Suppose we are given a vector $f$ in $\R^N$. How many linear measurements do
we need to make about $f$ to be able to recover $f$ to within precision
$\epsilon$ in the Euclidean ($\ell_2$) metric? Or more exactly, suppose we are
interested in a class ${\cal F}$ of such objects--discrete digital signals,
images, etc; how many linear measurements do we need to recover objects from
this class to within accuracy $\epsilon$? This paper shows that if the objects
of interest are sparse or compressible in the sense that the reordered entries
of a signal $f \in {\cal F}$ decay like a power-law (or if the coefficient
sequence of $f$ in a fixed basis decays like a power-law), then it is possible
to reconstruct $f$ to within very high accuracy from a small number of random
measurements.Comment: 39 pages; no figures; to appear. Bernoulli ensemble proof has been
  corrected; other expository and bibliographical changes made, incorporating
  referee's suggestion

Candes, Emmanuel

Tao, Terence

English

arXiv

Suppose we are given a vector f in a class F ⊂ ℝN,  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 ε in the Euclidean (ℓ2) 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 |f| (or of its coefficients in a fixed basis) obeys |f|(n) ≤ R  n^(1-p), where R > 0 and p > 0. Suppose that we take measurements yk = {f,Xk}, k = 1, . . .,K, where the Xk are N-dimensional 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 yk = 〈f#, Xk〉 with minimal ℓ1 norm, obeys [equation]. 



There is a sense in which this result is optimal; it is generally impossible to obtain a higher accuracy from any set of K measurements whatsoever. The methodology extends to various other random measurement ensembles; for example, we show that similar results hold if one observes a few randomly sampled Fourier coefficients of f. In fact, the results are quite general and require only two hypotheses on the measurement ensemble which are detailed

Candès, Emmanuel J.

Caltech Authors - Main

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Near Optimal Signal Recovery From Random Projections: Universal Encoding
  Strategies?

Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?

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