上限$\ell_1$ -优化截面阈值
Upper-bounding $\ell_1$-optimization sectional thresholds
摘要：
在本文中，我们研究与具有稀疏解的欠定线性方程组相关的一个特定问题。$\ell_1$ -最小化是一个相当成功的多项式技术，可以在某些统计情况下找到足够稀疏的解决方案，这样的系统。$\ell_1$性能的障碍通常称为其阈值。取决于是否对典型或最坏情况行为感兴趣，然后区分与一侧的典型行为相关的\emph{weak}阈值和与另一侧的最坏情况行为相关的\emph{sectional}和\emph{strong}阈值。从开创性的工作\cite{CRT,DonohoPol,DOnoho06CS}开始，在$\ell_1$最小化统计阈值的理论表征方面取得了实质性进展。更准确地说，\cite{CRT,DOnoho06CS}首次提出了所有这些阈值的线性下限。Donoho的工作\cite{DonohoPol}（以及我们自己的\cite{StojnicCSetam09,StojnicUpper10}）更进一步，基本上解决了$\ell_1$的\emph{weak}阈值。同时，它们还为\emph{sectional}和\emph{strong}阈值的值提供了相当好的下限。在本文中，我们重新审视了\emph{sectional}阈值，并提出了一个简单的机制，可以用来创建坚实的上限。我们提出的方法依赖于我们在\cite{StojnicHopBnds10}中研究Hopfield模型所取得的看似简单但实质性的进展。
英文摘要：
In this paper we look at a particular problem related to under-determined linear systems of equations with sparse solutions. $\ell_1$-minimization is a fairly successful polynomial technique that can in certain statistical scenarios find sparse enough solutions of such systems. Barriers of $\ell_1$ performance are typically referred to as its thresholds. Depending if one is interested in a typical or worst case behavior one then distinguishes between the \emph{weak} thresholds that relate to a typical behavior on one side and the \emph{sectional} and \emph{strong} thresholds that relate to the worst case behavior on the other side. Starting with seminal works \cite{CRT,DonohoPol,DOnoho06CS} a substantial progress has been achieved in theoretical characterization of $\ell_1$-minimization statistical thresholds. More precisely, \cite{CRT,DOnoho06CS} presented for the first time linear lower bounds on all of these thresholds. Donoho's work \cite{DonohoPol} (and our own \cite{StojnicCSetam09,StojnicUpper10}) went a bit further and essentially settled the $\ell_1$'s \emph{weak} thresholds. At the same time they also provided fairly good lower bounds on the values on the \emph{sectional} and \emph{strong} thresholds. In this paper, we revisit the \emph{sectional} thresholds and present a simple mechanism that can be used to create solid upper bounds as well. The method we present relies on a seemingly simple but substantial progress we made in studying Hopfield models in \cite{StojnicHopBnds10}.
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