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70
Computational methods for sparse solution of linear inverse problems
, 2009
"... The goal of sparse approximation problems is to represent a target signal approximately as a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, ..."
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Cited by 60 (0 self)
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The goal of sparse approximation problems is to represent a target signal approximately as a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, to the circumstances in which individual methods tend to perform well, and to the theoretical guarantees available. Many fundamental questions in electrical engineering, statistics, and applied mathematics can be posed as sparse approximation problems, making these algorithms versatile and relevant to a wealth of applications.
Sparse recovery using sparse random matrices
, 2008
"... We consider the approximate sparse recovery problem, where the goal is to (approximately) recover a highdimensional vector x from its lowerdimensional sketch Ax. A popular way of performing this recovery is by finding x # such that Ax = Ax # , and �x # �1 is minimal. It is known that this approach ..."
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Cited by 41 (4 self)
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We consider the approximate sparse recovery problem, where the goal is to (approximately) recover a highdimensional vector x from its lowerdimensional sketch Ax. A popular way of performing this recovery is by finding x # such that Ax = Ax # , and �x # �1 is minimal. It is known that this approach “works” if A is a random dense matrix, chosen from a proper distribution. In this paper, we investigate this procedure for the case where A is binary and very sparse. We show that, both in theory and in practice, sparse matrices are essentially as “good” as the dense ones. At the same time, sparse binary matrices provide additional benefits, such as reduced encoding and decoding time.
Observed universality of phase transitions in highdimensional geometry, with implications for modern data analysis and signal
, 1906
"... Abstract. We review connections between phase transitions in highdimensional combinatorial geometry and phase transitions occurring in modern highdimensional data analysis and signal processing. In data analysis, such transitions arise as abrupt breakdown of linear model selection, robust data fit ..."
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Cited by 31 (6 self)
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Abstract. We review connections between phase transitions in highdimensional combinatorial geometry and phase transitions occurring in modern highdimensional data analysis and signal processing. In data analysis, such transitions arise as abrupt breakdown of linear model selection, robust data fitting or compressed sensing reconstructions, when the complexity of the model or the number of outliers increases beyond a threshold. In combinatorial geometry these transitions appear as abrupt changes in the properties of face counts of convex polytopes when the dimensions are varied. The thresholds in these very different problems appear in the same critical locations after appropriate calibration of variables. These thresholds are important in each subject area: for linear modelling, they place hard limits on the degree to which the nowubiquitous highthroughput data analysis can be successful; for robustness, they place hard limits on the degree to which standard robust fitting methods can tolerate outliers before breaking down; for compressed sensing, they define the sharp boundary of the
Lower Bounds for Sparse Recovery
"... We consider the following ksparse recovery problem: design an m × n matrix A, such that for any signal x, given Ax we can efficiently recover ˆx satisfying ..."
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Cited by 29 (14 self)
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We consider the following ksparse recovery problem: design an m × n matrix A, such that for any signal x, given Ax we can efficiently recover ˆx satisfying
1 Sparse Recovery Using Sparse Matrices
"... Abstract—We survey algorithms for sparse recovery problems that are based on sparse random matrices. Such matrices has several attractive properties: they support algorithms with low computational complexity, and make it easy to perform incremental updates to signals. We discuss applications to seve ..."
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Cited by 26 (7 self)
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Abstract—We survey algorithms for sparse recovery problems that are based on sparse random matrices. Such matrices has several attractive properties: they support algorithms with low computational complexity, and make it easy to perform incremental updates to signals. We discuss applications to several areas, including compressive sensing, data stream computing and group testing. I.
Compressive Sensing
, 2010
"... Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1minimization can be used for recovery. The theory has many poten ..."
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Cited by 23 (8 self)
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Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1minimization can be used for recovery. The theory has many potential applications in signal processing and imaging. This chapter gives an introduction and overview on both theoretical and numerical aspects of compressive sensing.
Dense error correction via ℓ1 minimization
, 2009
"... This paper studies the problem of recovering a nonnegative sparse signal x ∈ Rn from highly corrupted linear measurements y = Ax + e ∈ Rm, where e is an unknown error vector whose nonzero entries may be unbounded. Motivated by an observation from face recognition in computer vision, this paper prov ..."
