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

TL;DR

This paper demonstrates that signals with power-law decay in their coefficients can be accurately reconstructed from a small number of random linear measurements, advancing universal encoding strategies for sparse and compressible signals.

Contribution

It establishes near-optimal measurement bounds for recovering signals with power-law decay, providing a universal approach for sparse and compressible data.

Findings

High-accuracy recovery from few measurements for power-law decaying signals

Universal encoding strategies applicable to various signal classes

Theoretical bounds on the number of measurements needed

Abstract

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.