A note on reductions between compressed sensing guarantees

TL;DR

This paper establishes a general hierarchy of recovery guarantees in compressed sensing and derives an efficient scheme for any 0<p<1 with minimal measurements, improving existing bounds.

Contribution

It provides a unifying theorem on the strength hierarchy of recovery guarantees and introduces an optimal measurement scheme for 0<p<1 compressed sensing.

Findings

Hierarchical relationships among recovery guarantees established

Efficient / recovery scheme for 0<p<1 derived

Number of measurements minimized compared to previous methods

Abstract

In compressed sensing, one wishes to acquire an approximately sparse high-dimensional signal $x\in\mathbb{R}^n$ via $m\ll n$ noisy linear measurements, then later approximately recover $x$ given only those measurement outcomes. Various guarantees have been studied in terms of the notion of approximation in recovery, and some isolated folklore results are known stating that some forms of recovery are stronger than others, via black-box reductions. In this note we provide a general theorem concerning the hierarchy of strengths of various recovery guarantees. As a corollary of this theorem, by reducing from well-known results in the compressed sensing literature, we obtain an efficient $\ell_p/\ell_p$ scheme for any $0<p<1$ with the fewest number of measurements currently known amongst efficient schemes, improving recent bounds of [SomaY16].