysis of thresholding and SOMP by imposing a probability model on the measured signals. In this paper, our main focus is on analysis of convex relaxation techniques. In particular, we focus on the mixed ℓ2,1 approach to multichannel recovery. We show that under a very mild condition on the sparsity and on the dictionary characteristics, measured for example by the coherence, the probability of recovery failure decays exponentially in the number of channels. This demonstrates that most of the time, multichannel sparse recovery is indeed superior to single channel methods. Our probability bounds are valid and meaningful even for a small number of signals. Using the tools we develop to analyze the convex relaxation method, we also tighten the previous bounds for thresholding and SOMP.
Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity
, 2010
"... A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAPEM algorithm. A dual mathematical interpretation of the proposed framework with structured sparse estimation is describe ..."
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Cited by 48 (8 self)
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A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAPEM algorithm. A dual mathematical interpretation of the proposed framework with structured sparse estimation is described, which shows that the resulting piecewise linear estimate stabilizes the estimation when compared to traditional sparse inverse problem techniques. This interpretation also suggests an effective dictionary motivated initialization for the MAPEM algorithm. We demonstrate that in a number of image inverse problems, including inpainting, zooming, and deblurring, the same algorithm produces either equal, often significantly better, or very small margin worse results than the best published ones, at a lower computational cost. 1 I.
CoherenceBased Performance Guarantees for Estimating a Sparse Vector Under Random Noise
"... We consider the problem of estimating a deterministic sparse vector x0 from underdetermined measurements Ax0 + w, where w represents white Gaussian noise and A is a given deterministic dictionary. We analyze the performance of three sparse estimation algorithms: basis pursuit denoising (BPDN), orth ..."
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Cited by 42 (15 self)
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We consider the problem of estimating a deterministic sparse vector x0 from underdetermined measurements Ax0 + w, where w represents white Gaussian noise and A is a given deterministic dictionary. We analyze the performance of three sparse estimation algorithms: basis pursuit denoising (BPDN), orthogonal matching pursuit (OMP), and thresholding. These algorithms are shown to achieve nearoracle performance with high probability, assuming that x0 is sufficiently sparse. Our results are nonasymptotic and are based only on the coherence of A, so that they are applicable to arbitrary dictionaries. Differences in the precise conditions required for the performance guarantees of each algorithm are manifested in the observed performance at high and low signaltonoise ratios. This provides insight on the advantages and drawbacks of ℓ1 relaxation techniques such as BPDN as opposed to greedy approaches such as OMP and thresholding.
Compressive Sensing on Manifolds Using a Nonparametric Mixture of Factor Analyzers: Algorithm and Performance Bounds 1
"... Nonparametric Bayesian methods are employed to constitute a mixture of lowrank Gaussians, for data x ∈ RN that are of high dimension N but are constrained to reside in a lowdimensional subregion of RN. The number of mixture components and their rank are inferred automatically from the data. The re ..."
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Cited by 40 (15 self)
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Nonparametric Bayesian methods are employed to constitute a mixture of lowrank Gaussians, for data x ∈ RN that are of high dimension N but are constrained to reside in a lowdimensional subregion of RN. The number of mixture components and their rank are inferred automatically from the data. The resulting algorithm can be used for learning manifolds and for reconstructing signals from manifolds, based on compressive sensing (CS) projection measurements. The statistical CS inversion is performed analytically. We derive the required number of CS random measurements needed for successful reconstruction, based on easily computed quantities, drawing on block–sparsity properties. The proposed methodology is validated on several synthetic and real datasets. I.
Various thresholds for ℓ1optimization in compressed sensing
, 2009
"... Recently, [14, 28] theoretically analyzed the success of a polynomial ℓ1optimization algorithm in solving an underdetermined system of linear equations. In a large dimensional and statistical context [14, 28] proved that if the number of equations (measurements in the compressed sensing terminolog ..."
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Cited by 33 (17 self)
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Recently, [14, 28] theoretically analyzed the success of a polynomial ℓ1optimization algorithm in solving an underdetermined system of linear equations. In a large dimensional and statistical context [14, 28] proved that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of the unknown vector then there is a sparsity (number of nonzero elements of the unknown vector) also proportional to the length of the unknown vector such that ℓ1optimization succeeds in solving the system. In this paper, we provide an alternative performance analysis of ℓ1optimization and obtain the proportionality constants that in certain cases match or improve on the best currently known ones from [28, 29].
Blocksparsity: Coherence and efficient recovery
 In ICASSP, The International Conference on Acoustics, Signal and Speech Processing
, 2009
"... We consider compressed sensing of blocksparse signals, i.e., sparse signals that have nonzero coefficients occuring in clusters. Based on an uncertainty relation for blocksparse signals, we define a blockcoherence measure and we show that a blockversion of the orthogonal matching pursuit algorit ..."
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Cited by 32 (5 self)
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We consider compressed sensing of blocksparse signals, i.e., sparse signals that have nonzero coefficients occuring in clusters. Based on an uncertainty relation for blocksparse signals, we define a blockcoherence measure and we show that a blockversion of the orthogonal matching pursuit algorithm recovers block ksparse signals in no more than k steps if the blockcoherence is sufficiently small. The same condition on blocksparsity is shown to guarantee successful recovery through a mixed ℓ2/ℓ1 optimization approach. The significance of the results lies in the fact that making explicit use of blocksparsity can yield better reconstruction properties than treating the signal as being sparse in the conventional sense thereby ignoring the additional structure in the problem. Index Terms — block sparsity, coherence, uncertainty relations 1.
Robust classification using structured sparse representation
 In CVPR
, 2011
"... In many problems in computer vision, data in multiple classes lie in multiple lowdimensional subspaces of a highdimensional ambient space. However, most of the existing classification methods do not explicitly take this structure into account. In this paper, we consider the problem of classificatio ..."
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Cited by 28 (5 self)
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In many problems in computer vision, data in multiple classes lie in multiple lowdimensional subspaces of a highdimensional ambient space. However, most of the existing classification methods do not explicitly take this structure into account. In this paper, we consider the problem of classification in the multisubspace setting using sparse representation techniques. We exploit the fact that the dictionary of all the training data has a block structure where the training data in each class form few blocks of the dictionary. We cast the classification as a structured sparse recovery problem where our goal is to find a representation of a test example that uses the minimum number of blocks from the dictionary. We formulate this problem using two different classes of nonconvex optimization programs. We propose convex relaxations for these two nonconvex programs and study conditions under which the relaxations are equivalent to the original problems. In addition, we show that the proposed optimization programs can be modified properly to also deal with corrupted data. To evaluate the proposed algorithms, we consider the problem of automatic face recognition. We show that casting the face recognition problem as a structured sparse recovery problem can improve the results of the stateoftheart face recognition algorithms, especially when we have relatively small number of training data for each class. In particular, we show that the new class of convex programs can improve the stateoftheart face recognition results by 10 % with only 25 % of the training data. In addition, we show that the algorithms are robust to occlusion, corruption, and disguise. 1.