标题： 关于随机半正定矩阵最小特征值的比较定理 显示英文标题

标题： Comparison theorems for the minimum eigenvalue of a random positive-semidefinite matrix

摘要： 本文建立了一个关于独立随机半正定矩阵之和的最小特征值的新比较原理。 该原理表明，矩阵和的最小特征值可以通过一个高斯随机矩阵的最小特征值来控制，而这个高斯随机矩阵继承了加数的统计特性。这种方法的强大之处在于处理高斯随机矩阵时拥有的大量工具。 作为应用，本文给出了高维统计学中一些新旧结果的简洁且概念性的证明，并解决了计算线性代数领域关于非常稀疏随机矩阵注入性质的一个长期悬而未决的问题。 显示英文摘要

摘要： This paper establishes a new comparison principle for the minimum eigenvalue of a sum of independent random positive-semidefinite matrices. The principle states that the minimum eigenvalue of the matrix sum is controlled by the minimum eigenvalue of a Gaussian random matrix that inherits its statistics from the summands. This methodology is powerful because of the vast arsenal of tools for treating Gaussian random matrices. As applications, the paper presents short, conceptual proofs of some old and new results in high-dimensional statistics. It also settles a long-standing open question in computational linear algebra about the injectivity properties of very sparse random matrices.