On the Scaling Law for Compressive Sensing and its Applications

TL;DR

This paper derives a closed-form scaling law for the stability of $ ext{l}_1$ minimization in compressive sensing, revealing how recovery stability degrades as sparsity approaches the weak threshold, and discusses implications for iterative reweighted algorithms.

Contribution

It provides a novel closed-form characterization of the tradeoff between sparsity and recovery stability in compressive sensing, based on Grassmann angle analysis.

Findings

Recovery stability scales as 1/√(1-ϖ) near the weak threshold.

The scaling law aids in analyzing iterative reweighted $ ext{l}_1$ algorithms.

Phase transition thresholds can be improved using amplitude distribution properties.

Abstract

$\ell_1$ minimization can be used to recover sufficiently sparse unknown signals from compressed linear measurements. In fact, exact thresholds on the sparsity (the size of the support set), under which with high probability a sparse signal can be recovered from i.i.d. Gaussian measurements, have been computed and are referred to as "weak thresholds" \cite{D}. It was also known that there is a tradeoff between the sparsity and the $\ell_1$ minimization recovery stability. In this paper, we give a \emph{closed-form} characterization for this tradeoff which we call the scaling law for compressive sensing recovery stability. In a nutshell, we are able to show that as the sparsity backs off $\varpi$ ($0<\varpi<1$) from the weak threshold of $\ell_1$ recovery, the parameter for the recovery stability will scale as $\frac{1}{\sqrt{1-\varpi}}$. Our result is based on a careful analysis through the Grassmann angle framework for the Gaussian measurement matrix. We will further discuss how this scaling law helps in analyzing the iterative reweighted $\ell_1$ minimization algorithms. If the nonzero elements over the signal support follow an amplitude probability density function (pdf) $f(\cdot)$ whose $t$-th derivative $f^{t}(0) \neq 0$ for some integer $t \geq 0$, then a certain iterative reweighted $\ell_1$ minimization algorithm can be analytically shown to lift the phase transition thresholds (weak thresholds) of the plain $\ell_1$ minimization algorithm.