Results 1 
2 of
2
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 ..."
Abstract

Cited by 1513 (20 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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