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60
Signal recovery from random measurements via Orthogonal Matching Pursuit
 IEEE Trans. Inform. Theory
, 2007
"... Abstract. 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 ove ..."
Abstract

Cited by 292 (9 self)
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Abstract. 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 results for OMP, which require O(m 2) measurements. The new results for OMP are comparable with recent results for another algorithm called Basis Pursuit (BP). The OMP algorithm is faster and easier to implement, which makes it an attractive alternative to BP for signal recovery problems. 1.
Signal recovery from partial information via Orthogonal Matching Pursuit.” Submitted to
 IEEE Trans. Inform. Theory
, 2005
"... Abstract. 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 previou ..."
Abstract

Cited by 149 (8 self)
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Abstract. 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 for OMP, which require O(m 2) measurements. The new results for OMP are comparable with recent results for another algorithm called Basis Pursuit (BP). The OMP algorithm is much faster and much easier to implement, which makes it an attractive alternative to BP for signal recovery problems. 1.
Gaussian processes: inequalities, small ball probabilities and applications. Stochastic processes: theory and methods
 Handbook of Statist
, 2001
"... Edited by C.R. Rao and D. Shanbhag. 1 ..."
Reconstruction and subgaussian operators in Asymptotic Geometric Analysis
 FUNCT. ANAL
"... We present a randomized method to approximate any vector v from some set T ⊂ R n. The data one is given is the set T, vectors (Xi) k i=1 of R n and k scalar products (〈Xi, v〉) k i=1, where (Xi) k i=1 are i.i.d. isotropic subgaussian random vectors in R n, and k ≪ n. We show that with high probabilit ..."
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Cited by 35 (5 self)
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We present a randomized method to approximate any vector v from some set T ⊂ R n. The data one is given is the set T, vectors (Xi) k i=1 of R n and k scalar products (〈Xi, v〉) k i=1, where (Xi) k i=1 are i.i.d. isotropic subgaussian random vectors in R n, and k ≪ n. We show that with high probability, any y ∈ T for which (〈Xi, y〉) k i=1 is close to the data vector (〈Xi, v〉) k i=1 will be a good approximation of v, and that the degree of approximation is determined by a natural geometric parameter associated with the set T. We also investigate a random method to identify exactly any vector which has a relatively short support using linear subgaussian measurements as above. It turns out that our analysis, when applied to {−1, 1}valued vectors with i.i.d, symmetric entries, yields new information on the geometry of faces of random {−1, 1}polytope; we show that a kdimensional random {−1, 1}polytope with n vertices is mneighborly for very large m ≤ ck / log(c ′ n/k). The proofs are � based on new estimates on the behavior of the empirical process supf∈F �k−1 �k i=1 f 2 (Xi) − Ef 2 � when F is a subset of the L2 sphere. The estimates are given in terms of the γ2 functional with respect to the ψ2 metric on F, and hold both in exponential probability and in expectation.
Existence and uniqueness of the integrated density of states for Schrödinger operators with magnetic fields and unbounded random potentials, Rev
 Math. Phys
"... The object of the present study is the integrated density of states of a quantum particle in multidimensional Euclidean space which is characterized by a Schrödinger operator with a constant magnetic field and a random potential which may be unbounded from above and from below. For an ergodic rando ..."
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Cited by 23 (7 self)
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The object of the present study is the integrated density of states of a quantum particle in multidimensional Euclidean space which is characterized by a Schrödinger operator with a constant magnetic field and a random potential which may be unbounded from above and from below. For an ergodic random potential satisfying a simple moment condition, we give a detailed proof that the infinitevolume limits of spatial eigenvalue concentrations of finitevolume operators with different boundary conditions exist almost surely. Since all these limits are shown to coincide with the expectation of the trace of the spatially localized spectral family of the infinitevolume operator, the integrated density of states is almost surely nonrandom and independent of the chosen boundary condition. Our proof of the
Existence of the density of states for multidimensional continuum Schrödinger operators with Gaussian random potentials
, 1997
"... : A Wegner estimate is proved for quantum systems in multidimensional Euclidean space which are characterized by oneparticle Schrodinger operators with random potentials that admit a certain oneparameter decomposition. In particular, the Wegner estimate applies to systems with rather general Gaus ..."
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Cited by 21 (4 self)
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: A Wegner estimate is proved for quantum systems in multidimensional Euclidean space which are characterized by oneparticle Schrodinger operators with random potentials that admit a certain oneparameter decomposition. In particular, the Wegner estimate applies to systems with rather general Gaussian random potentials. As a consequence, these systems possess an absolutely continuous integrated density of states, whose derivative, the density of states, is locally bounded. An explicit upper bound is derived. 1. Introduction The integrated density of states is a quantity of primary interest in the theory and applications of oneparticle random Schrodinger operators [SE, BEE+, LGP, CL, PF]. For example, the topological support of the associated measure coincides with the almostsure spectrum of the infinitevolume operator. Moreover, its knowledge allows to compute the free energy and hence all basic thermostatic quantities of the corresponding noninteracting manyparticle system. An ...