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Cited by 17 (5 self)
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This paper studies the problem of recovering a nonnegative sparse signal x ∈ Rn from highly corrupted linear measurements y = Ax + e ∈ Rm, where e is an unknown error vector whose nonzero entries may be unbounded. Motivated by an observation from face recognition in computer vision, this paper proves that for highly correlated (and possibly overcomplete) dictionaries A, any nonnegative, sufficiently sparse signal x can be recovered by solving an ℓ1minimization problem: min ‖x‖1 + ‖e‖1 subject to y = Ax + e. More precisely, if the fraction ρ of errors is bounded away from one and the support of x grows sublinearly in the dimension m of the observation, then as m goes to infinity, the above ℓ1minimization succeeds for all signals x and almost all signandsupport patterns of e. This result suggests that accurate recovery of sparse signals is possible and computationally feasible even with nearly 100 % of the observations corrupted. The proof relies on a careful characterization of the faces of a convex polytope spanned together by the standard crosspolytope and a set of iid Gaussian vectors with nonzero mean and small variance, which we call the “crossandbouquet ” model. Simulations and experimental results corroborate the findings, and suggest extensions to the result.
Precise Undersampling Theorems
"... Undersampling Theorems state that we may gather far fewer samples than the usual sampling theorem while exactly reconstructing the object of interest – provided the object in question obeys a sparsity condition, the samples measure appropriate linear combinations of signal values, and we reconstruc ..."
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Cited by 17 (2 self)
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Undersampling Theorems state that we may gather far fewer samples than the usual sampling theorem while exactly reconstructing the object of interest – provided the object in question obeys a sparsity condition, the samples measure appropriate linear combinations of signal values, and we reconstruct with a particular nonlinear procedure. While there are many ways to crudely demonstrate such undersampling phenomena, we know of only one approach which precisely quantifies the true sparsityundersampling tradeoff curve of standard algorithms and standard compressed sensing matrices. That approach, based on combinatorial geometry, predicts the exact location in sparsityundersampling domain where standard algorithms exhibit phase transitions in performance. We review the phase transition approach here and describe the broad range of cases where it applies. We also mention exceptions and state challenge problems for future research. Sample result: one can efficiently reconstruct a ksparse signal of length N from n measurements, provided n � 2k · log(N/n), for (k, n, N) large, k ≪ N.
Calderbank R., Efficient and Robust Compressive Sensing using HighQuality Expander Graphs. Submitted to the IEEE transaction on Information Theory
, 2008
"... Abstract—Expander graphs have been recently proposed to construct efficient compressed sensing algorithms. In particular, it has been shown that any ndimensional vector that is ksparse (with k ≪ n) can be fully recovered using O(k log n k) measurements and only O(k log n) simple recovery iteration ..."
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Cited by 12 (1 self)
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Abstract—Expander graphs have been recently proposed to construct efficient compressed sensing algorithms. In particular, it has been shown that any ndimensional vector that is ksparse (with k ≪ n) can be fully recovered using O(k log n k) measurements and only O(k log n) simple recovery iterations. In this paper we improve upon this result by considering expander graphs with expansion coefficient beyond 3 and show that, with 4 the same number of measurements, only O(k) recovery iterations are required, which is a significant improvement when n is large. In fact, full recovery can be accomplished by at most 2k very simple iterations. The number of iterations can be made arbitrarily close to k, and the recovery algorithm can be implemented very efficiently using a simple binary search tree. We also show that by tolerating a small penalty on the number of measurements, and not on the number of recovery iterations, one can use the efficient construction of a family of expander graphs to come up with explicit measurement matrices for this method. We compare our result with other recently developed expandergraphbased methods and argue that it compares favorably both in terms of the number of required measurements and in terms of the recovery time complexity. Finally we will show how our analysis extends to give a robust algorithm that finds the position and sign of the k significant elements of an almost ksparse signal and then, using very simple optimization techniques, finds in sublinear time a ksparse signal which approximates the original signal with very high precision. I.