New Bounds for Restricted Isometry Constants

TL;DR

This paper establishes a new upper bound of 0.307 on the restricted isometry constant for guaranteed sparse signal recovery in compressed sensing, providing both theoretical limits and explicit counterexamples.

Contribution

It introduces a tighter bound on the restricted isometry constant for exact and stable recovery of sparse signals, and demonstrates the bound's optimality with explicit examples.

Findings

Recovery guaranteed if δ_k < 0.307

Bound cannot be substantially improved

Counterexample with δ_k < 0.5 shows limits of recovery

Abstract

In this paper we show that if the restricted isometry constant $\delta_k$ of the compressed sensing matrix satisfies \[ \delta_k < 0.307, \] then $k$-sparse signals are guaranteed to be recovered exactly via $\ell_1$ minimization when no noise is present and $k$-sparse signals can be estimated stably in the noisy case. It is also shown that the bound cannot be substantively improved. An explicitly example is constructed in which $\delta_{k}=\frac{k-1}{2k-1} < 0.5$, but it is impossible to recover certain $k$-sparse signals.