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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 291 (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.
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images
, 2007
"... A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinato ..."
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Cited by 202 (31 self)
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A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easilyverifiable conditions under which optimallysparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several wellknown signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable. There is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems energizes research on such signal and image processing problems – to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical
Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit
, 2006
"... Finding the sparsest solution to underdetermined systems of linear equations y = Φx is NPhard in general. We show here that for systems with ‘typical’/‘random ’ Φ, a good approximation to the sparsest solution is obtained by applying a fixed number of standard operations from linear algebra. Our pr ..."
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Cited by 172 (20 self)
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Finding the sparsest solution to underdetermined systems of linear equations y = Φx is NPhard in general. We show here that for systems with ‘typical’/‘random ’ Φ, a good approximation to the sparsest solution is obtained by applying a fixed number of standard operations from linear algebra. Our proposal, Stagewise Orthogonal Matching Pursuit (StOMP), successively transforms the signal into a negligible residual. Starting with initial residual r0 = y, at the sth stage it forms the ‘matched filter ’ Φ T rs−1, identifies all coordinates with amplitudes exceeding a speciallychosen threshold, solves a leastsquares problem using the selected coordinates, and subtracts the leastsquares fit, producing a new residual. After a fixed number of stages (e.g. 10), it stops. In contrast to Orthogonal Matching Pursuit (OMP), many coefficients can enter the model at each stage in StOMP while only one enters per stage in OMP; and StOMP takes a fixed number of stages (e.g. 10), while OMP can take many (e.g. n). StOMP runs much faster than competing proposals for sparse solutions, such as ℓ1 minimization and OMP, and so is attractive for solving largescale problems. We use phase diagrams to compare algorithm performance. The problem of recovering a ksparse vector x0 from (y, Φ) where Φ is random n × N and y = Φx0 is represented by a point (n/N, k/n)
Sparse Representations are Most Likely to be the Sparsest Possible
 EURASIP Journal on Applied Signal Processing, Paper No. 96247
, 2004
"... and a full rank matrix D with N < L, we define the signal's overcomplete representations as all # satisfying S = D#. Among all the possible solutions, we have special interest in the sparsest one  the one minimizing 0 . Previous work has established that a representation is unique if i ..."
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Cited by 12 (2 self)
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and a full rank matrix D with N < L, we define the signal's overcomplete representations as all # satisfying S = D#. Among all the possible solutions, we have special interest in the sparsest one  the one minimizing 0 . Previous work has established that a representation is unique if it is sparse enough, requiring 0 < Spark(D)/2.
Sparse solution of underdetermined linear systems: algorithms and applications
, 2007
"... in my opinion, it ..."