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Signal Recovery from Random Measurements Via Orthogonal Matching Pursuit: The Gaussian Case
Joel A. Tropp,Anna C. Gilbert +1 more
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TLDR
In this paper, a greedy algorithm called Orthogonal Matching Pursuit (OMP) was proposed to recover a signal with m nonzero entries in dimension 1 given O(m n d) random linear measurements of that signal.Abstract:
This 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(mln d) random linear measurements of that signal. This is a massive improvement
over previous results, which require O(m2) measurements. The new results for OMP are comparable
with recent results for another approach called Basis Pursuit (BP). In some settings, the
OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal
recovery problems.read more
Citations
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Proceedings Article
Submodular Dictionary Selection for Sparse Representation
Andreas Krause,Volkan Cevher +1 more
TL;DR: An efficient learning framework to construct signal dictionaries for sparse representation by selecting the dictionary columns from multiple candidate bases is developed and it is shown that if the available dictionary column vectors are incoherent, the objective function satisfies approximate submodularity.
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Sparsity Averaging Reweighted Analysis (SARA): a novel algorithm for radio‐interferometric imaging
TL;DR: It is shown through simulations that the proposed approach outperforms state-of-the-art imaging methods in the field, which are based on the assumption of signal sparsity in a single basis only.
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Fourier-domain beamforming: the path to compressed ultrasound imaging
TL;DR: In this paper, the authors extend the concept of beamforming in frequency to a general concept, which allows exploitation of the low bandwidth of the ultrasound signal and bypassing of the oversampling dictated by digital implementation of beamformers in time.
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Best Basis Compressed Sensing
TL;DR: A best basis extension of compressed sensing recovery is proposed that makes use of sparsity in a tree-structured dictionary of orthogonal bases and improves the recovery with respect to fixed sparsity priors.
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Dictionary learning method for joint sparse representation-based image fusion
TL;DR: This work proposes a novel dictionary learning method (MODJSR) whose dictionary updating procedure is derived by employing the JSR structure one time with singular value decomposition (SVD), which has lower complexity than the K-SVD algorithm which is often used in previous JSR-based fusion algorithms.
References
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Book
Compressed sensing
TL;DR: It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients, and a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing.
Journal ArticleDOI
Atomic Decomposition by Basis Pursuit
TL;DR: Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions.
Journal ArticleDOI
Matching pursuits with time-frequency dictionaries
Stéphane Mallat,Zhifeng Zhang +1 more
TL;DR: The authors introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions, chosen in order to best match the signal structures.
Journal ArticleDOI
Least angle regression
Bradley Efron,Trevor Hastie,Iain M. Johnstone,Robert Tibshirani,Hemant Ishwaran,Keith Knight,Jean-Michel Loubes,Jean-Michel Loubes,Pascal Massart,Pascal Massart,David Madigan,David Madigan,Greg Ridgeway,Greg Ridgeway,Saharon Rosset,Saharon Rosset,Ji Zhu,Robert A. Stine,Berwin A. Turlach,Sanford Weisberg +19 more
TL;DR: A publicly available algorithm that requires only the same order of magnitude of computational effort as ordinary least squares applied to the full set of covariates is described.