Lifting $\ell_1$-optimization strong and sectional thresholds
Abstract
In this paper we revisit under-determined linear systems of equations with sparse solutions. As is well known, these systems are among core mathematical problems of a very popular compressed sensing field. The popularity of the field as well as a substantial academic interest in linear systems with sparse solutions are in a significant part due to seminal results \cite{CRT,DonohoPol}. Namely, working in a statistical scenario, \cite{CRT,DonohoPol} provided substantial mathematical progress in characterizing relation between the dimensions of the systems and the sparsity of unknown vectors recoverable through a particular polynomial technique called -minimization. In our own series of work \cite{StojnicCSetam09,StojnicUpper10,StojnicEquiv10} we also provided a collection of mathematical results related to these problems. While, Donoho's work \cite{DonohoPol,DonohoUnsigned} established (and our own work \cite{StojnicCSetam09,StojnicUpper10,StojnicEquiv10} reaffirmed) the typical or the so-called \emph{weak threshold} behavior of -minimization many important questions remain unanswered. Among the most important ones are those that relate to non-typical or the so-called \emph{strong threshold} behavior. These questions are usually combinatorial in nature and known techniques come up short of providing the exact answers. In this paper we provide a powerful mechanism that that can be used to attack the "tough" scenario, i.e. the \emph{strong threshold} (and its a similar form called \emph{sectional threshold}) of -minimization.
Cite
@article{arxiv.1306.3770,
title = {Lifting $\ell_1$-optimization strong and sectional thresholds},
author = {Mihailo Stojnic},
journal= {arXiv preprint arXiv:1306.3770},
year = {2013}
}