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

Abstract

Suppose we are given a vector $f$ in ${\mathbb{R}}^N$. 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 $\mathcal{F}$ 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 $f\in \mathcal{F}$ 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.

39 pages; no figures; to appear. Bernoulli ensemble proof has been corrected; other expository and bibliographical changes made, incorporating referee's suggestions

References in corpus (1)

• Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions