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

Paper

Abstract

Suppose we are given a vector

. How many linear measurements do we need to make about

to be able to recover

to within precision

in the Euclidean (

) metric? Or more exactly, suppose we are interested in a class

of such objects--discrete digital signals, images, etc; how many linear measurements do we need to recover objects from this class to within accuracy

? This paper shows that if the objects of interest are sparse or compressible in the sense that the reordered entries of a signal

decay like a power-law (or if the coefficient sequence of

in a fixed basis decays like a power-law), then it is possible to reconstruct

to within very high accuracy from a small number of random measurements.