Sparse representations in unions of bases

By Rémi Gribonval, Morten Nielsen. IEEE Transactions on Information Theory, 2003

1,012 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics, Signal Processing

View Paper: https://www.semanticscholar.org/paper/c6a377d34d84a1c988070e193897623bb02dba69

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Citations in the corpus

Sparse nonnegative solution of underdetermined linear equations by linear programming. (2005, background): “There is very active research on such problems; for a sampling of different perspectives, see [3, 9, 12, 13, 17, 26, 23] and related work.”
Decoding by linear programming (2005, methodology): “Using very diﬀerent ideas and together with Romberg [4], the authors proved that the equivalence holds with overwhelming probability for various types of random matrices provided that provided that the number of nonzero entries in the vector e be of the order of m/ log m [5, 6].”
Sparse Solution of Underdetermined Systems of Linear Equations by Stagewise Orthogonal Matching Pursuit (2012, methodology): “As the papers [18] and [19] may help clarify, much of the literature discusses the notion of strong equivalence of and ; in this notion that for a given , every sparse generates a problem for which the solution is the unique sparsest solution [8], [9], [16], [20], [21], [32], [35], [48], [55].”
An efficient algorithm for total variation regularization with applications to the single pixel camera and compressive sensing (2010, background): “The question why £i minimization could work in some special setups was further investigated and answered in a series of paper [10, 11, 12, 13, 14, 15].”
Theoretical Results on Sparse Representations of Multiple-Measurement Vectors (2006, background): “The introduction of Donoho, Elad, and Temlyakov [15] gives a comprehensive depiction on many important applications.”
For most large underdetermined systems of equations, the minimal 𝓁1‐norm near‐solution approximates the sparsest near‐solution (2006, background): “In recent years, there has been rapid development in the theory of sparse overcomplete signal representations [5] [13] [14] [29] [30] [16].”
Recovery of exact sparse representations in the presence of bounded noise (2005, background): “[ 4 ] T. Kailath, “Application of a resolvent identity to a linear smoothing”
Approximation accuracy, gradient methods, and error bound for structured convex optimization (2010, background): “This μ is central to the analysis in [40, 41, 42, 56, 57, 59, 62, 83, 133, 134, 143].”
The null space property for sparse recovery from multiple measurement vectors (2011, positive): “[14] and [20]) for the standard 1 minimization has been extended to this setting (2) for multiple measurement vectors.”
High-Resolution Radar via Compressed Sensing (2008, positive): “Proof: Theorem B in [7] (which incorporates results from [27], [28], and [ 29 ]) concludes for general dictionary that every -sparse signal with is the unique sparsest representation, and is guaranteed to be recovered by both BP and OMP when observing . Set and assume the hypothesis of Theorem 1. Eq”
Structured Compressed Sensing: From Theory to Applications (2011, background): “Theorem 4 [28], [30], [31]: then for each measurement vector there exists at most one signal such that .”
New Algorithms for Learning Incoherent and Overcomplete Dictionaries (2013, background): “The pioneering work of [18], [17] and [29] (building on the uncertainty principle of [19]) presented a number of important examples (in fact, the ones we used above) of dictionaries that are incoherent and showed that ℓ 1 -minimization can ﬁnd a sparse representation in a known, incoherent…”
Image Compressed Sensing Using Convolutional Neural Network (2020, background): “to determine the distribution of nonzero elements of binary sampling matrix by minimizing the coherence parameter [38]”
The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing (2012, background): “r tool for guaranteeing ℓ 0-ℓ 1-equivalence is thenullspace property (NSP), see, e.g., [24], [25], [26], which characterizes recoverability by (P 1) (in fact, via ℓp-minimization with0< p ≤ 1, see [27], [28]) for sufﬁciently sparse solutions of (P 0). The NSP of order k is satisﬁed with constant ”
On the Reconstruction of Block-Sparse Signals With an Optimal Number of Measurements (2008, methodology): “Then we have Given the symmetry of the problem (i.e. the rotaional invariance of the Z i ) it should be noted that, without loss of generality, we can assume z t = [ z t || 2 , 0, 0, . . . of || Further, using the results from [23] we have that η d ( n − m ) − 1 points can be located on the sphere…”
Sparse Solution Of Underdetermined Linear Equations By Stagewise Orthogonal Matching Pursuit (2006, background): “…some particularly striking theoretical properties, such as rigorous proofs of exact reconstruction under seemingly quite general circumstances [21] [35] [32] [7] [16] [8] [17] [18] Unfortunately, some of the most powerful theoretical results are associated with fairly heavy computationally…”