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134
For Most Large Underdetermined Systems of Linear Equations the Minimal ℓ1norm Solution is also the Sparsest Solution
 Comm. Pure Appl. Math
, 2004
"... We consider linear equations y = Φα where y is a given vector in R n, Φ is a given n by m matrix with n < m ≤ An, and we wish to solve for α ∈ R m. We suppose that the columns of Φ are normalized to unit ℓ 2 norm 1 and we place uniform measure on such Φ. We prove the existence of ρ = ρ(A) so that ..."
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Cited by 568 (10 self)
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that for large n, and for all Φ’s except a negligible fraction, the following property holds: For every y having a representation y = Φα0 by a coefficient vector α0 ∈ R m with fewer than ρ · n nonzeros, the solution α1 of the ℓ 1 minimization problem min �x�1 subject to Φα = y is unique and equal to α0
Sparsest solutions of underdetermined linear systems via ℓ
"... We present a condition on the matrix of an underdetermined linear system which guarantees that the solution of the system with minimal ℓqquasinorm is also the sparsest one. This generalizes, and sightly improves, a similar result for the ℓ1norm. We then introduce a simple numerical scheme to compu ..."
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Cited by 192 (11 self)
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We present a condition on the matrix of an underdetermined linear system which guarantees that the solution of the system with minimal ℓqquasinorm is also the sparsest one. This generalizes, and sightly improves, a similar result for the ℓ1norm. We then introduce a simple numerical scheme
New and improved conditions for uniqueness of sparsest solutions of underdetermined linear systems
 Applied Mathematics and Computation
"... Abstract. The uniqueness of sparsest solutions of underdetermined linear systems plays a fundamental role in compressed sensing theory. Several new algebraic concepts, including the submutual coherence, scaled mutual coherence, coherence rank, and subcoherence rank, are introduced in this paper in ..."
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Cited by 1 (1 self)
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Abstract. The uniqueness of sparsest solutions of underdetermined linear systems plays a fundamental role in compressed sensing theory. Several new algebraic concepts, including the submutual coherence, scaled mutual coherence, coherence rank, and subcoherence rank, are introduced in this paper
Features of Big Data and sparsest solution in high confidence set
"... This chapter summarizes some of the unique features of Big Data analysis. These features are shared neither by lowdimensional data nor by small samples. Big Data pose new computational challenges and hold great promises for understanding population heterogeneity as in personalized medicine or ser ..."
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Cited by 2 (2 self)
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vices. High dimensionality introduces spurious correlations, incidental endogeneity, noise accumulation, and measurement error. These unique features are very distinguished and statistical procedures should be designed with these issues in mind. To illustrate, a method called a sparsest solution in high
An upper bound on the estimation error of the sparsest solution of underdetermined linear systems
 SPARS'09 SIGNAL PROCESSING WITH ADAPTIVE SPARSE STRUCTURED REPRESENTATIONS
, 2009
"... Let A be an n×m matrix with m> n, and suppose the underdetermined linear system As = x admits a unique sparse solution s0 (i.e. it has a solution s0 for which ‖s0‖0 < 1/2 spark(A)). Suppose that we have somehow a solution (sparse or nonsparse) ˆs of this system as an estimation of the true sp ..."
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sparsest solution s0. Is it possible to construct an upper bound on the estimation error ‖ˆs − s0‖2 without knowing s0? The answer is positive, and in this paper we construct such a bound which, in the case A has Unique Representation Property (URP), depends on the smallest singular value of all n × n
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
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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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
The kSparsest Subgraph Problem
, 2013
"... Given a simple undirected graph G = (V, E) and an integer k ≤ V , the ksparsest subgraph problem asks for a set of k vertices that induce the minimum number of edges. As a generalization of the classical independent set problem, ksparsest subgraph cannot admit (unless P = N P) neither an approx ..."
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an approximation nor an FPT algorithm (parameterized by the number of edges in the solution) in all graph classes where independent set is N Phard. Thus, it appears natural to investigate the approximability and fixed parameterized tractability of ksparsest subgraph in graph classes where independent set
Results 1  10
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134