论文标题
在对称随机矩阵的最小奇异值上
On the smallest singular value of symmetric random matrices
论文作者
论文摘要
我们表明，对于$ n \ times n $随机对称矩阵$ a_n $，其对角的条目是次高斯随机变量的独立副本，该副本具有平均$ 0 $和方差$ 1 $，\ [\ mathbb {p} [p} [s_n（s_n（a_n）\ le l le pe sq.sq rt { O_ξ（ε^{1/8} + \ exp（-Ω_配给（n^{{1/2}）））\ quad \ text {对于所有}ε\ ge 0. \ \ge0。 $（1/8） +η$（带有隐式常数也取决于$η> 0 $）。此外，当$ξ$是一个rademacher随机变量时，我们证明\ [\ mathbb {p} [s_n（a_n）\ leas/\ sqrt {n} {n}] \ le o（ε^{1/8} {1/8} \ quad \ text {对于所有}ε\ ge 0. \ \]特殊情况$ε= 0 $改善了Campos，Mattos，Morris和Morrison的最新结果，这表明$ \ Mathbb {p} [s_n（a_n）= 0] 我们工作中的主要创新是算术结构的新概念 - 中位正规的最小共同点和中值阈值，我们认为，在需要结合矢量不同部分的抗调节信息的情况下，我们认为这应该更有用。
We show that for an $n\times n$ random symmetric matrix $A_n$, whose entries on and above the diagonal are independent copies of a sub-Gaussian random variable $ξ$ with mean $0$ and variance $1$, \[\mathbb{P}[s_n(A_n) \le ε/\sqrt{n}] \le O_ξ(ε^{1/8} + \exp(-Ω_ξ(n^{1/2}))) \quad \text{for all } ε\ge 0.\] This improves a result of Vershynin, who obtained such a bound with $n^{1/2}$ replaced by $n^{c}$ for a small constant $c$, and $1/8$ replaced by $(1/8) + η$ (with implicit constants also depending on $η> 0$). Furthermore, when $ξ$ is a Rademacher random variable, we prove that \[\mathbb{P}[s_n(A_n) \le ε/\sqrt{n}] \le O(ε^{1/8} + \exp(-Ω((\log{n})^{1/4}n^{1/2}))) \quad \text{for all } ε\ge 0.\] The special case $ε= 0$ improves a recent result of Campos, Mattos, Morris, and Morrison, which showed that $\mathbb{P}[s_n(A_n) = 0] \le O(\exp(-Ω(n^{1/2}))).$ The main innovation in our work are new notions of arithmetic structure -- the Median Regularized Least Common Denominator and the Median Threshold, which we believe should be more generally useful in contexts where one needs to combine anticoncentration information of different parts of a vector.
