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Testing Fourier dimensionality and sparsity
"... Abstract. We present a range of new results for testing properties of Boolean functions that are defined in terms of the Fourier spectrum. Broadly speaking, our results show that the property of a Boolean function having a concise Fourier representation is locally testable. We first give an efficien ..."
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Cited by 22 (3 self)
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an efficient algorithm for testing whether the Fourier spectrum of a Boolean function is supported in a lowdimensional subspace of F n 2 (equivalently, for testing whether f is a junta over a small number of parities). We next give an efficient algorithm for testing whether a Boolean function has a sparse
Chebyshev and Fourier Spectral Methods
, 1999
"... Contents PREFACE x Acknowledgments xiv Errata and ExtendedBibliography xvi 1 Introduction 1 1.1 Series expansions .................................. 1 1.2 First Example .................................... 2 1.3 Comparison with finite element methods .................... 4 1.4 Comparisons with ..."
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Cited by 784 (12 self)
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.9 Nonlinearity ..................................... 14 1.10 Timedependent problems ............................. 16 1.11 FAQ: Frequently Asked Questions ........................ 17 1.12 The Chrysalis .................................... 18 2 Chebyshev & Fourier Spectral Methods 19 2.1 Introduction
Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information
, 2006
"... This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discretetime signal and a randomly chosen set of frequencies. Is it possible to reconstruct from the partial knowledge of its Fourier coefficients on the set? A typical result of this pa ..."
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Cited by 2606 (51 self)
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This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discretetime signal and a randomly chosen set of frequencies. Is it possible to reconstruct from the partial knowledge of its Fourier coefficients on the set? A typical result
ATOMIC DECOMPOSITION BY BASIS PURSUIT
, 1995
"... The TimeFrequency and TimeScale communities have recently developed a large number of overcomplete waveform dictionaries  stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for d ..."
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Cited by 2739 (61 self)
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the smallest l 1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP and BOB, including better sparsity, and superresolution. BP has interesting relations to ideas in areas as diverse as illposed problems, in abstract harmonic analysis, total
KSVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation
, 2006
"... In recent years there has been a growing interest in the study of sparse representation of signals. Using an overcomplete dictionary that contains prototype signalatoms, signals are described by sparse linear combinations of these atoms. Applications that use sparse representation are many and inc ..."
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Cited by 935 (41 self)
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signal representations. Given a set of training signals, we seek the dictionary that leads to the best representation for each member in this set, under strict sparsity constraints. We present a new method—the KSVD algorithm—generalizing the umeans clustering process. KSVD is an iterative method
Compressive sampling
, 2006
"... Conventional wisdom and common practice in acquisition and reconstruction of images from frequency data follow the basic principle of the Nyquist density sampling theory. This principle states that to reconstruct an image, the number of Fourier samples we need to acquire must match the desired res ..."
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Cited by 1434 (15 self)
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Conventional wisdom and common practice in acquisition and reconstruction of images from frequency data follow the basic principle of the Nyquist density sampling theory. This principle states that to reconstruct an image, the number of Fourier samples we need to acquire must match the desired
Robust face recognition via sparse representation
 IEEE TRANS. PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2008
"... We consider the problem of automatically recognizing human faces from frontal views with varying expression and illumination, as well as occlusion and disguise. We cast the recognition problem as one of classifying among multiple linear regression models, and argue that new theory from sparse signa ..."
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Cited by 920 (41 self)
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extraction and robustness to occlusion. For feature extraction, we show that if sparsity in the recognition problem is properly harnessed, the choice of features is no longer critical. What is critical, however, is whether the number of features is sufficiently large and whether the sparse representation
Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems
 IEEE Journal of Selected Topics in Signal Processing
, 2007
"... Abstract—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 wi ..."
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Cited by 527 (15 self)
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constrained quadratic programming (BCQP) formulation of these problems. We test variants of this approach that select the line search parameters in different ways, including techniques based on the BarzilaiBorwein method. Computational experiments show that these GP approaches perform well in a wide range
An introduction to kernelbased learning algorithms
 IEEE TRANSACTIONS ON NEURAL NETWORKS
, 2001
"... This paper provides an introduction to support vector machines (SVMs), kernel Fisher discriminant analysis, and ..."
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Cited by 590 (54 self)
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This paper provides an introduction to support vector machines (SVMs), kernel Fisher discriminant analysis, and
The Dantzig Selector: Statistical Estimation When p Is Much Larger Than n
, 2007
"... In many important statistical applications, the number of variables or parameters p is much larger than the number of observations n. Suppose then that we have observations y = Xβ + z, where β ∈ Rp is a parameter vector of interest, X is a data matrix with possibly far fewer rows than columns, n ≪ p ..."
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Cited by 881 (14 self)
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In many important statistical applications, the number of variables or parameters p is much larger than the number of observations n. Suppose then that we have observations y = Xβ + z, where β ∈ Rp is a parameter vector of interest, X is a data matrix with possibly far fewer rows than columns, n ≪ p, and the zi’s are i.i.d. N(0,σ2). Is it possible to estimate β reliably based on the noisy data y? To estimate β, we introduce a new estimator—we call it the Dantzig selector—which is a solution to the ℓ1regularization problem min ˜β∈R p ‖ ˜β‖ℓ1 subject to ‖X ∗ r‖ℓ ∞ ≤ (1 + t−1 √) 2logp · σ, where r is the residual vector y − X ˜β and t is a positive scalar. We show that if X obeys a uniform uncertainty principle (with unitnormed columns) and if the true parameter vector β is sufficiently sparse (which here roughly guarantees that the model is identifiable), then with very large probability,
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