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814
Signal recovery from random measurements via Orthogonal Matching Pursuit
 IEEE TRANS. INFORM. THEORY
, 2007
"... This technical report demonstrates theoretically and empirically that a greedy algorithm called Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. This is a massive improvement over previous ..."
Abstract

Cited by 802 (9 self)
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This technical report demonstrates theoretically and empirically that a greedy algorithm called Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. This is a massive improvement over
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 combin ..."
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Cited by 427 (36 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
Sparsity and Incoherence in Compressive Sampling
, 2006
"... We consider the problem of reconstructing a sparse signal x 0 ∈ R n from a limited number of linear measurements. Given m randomly selected samples of Ux 0, where U is an orthonormal matrix, we show that ℓ1 minimization recovers x 0 exactly when the number of measurements exceeds m ≥ Const · µ 2 (U) ..."
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Cited by 238 (13 self)
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) · S · log n, where S is the number of nonzero components in x 0, and µ is the largest entry in U properly normalized: µ(U) = √ n · maxk,j Uk,j. The smaller µ, the fewer samples needed. The result holds for “most ” sparse signals x 0 supported on a fixed (but arbitrary) set T. Given T, if the sign of x 0
Parallel Preconditioning with Sparse Approximate Inverses
 SIAM J. Sci. Comput
, 1996
"... A parallel preconditioner is presented for the solution of general sparse linear systems of equations. A sparse approximate inverse is computed explicitly, and then applied as a preconditioner to an iterative method. The computation of the preconditioner is inherently parallel, and its application o ..."
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Cited by 226 (10 self)
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only requires a matrixvector product. The sparsity pattern of the approximate inverse is not imposed a priori but captured automatically. This keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum. Rigorous bounds on the clustering of the eigenvalues
Signal recovery from partial information via Orthogonal Matching Pursuit
 IEEE TRANS. INFORM. THEORY
, 2005
"... This article demonstrates theoretically and empirically that a greedy algorithm called Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. This is a massive improvement over previous results ..."
Abstract

Cited by 191 (8 self)
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This article demonstrates theoretically and empirically that a greedy algorithm called Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. This is a massive improvement over previous
Coil sensitivity encoding for fast MRI. In:
 Proceedings of the ISMRM 6th Annual Meeting,
, 1998
"... New theoretical and practical concepts are presented for considerably enhancing the performance of magnetic resonance imaging (MRI) by means of arrays of multiple receiver coils. Sensitivity encoding (SENSE) is based on the fact that receiver sensitivity generally has an encoding effect complementa ..."
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Cited by 193 (3 self)
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n K . Assembling sample and image values in vectors, image reconstruction may be rewritten in matrix notation: With such linear mapping the propagation of noise from sample values into image values is conveniently described by noise matrices. The th diagonal entry of the image noise matrix X
Sparse matrices in Matlab: Design and implementation
, 1991
"... We have extended the matrix computation language and environment Matlab to include sparse matrix storage and operations. The only change to the outward appearance of the Matlab language is a pair of commands to create full or sparse matrices. Nearly all the operations of Matlab now apply equally to ..."
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Cited by 164 (22 self)
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to full or sparse matrices, without any explicit action by the user. The sparse data structure represents a matrix in space proportional to the number of nonzero entries, and most of the operations compute sparse results in time proportionaltothenumber of arithmetic operations on nonzeros.
Adapting to unknown sparsity by controlling the false discovery rate
, 2000
"... We attempt to recover a highdimensional vector observed in white noise, where the vector is known to be sparse, but the degree of sparsity is unknown. We consider three different ways of defining sparsity of a vector: using the fraction of nonzero terms; imposing powerlaw decay bounds on the order ..."
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Cited by 183 (23 self)
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We attempt to recover a highdimensional vector observed in white noise, where the vector is known to be sparse, but the degree of sparsity is unknown. We consider three different ways of defining sparsity of a vector: using the fraction of nonzero terms; imposing powerlaw decay bounds
Lower Bound Theory of Nonzero Entries in Solutions of ℓ2ℓp Minimization
, 2009
"... Abstract. Recently, variable selection and sparse reconstruction are solved by finding an optimal solution of a minimization model where the objective function is the sum of a datafitting term in ℓ2 norm and a regularization term in ℓp norm (0 < p < 1). In this model, being able to classify ze ..."
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Cited by 25 (6 self)
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zero and nonzero entries in its local solutions is a very important task. However, most algorithms for solving the problem can only provide an approximate local optimal solution, where nonzero entries in the solution cannot be identified theoretically. In this paper, we establish lower bounds
Fast Computation of Low Rank Matrix Approximations
, 2001
"... In many practical applications, given an m n matrix A it is of interest to nd an approximation to A that has low rank. We introduce a technique that exploits spectral structure in A to accelerate Orthogonal Iteration and Lanczos Iteration, the two most common methods for computing such approximat ..."
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Cited by 165 (5 self)
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such approximations. Our technique amounts to independently sampling and/or quantizing the entries of the input matrix A, thus speeding up computation by reducing the number of nonzero entries and/or the length of their representation. Our analysis s based on observing that both sampling and quantization can
Results 1  10
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814