提升$\ell_q$ -优化阈值
Lifting $\ell_q$-optimization thresholds
摘要：
在本文中，我们期待在$\ell_q,0\leq q\leq 1$，优化和欠定线性方程组与稀疏解之间的连接。$q=1$的情况下，或换句话说$\ell_1$优化及其与线性系统的连接已被彻底研究在过去几十年中，事实上，尤其是在过去十年后，开创性的作品\cite{CRT,DOnoho06CS}出现。虽然目前对$\ell_1$优化-线性系统连接的理解是相当已知的，但对于一般的$\ell_q,0<q<1$优化的情况则知之甚少。在我们最近的工作\cite{StojnicLqThrBnds10}，我们提供了一个研究在这个方向。因此，我们能够获得各种$\ell_q,0\leq q\leq 1$优化阈值的下限集合。在本文中，我们提供了一个实质性的概念改进的方法，在\cite{StojnicLqThrBnds10}。此外，在可达到的阈值方面取得的实际成果也令人鼓舞。通常情况下，这些和类似的问题，我们开发的方法强调他们的组合性质，并试图以某种方式处理它。虽然我们的结果的主要贡献应该是在概念层面上，他们已经给了一个非常强烈的建议，$\ell_q$优化实际上可以提供一个更好的性能比$\ell_1$，这一事实长期以来被认为是真实的，因为它向原始的面向$\ell_0$稀疏性发现的原始问题公式提供了更严格的优化松弛。因此，它们在某种程度上为进一步探索能够处理$\ell_q,0<q<1$的算法设计提供了坚实的推动力，在合理的（如果不是多项式的话）时间内进行优化。
英文摘要：
In this paper we look at a connection between the $\ell_q,0\leq q\leq 1$, optimization and under-determined linear systems of equations with sparse solutions. The case $q=1$, or in other words $\ell_1$ optimization and its a connection with linear systems has been thoroughly studied in last several decades; in fact, especially so during the last decade after the seminal works \cite{CRT,DOnoho06CS} appeared. While current understanding of $\ell_1$ optimization-linear systems connection is fairly known, much less so is the case with a general $\ell_q,0<q<1$, optimization. In our recent work \cite{StojnicLqThrBnds10} we provided a study in this direction. As a result we were able to obtain a collection of lower bounds on various $\ell_q,0\leq q\leq 1$, optimization thresholds. In this paper, we provide a substantial conceptual improvement of the methodology presented in \cite{StojnicLqThrBnds10}. Moreover, the practical results in terms of achievable thresholds are also encouraging. As is usually the case with these and similar problems, the methodology we developed emphasizes their a combinatorial nature and attempts to somehow handle it. Although our results' main contributions should be on a conceptual level, they already give a very strong suggestion that $\ell_q$ optimization can in fact provide a better performance than $\ell_1$, a fact long believed to be true due to a tighter optimization relaxation it provides to the original $\ell_0$ sparsity finding oriented original problem formulation. As such, they in a way give a solid boost to further exploration of the design of the algorithms that would be able to handle $\ell_q,0<q<1$, optimization in a reasonable (if not polynomial) time.
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