发表机构
University of Southern California; Georgia Institute of Technology（南加州大学; 佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在二阶矩与反集中假设下，证明了随机矩阵中间奇异值的最优上界及指数尾估计，结合下界得到其渐近阶，还推广到矩形随机矩阵的情形。
AI 中文摘要
设$A=(\xi_{ij})$为$n\times n$随机矩阵，其独立且未必同分布的实元素满足$\mathbb E\xi_{ij}=0$、$\mathbb E\xi_{ij}^{2}=1$，且对固定的$a>0$和$b\in(0,1)$，有$\sup_{z\in\mathbb R}\mathbb P(|\xi_{ij}-z|<a)\le b$。我们证明，对每个$\delta\in(0,1)$，存在仅依赖于$a,b,\delta$的常数$c,C>0$，使得对所有$t\ge1$和$1\le l\le(1-\delta)n$，有$\mathbb P\left( s_{n+1-l}(A)>Ct\frac l{\sqrt n} \right) \le \exp\left(-c\min\{tl,n\}\right)$。因此，在仅需方差的矩假设下，除固定比例的最大奇异值外，其余奇异值均满足阶为$l/\sqrt n$的最优上界，并带有指数型上尾估计。结合矩形最小奇异值的下界，可得$s_{n+1-l}(A)\asymp l/\sqrt n$，其失败概率随$l$呈指数级小。当$N\times n$矩阵满足$N-n+l\le(1-\delta)N$时，相同论证给出矩形尺度为$\sqrt{N+1}-\sqrt{n-l+1}$。
英文摘要
Let $A=(ξ_{ij})$ be an $n\times n$ random matrix with independent, not necessarily identically distributed, real entries satisfying \[ \mathbb Eξ_{ij}=0,\qquad \mathbb Eξ_{ij}^{2}=1,\qquad \sup_{z\in\mathbb R}\mathbb P(|ξ_{ij}-z|<a)\le b \] for fixed $a>0$ and $b\in(0,1)$. We prove that, for every $δ\in(0,1)$, there are constants $c,C>0$, depending only on $a,b,δ$, such that \[ \mathbb P\left( s_{n+1-l}(A)>Ct\frac l{\sqrt n} \right) \le \exp\!\left(-c\min\{tl,n\}\right) \] for every $t\ge1$ and every $1\le l\le(1-δ)n$. Thus, with no moment assumption beyond variance, all but a fixed proportion of the largest singular values satisfy the optimal upper bound of order $l/\sqrt n$ with an exponential upper-tail estimate. Combined with the lower bound of the rectangular least singular value bound, this gives $s_{n+1-l}(A)\asymp l/\sqrt n$ with failure probability exponentially small in $l$. The same argument gives the rectangular scale $\sqrt{N+1}-\sqrt{n-l+1}$ for $N\times n$ matrices whenever $N-n+l\le(1-δ)N$.
Comments35 pages
