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A Simple Proof of the Restricted Isometry Property for Random Matrices
Abstract We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candès and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main
Constructive Approximation
TLDR
This paper works on [-1, 1 ] and obtains Markov-type estimates for the derivatives of polynomials from a rather wide family of classes of constrained polynomes and results turn out to be sharp.
Iteratively reweighted least squares minimization for sparse recovery
Under certain conditions (known as the restricted isometry property, or RIP) on the m × N matrix Φ (where m < N), vectors x ∈ ℝN that are sparse (i.e., have most of their entries equal to 0) can be
Compressed sensing and best k-term approximation
The typical paradigm for obtaining a compressed version of a discrete signal represented by a vector x ∈ R is to choose an appropriate basis, compute the coefficients of x in this basis, and then
Nonlinear approximation
TLDR
This is a survey of nonlinear approximation, especially that part of the subject which is important in numerical computation, and emphasis will be placed on approximation by piecewise polynomials and wavelets as well as their numerical implementation.
Adaptive wavelet methods for elliptic operator equations: Convergence rates
TLDR
The main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error O(N -s ) in the energy norm, whenever such a rate is possible by N-term approximation.
Adaptive Finite Element Methods with convergence rates
Summary.Adaptive Finite Element Methods for numerically solving elliptic equations are used often in practice. Only recently [12], [17] have these methods been shown to converge. However, this
Some remarks on greedy algorithms
TLDR
Three greedy algorithms are discussed: the Pure GreedyAlgorithm, an Orthogonal Greedy Algorithm, and a Relaxed Gre greedy Algorithm.
Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage
TLDR
Extensive computations are presented that support the hypothesis that near-optimal shrinkage parameters can be derived if one knows (or can estimate) only two parameters about an image F: the largest alpha for which FinEpsilon(q)(alpha )(L( q)(I)),1/q=alpha/2+1/2, and the norm |F|B(q) alpha)(L(Q)(I)).
Approximation and learning by greedy algorithms
TLDR
This work improves on the existing theory of convergence rates for both the orthogonal greedy algorithm and the relaxed greedy algorithm, as well as for the forward stepwise projection algorithm, and proves convergence results for a variety of function classes and not simply those that are related to the convex hull of the dictionary.
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