A Rice method proof of the Null-Space Property over the Grassmannian

TL;DR

This paper proves the Null-Space Property (NSP) using the Rice method, providing non-asymptotic bounds and a simple sufficient condition, which advances understanding of NSP in high-dimensional statistics and related fields.

Contribution

First proof of NSP using the Rice method, offering non-asymptotic bounds and a new sufficient condition for NSP over the Grassmannian.

Findings

Provides non-asymptotic bounds for NSP

Derives a simple sufficient condition for NSP

Uses the Rice method to analyze NSP

Abstract

The Null-Space Property (NSP) is a necessary and sufficient condition for the recovery of the largest coefficients of solutions to an under-determined system of linear equations. Interestingly, this property governs also the success and the failure of recent developments in high-dimensional statistics, signal processing, error-correcting codes and the theory of polytopes. Although this property is the keystone of $\ell\_{1}$-minimization techniques, it is an open problem to derive a closed form for the phase transition on NSP. In this article, we provide the first proof of NSP using random processes theory and the Rice method. As a matter of fact, our analysis gives non-asymptotic bounds for NSP with respect to unitarily invariant distributions. Furthermore, we derive a simple sufficient condition for NSP.