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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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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
Sparse solutions to linear inverse problems with multiple measurement vectors
 IEEE Trans. Signal Processing
, 2005
"... Abstract—We address the problem of finding sparse solutions to an underdetermined system of equations when there are multiple measurement vectors having the same, but unknown, sparsity structure. The single measurement sparse solution problem has been extensively studied in the past. Although known ..."
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Cited by 272 (22 self)
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Abstract—We address the problem of finding sparse solutions to an underdetermined system of equations when there are multiple measurement vectors having the same, but unknown, sparsity structure. The single measurement sparse solution problem has been extensively studied in the past. Although known
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 274 (22 self)
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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
Sparse Solutions to Complex Models
"... Abstract Recent years witnessed the proliferation of the notion of sparsity and its applications in operations research models. To bring to the attention and raise the interest of the operations research community on this topic, we present in this tutorial a wide range of complex models that admit s ..."
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sparse yet effective solutions. Our examples range from compressed sensing and process flexibility to queuing applications, and from equation systems and optimization problems to game theory models. Keywords sparse; complex; equations; quadratic program; game theory; twostage stochastic pro
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 167 (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
On Sparse Solutions of Underdetermined Linear Systems
, 2009
"... We first explain the research problem of finding the sparse solution of underdetermined linear systems with some applications. Then we explain three different approaches how to solve the sparse solution: the ℓ1 approach, the orthogonal greedy approach, and the ℓq approach with 0 < q ≤ 1. We mainl ..."
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Cited by 5 (1 self)
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We first explain the research problem of finding the sparse solution of underdetermined linear systems with some applications. Then we explain three different approaches how to solve the sparse solution: the ℓ1 approach, the orthogonal greedy approach, and the ℓq approach with 0 < q ≤ 1. We
Neighborly Polytopes and Sparse Solutions of Underdetermined Linear Equations
, 2005
"... Consider a d × n matrix A, with d < n. The problem of solving for x in y = Ax is underdetermined, and has many possible solutions (if there are any). In several fields it is of interest to find the sparsest solution – the one with fewest nonzeros – but in general this involves combinatorial optim ..."
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Cited by 130 (13 self)
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show that if P is kneighborly, then if a system y = Ax has a solution with at most k nonzeros, that solution is also the unique solution of the convex optimization problem min �x�1 subject to y = Ax; the converse holds as well. This complete equivalence between the study of sparse solution by ℓ 1
Sparse Bayesian Learning and the Relevance Vector Machine
, 2001
"... This paper introduces a general Bayesian framework for obtaining sparse solutions to regression and classification tasks utilising models linear in the parameters. Although this framework is fully general, we illustrate our approach with a particular specialisation that we denote the `relevance vect ..."
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Cited by 966 (5 self)
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This paper introduces a general Bayesian framework for obtaining sparse solutions to regression and classification tasks utilising models linear in the parameters. Although this framework is fully general, we illustrate our approach with a particular specialisation that we denote the `relevance
Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems
 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING
, 2007
"... Many problems in signal processing and statistical inference involve finding sparse solutions to underdetermined, or illconditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared ℓ2) error term combined with a spa ..."
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Cited by 539 (17 self)
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Many problems in signal processing and statistical inference involve finding sparse solutions to underdetermined, or illconditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared ℓ2) error term combined with a
On the Uniqueness of Nonnegative Sparse Solutions to Underdetermined Systems of Equations
, 2008
"... An underdetermined linear system of equations Ax = b with nonnegativity constraint x 0 is considered. It is shown that for matrices A with a rowspan intersecting the positive orthant, if this problem admits a sufficiently sparse solution, it is necessarily unique. The bound on the required sparsity ..."
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Cited by 44 (0 self)
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An underdetermined linear system of equations Ax = b with nonnegativity constraint x 0 is considered. It is shown that for matrices A with a rowspan intersecting the positive orthant, if this problem admits a sufficiently sparse solution, it is necessarily unique. The bound on the required
Results 1  10
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4,482