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Elad M 2003 Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization
 Proc. Natl Acad. Sci. USA 100 2197–202
"... Given a ‘dictionary ’ D = {dk} of vectors dk, we seek to represent a signal S as a linear combination S = ∑ k γ(k)dk, with scalar coefficients γ(k). In particular, we aim for the sparsest representation possible. In general, this requires a combinatorial optimization process. Previous work considere ..."
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Cited by 382 (32 self)
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Given a ‘dictionary ’ D = {dk} of vectors dk, we seek to represent a signal S as a linear combination S = ∑ k γ(k)dk, with scalar coefficients γ(k). In particular, we aim for the sparsest representation possible. In general, this requires a combinatorial optimization process. Previous work considered the special case where D is an overcomplete system consisting of exactly two orthobases, and has shown that, under a condition of mutual incoherence of the two bases, and assuming that S has a sufficiently sparse representation, this representation is unique and can be found by solving a convex optimization problem: specifically, minimizing the ℓ1 norm of the coefficients γ. In this paper, we obtain parallel results in a more general setting, where the dictionary D can arise from two or several bases, frames, or even less structured systems. We introduce the Spark, ameasure of linear dependence in such a system; it is the size of the smallest linearly dependent subset (dk). We show that, when the signal S has a representation using less than Spark(D)/2 nonzeros, this representation is necessarily unique.
Stable recovery of sparse overcomplete representations in the presence of noise
 IEEE TRANS. INFORM. THEORY
, 2006
"... Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes t ..."
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Cited by 309 (20 self)
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Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes the possibility of stable recovery under a combination of sufficient sparsity and favorable structure of the overcomplete system. Considering an ideal underlying signal that has a sufficiently sparse representation, it is assumed that only a noisy version of it can be observed. Assuming further that the overcomplete system is incoherent, it is shown that the optimally sparse approximation to the noisy data differs from the optimally sparse decomposition of the ideal noiseless signal by at most a constant multiple of the noise level. As this optimalsparsity method requires heavy (combinatorial) computational effort, approximation algorithms are considered. It is shown that similar stability is also available using the basis and the matching pursuit algorithms. Furthermore, it is shown that these methods result in sparse approximation of the noisy data that contains only terms also appearing in the unique sparsest representation of the ideal noiseless sparse signal.
Uncertainty Principles and Ideal Atomic Decomposition
"... is a superposition of atoms taken from a combined time–frequency dictionary made of spike sequences I a and sinusoids �� � P A. Can one recover, from knowledge of alone, the precise collection of atoms going to make up? Because every discretetime signal can be represented as a superposition of spik ..."
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is a superposition of atoms taken from a combined time–frequency dictionary made of spike sequences I a and sinusoids �� � P A. Can one recover, from knowledge of alone, the precise collection of atoms going to make up? Because every discretetime signal can be represented as a superposition of spikes alone, or as a superposition of sinusoids alone, there is no unique way of writing as a sum of spikes and sinusoids in general. We prove that if is representable as a highly sparse superposition of atoms from this time–frequency dictionary, then there is only one such highly sparse representation of, and it can be obtained by solving the convex optimization problem of minimizing the I norm of the coefficients among all decompositions. Here “highly sparse ” means that C P where is the number of time atoms, is the number of frequency atoms, and