随机矩阵限制isometry常数的界
Bounds on restricted isometry constants of random matrices
摘要：
在本文中，我们研究随机矩阵的等距性质。在过去的十年中，这些属性获得了大量的关注，在一个领域称为压缩传感摆在首位，由于他们最初使用的\cite{CRT,CT}。也就是说，在\cite{CRT,CT}中，这些量被用作严格分析$\ell_1$优化求解具有稀疏解的欠定线性方程组系统的能力的关键工具。在这样一个框架中，一种特殊类型的等距，称为限制等距，起着关键作用。然后，人们通常会引入一对数量，称为上限和下限限制等距常数来表征随机矩阵的等距性质。这些常数通常被视为数学对象的兴趣和他们的精确表征是可取的。压缩感知中这些量的首次估计在\cite{CRT,CT}中给出。随着对精确估计的需求进一步增长，这些初始估计值得到了更精细的改进，例如\cite{BCTsharp09,BT10}。这些通常通过联合绑定策略和Wishart（高斯）矩阵的极端特征值的强大尾部估计的组合来获得（参见例如\cite{Edelman88}）。在本文中，我们试图绕过这种方法，并提供了一种替代方法来获得类似的估计。
英文摘要：
In this paper we look at isometry properties of random matrices. During the last decade these properties gained a lot attention in a field called compressed sensing in first place due to their initial use in \cite{CRT,CT}. Namely, in \cite{CRT,CT} these quantities were used as a critical tool in providing a rigorous analysis of $\ell_1$ optimization's ability to solve an under-determined system of linear equations with sparse solutions. In such a framework a particular type of isometry, called restricted isometry, plays a key role. One then typically introduces a couple of quantities, called upper and lower restricted isometry constants to characterize the isometry properties of random matrices. Those constants are then usually viewed as mathematical objects of interest and their a precise characterization is desirable. The first estimates of these quantities within compressed sensing were given in \cite{CRT,CT}. As the need for precisely estimating them grew further a finer improvements of these initial estimates were obtained in e.g. \cite{BCTsharp09,BT10}. These are typically obtained through a combination of union-bounding strategy and powerful tail estimates of extreme eigenvalues of Wishart (Gaussian) matrices (see, e.g. \cite{Edelman88}). In this paper we attempt to circumvent such an approach and provide an alternative way to obtain similar estimates.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
