Analysis of The Ratio of $\ell_1$ and $\ell_2$ Norms in Compressed Sensing

TL;DR

This paper introduces a new criterion for sparse signal recovery using the ratio of  and  norms, provides geometric recovery conditions, analyzes robustness to noise, and proposes an initialization method to enhance algorithm performance.

Contribution

It presents the first uniform recovery condition based on null space geometry and introduces a support selection initialization to improve /-based compressed sensing algorithms.

Findings

The proposed criterion guarantees local optimality for sparse signals.

The geometric null space condition is satisfied by certain random matrices.

Support selection improves convergence and recovery performance.

Abstract

We first propose a novel criterion that guarantees that an $s$-sparse signal is the local minimizer of the $\ell_1/\ell_2$ objective; our criterion is interpretable and useful in practice. We also give the first uniform recovery condition using a geometric characterization of the null space of the measurement matrix, and show that this condition is easily satisfied for a class of random matrices. We also present analysis on the robustness of the procedure when noise pollutes data. Numerical experiments are provided that compare $\ell_1/\ell_2$ with some other popular non-convex methods in compressed sensing. Finally, we propose a novel initialization approach to accelerate the numerical optimization procedure. We call this initialization approach \emph{support selection}, and we demonstrate that it empirically improves the performance of existing $\ell_1/\ell_2$ algorithms.