标题： 关于对数凹随机矩阵的最小奇异值 显示英文标题

标题： On the Smallest Singular Value of Log-Concave Random Matrices

摘要： 设$A$为一个$N\times n$随机矩阵，其元素为$\mathbb{R}^{Nn}$中一个各向同性对数凹随机向量的坐标。 我们证明了在以下情况下，$A$最小奇异值的尖锐下尾估计： (1) 当$N=n$和$A$来自一个无条件分布，且没有独立性假设；(2) 当$A$的列是独立的且$N\geq n$；(3) 当$A$足够高，即对于任何正数常量$\lambda$，$N\geq (1+\lambda)n$。 显示英文摘要

摘要： Let $A$ be an $N\times n$ random matrix whose entries are coordinates of an isotropic log-concave random vector in $\mathbb{R}^{Nn}$. We prove sharp lower tail estimates for the smallest singular value of $A$ in the following cases: (1) when $N=n$ and $A$ is drawn from an unconditional distribution, with no independence assumption; (2) when the columns of $A$ are independent and $N\geq n$; (3) when $A$ is sufficiently tall, that is $N\geq (1+\lambda)n$ for any positive constant $\lambda$.