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Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
, 2004
"... Suppose we are given a vector f in RN. How many linear measurements do we need to make about f to be able to recover f to within precision ɛ in the Euclidean (ℓ2) metric? Or more exactly, suppose we are interested in a class F of such objects— discrete digital signals, images, etc; how many linear m ..."
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Suppose we are given a vector f in RN. How many linear measurements do we need to make about f to be able to recover f to within precision ɛ in the Euclidean (ℓ2) metric? Or more exactly, suppose we are interested in a class F of such objects— discrete digital signals, images, etc; how many linear measurements do we need to recover objects from this class to within accuracy ɛ? This paper shows that if the objects of interest are sparse or compressible in the sense that the reordered entries of a signal f ∈ F decay like a powerlaw (or if the coefficient sequence of f in a fixed basis decays like a powerlaw), then it is possible to reconstruct f to within very high accuracy from a small number of random measurements. typical result is as follows: we rearrange the entries of f (or its coefficients in a fixed basis) in decreasing order of magnitude f  (1) ≥ f  (2) ≥... ≥ f  (N), and define the weakℓp ball as the class F of those elements whose entries obey the power decay law f  (n) ≤ C · n −1/p. We take measurements 〈f, Xk〉, k = 1,..., K, where the Xk are Ndimensional Gaussian
Nearly optimal signal recovery from random projections: Universal encoding strategies?
 IEEE TRANS. INFO. THEORY
, 2006
"... Suppose we are given a vector f in a class F, e.g., a class of digital signals or digital images. How many linear measurements do we need to make about f to be able to recover f to within precision in the Euclidean (`2) metric? This paper shows that if the objects of interest are sparse in a fixed ..."
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Suppose we are given a vector f in a class F, e.g., a class of digital signals or digital images. How many linear measurements do we need to make about f to be able to recover f to within precision in the Euclidean (`2) metric? This paper shows that if the objects of interest are sparse in a fixed basis or compressible, then it is possible to reconstruct f to within very high accuracy from a small number of random measurements by solving a simple linear program. More precisely, suppose that the nth largest entry of the vector jfj (or of its coefficients in a fixed basis) obeys jfj(n) R 1 n01=p, where R>0 and p>0. Suppose that we take measurements yk = hf; Xki;k =1;...;K, where the Xk are Ndimensional Gaussian vectors with independent standard normal entries. Then for each f obeying the decay estimate above for some 0 < p < 1 and with overwhelming probability, our reconstruction f] , defined as the solution to the constraints
Highly Robust Error Correction by Convex Programming
"... Abstract—This paper discusses a stylized communications problem where one wishes to transmit a realvalued signal x 2 n (a block of n pieces of information) to a remote receiver. We ask whether it is possible to transmit this information reliably when a fraction of the transmitted codeword is corrup ..."
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Abstract—This paper discusses a stylized communications problem where one wishes to transmit a realvalued signal x 2 n (a block of n pieces of information) to a remote receiver. We ask whether it is possible to transmit this information reliably when a fraction of the transmitted codeword is corrupted by arbitrary gross errors, and when in addition, all the entries of the codeword are contaminated by smaller errors (e.g., quantization errors). We show that if one encodes the information as Ax where A 2 m2n (m n) is a suitable coding matrix, there are two decoding schemes that allow the recovery of the block of n pieces of information x with nearly the same accuracy as if no gross errors occurred upon transmission (or equivalently as if one had an oracle supplying perfect information about the sites and amplitudes of the gross errors). Moreover, both decoding strategies are very concrete and only involve solving simple convex optimization programs, either a linear program or a secondorder cone program. We complement our study with numerical simulations showing that the encoder/decoder pair performs remarkably well. Index Terms—Decoding of (random) linear codes, Gaussian random matrices and random projections, linear codes, `1 minimization, linear programming, restricted orthonormality, secondorder cone programming, sparse solutions to underdetermined systems, the Dantzig selector. I.
ROBUST COMPUTATION OF LINEAR MODELS, OR HOW TO FIND A NEEDLE IN A HAYSTACK
"... Abstract. Consider a dataset of vectorvalued observations that consists of a modest number of noisy inliers, which are explained well by a lowdimensional subspace, along with a large number of outliers, which have no linear structure. This work describes a convex optimization problem, called reape ..."
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Abstract. Consider a dataset of vectorvalued observations that consists of a modest number of noisy inliers, which are explained well by a lowdimensional subspace, along with a large number of outliers, which have no linear structure. This work describes a convex optimization problem, called reaper, that can reliably fit a lowdimensional model to this type of data. The paper provides an efficient algorithm for solving the reaper problem, and it documents numerical experiments which confirm that reaper can dependably find linear structure in synthetic and natural data. In addition, when the inliers are contained in a lowdimensional subspace, there is a rigorous theory that describes when reaper can recover the subspace exactly. 1.
Distributed noiseshaping quantization: I. Beta duals of finite frames and nearoptimal quantization of random measurements
, 2014
"... This paper introduces a new algorithm for the socalled “Analysis Problem ” in quantization of finite frame representations which provides a nearoptimal solution in the case of random measurements. The main contributions include the development of a general quantization framework called distribute ..."
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This paper introduces a new algorithm for the socalled “Analysis Problem ” in quantization of finite frame representations which provides a nearoptimal solution in the case of random measurements. The main contributions include the development of a general quantization framework called distributed noiseshaping, and in particular, beta duals of frames, as well as the performance analysis of beta duals in both deterministic and probabilistic settings. It is shown that for Gaussian random frames, using beta duals result in nearoptimally accurate reconstructions with respect to both the frame redundancy and the number of levels that the frame coefficients are quantized at. More specifically, if L quantization levels per measurement are used to encode the unit ball in Rk via a Gaussian frame of m vectors, then with overwhelming probability the betadual reconstruction error is shown to be bounded by kL−(1−η)m/k where η is arbitrarily small for sufficiently large problems. Additional features of the proposed algorithm include low computational cost and parallel implementability.
ROBUST COMPUTATION OF LINEAR MODELS, OR HOW TO FIND A NEEDLE IN A HAYSTACK
"... Abstract. Consider a dataset of vectorvalued observations that consists of a modest number of noisy inliers, which are explained well by a lowdimensional subspace, along with a large number of outliers, which have no linear structure. This work describes a convex optimization problem, called reape ..."
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Abstract. Consider a dataset of vectorvalued observations that consists of a modest number of noisy inliers, which are explained well by a lowdimensional subspace, along with a large number of outliers, which have no linear structure. This work describes a convex optimization problem, called reaper, that can reliably fit a lowdimensional model to this type of data. The paper provides an efficient algorithm for solving the reaper problem, and it documents numerical experiments which confirm that reaper can dependably find linear structure in synthetic and natural data. In addition, when the inliers are contained in a lowdimensional subspace, there is a rigorous theory that describes when reaper can recover the subspace exactly. 1.