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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
Geometric approach to error correcting codes and reconstruction of signals
 INT. MATH. RES. NOT
, 2005
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A remark on compressed sensing
, 2007
"... Abstract—Recently, a new direction in signal processing – “Compressed Sensing " is being actively developed. A number of authors have pointed out a connection between the Compressed Sensing problem and the problem of estimating the Kolmogorov widths, studied in the seventies and eighties of the ..."
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Abstract—Recently, a new direction in signal processing – “Compressed Sensing " is being actively developed. A number of authors have pointed out a connection between the Compressed Sensing problem and the problem of estimating the Kolmogorov widths, studied in the seventies and eighties of the last century. In this paper we make the above mentioned connection more precise. DOI: 10.1134/S0001434607110193
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
Compressive sensing: a paradigm shift in signal processing
, 2008
"... We survey a new paradigm in signal processing known as "compressive sensing". Contrary to old practices of data acquisition and reconstruction based on the ShannonNyquist sampling principle, the new theory shows that it is possible to reconstruct images or signals of scientific interest a ..."
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We survey a new paradigm in signal processing known as "compressive sensing". Contrary to old practices of data acquisition and reconstruction based on the ShannonNyquist sampling principle, the new theory shows that it is possible to reconstruct images or signals of scientific interest accurately and even exactly from a number of samples which is far smaller than the desired resolution of the image/signal, e.g., the number of pixels in the image. This new technique draws from results in several fields of mathematics, including algebra, optimization, probability theory, and harmonic analysis. We will discuss some of the key mathematical ideas behind compressive sensing, as well as its implications to other fields: numerical analysis, information theory, theoretical computer science, and engineering.
Linear Systems
, 2013
"... Our main goal here is to discuss perspectives of applications based on solving underdetermined systems of linear equations (SLE). This discussion will include interconnection of underdetermined SLE with global problems of Information Theory and with data measuring and representation. The preference ..."
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Our main goal here is to discuss perspectives of applications based on solving underdetermined systems of linear equations (SLE). This discussion will include interconnection of underdetermined SLE with global problems of Information Theory and with data measuring and representation. The preference will be given to the description of the hypothetic destination point of the undertaken efforts, the current status of the problem, and possible methods to overcome difficulties on the way to that destination point. We do not pretend on a full survey of the current state of the theoretic researches which are very extensive now. We are going to discuss only some fundamental theoretical results justifying main applied ideas. In the end of the chapter we give numerical results related to the