提升$\ell_1$ -优化强阈值和分段阈值
Lifting $\ell_1$-optimization strong and sectional thresholds
摘要：
在本文中，我们重温欠定线性方程组的稀疏解。众所周知，这些系统是非常流行的压缩传感领域的核心数学问题之一。该领域的普及，以及大量的学术兴趣，在线性系统的稀疏解决方案是在一个显着的一部分，由于开创性的成果\cite{CRT,DonohoPol}。也就是说，工作在一个统计的情况下，\cite{CRT,DonohoPol}提供了大量的数学进步，在表征之间的关系的尺寸的系统和稀疏的未知向量可恢复通过一个特定的多项式技术称为$\ell_1$ -最小化。在我们自己的一系列工作\cite{StojnicCSetam09,StojnicUpper10,StojnicEquiv10}，我们还提供了一个收集的数学结果有关这些问题。虽然Donoho的工作\cite{DonohoPol,DonohoUnsigned}建立（和我们自己的工作\cite{StojnicCSetam09,StojnicUpper10,StojnicEquiv10}重申）典型的或所谓的\emph{weak threshold}行为的$\ell_1$ -最小化许多重要的问题仍然没有答案。其中最重要的是那些涉及非典型或所谓的\emph{strong threshold}行为。这些问题通常在本质上是组合的，并且已知的技术无法提供确切的答案。在本文中，我们提供了一个强大的机制，可以用来攻击“艰难”的情况下，即\emph{strong threshold}（和它的类似形式称为\emph{sectional threshold}）的$\ell_1$ -最小化。
英文摘要：
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 $\ell_1$-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 $\ell_1$-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 $\ell_1$-minimization.
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