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

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

2004-10-25

arxiv.org/abs/math/0410542v3

Mathematics

Classical Analysis and ODEs

39 pages; no figures; to appear. Bernoulli ensemble proof has been corrected; other expository and bibliographical changes mad

Scientific paper

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.

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