We provide a polynomial lower bound on the minimum singular value of an $m\times m$ random matrix $M$ with jointly Gaussian entries, under a polynomial bound on the matrix norm and a global small-ball probability bound $$\inf_{x,y\in S^{m-1}}\mathbb{P}\left(\left|x^* M y\right|>m^{-O(1)}\right)\ge \frac{1}{2}.$$ With the additional assumption that $M$ is self-adjoint, the global small-ball probability bound can be replaced by a weaker version. We establish two matrix anti-concentration inequalities, which lower bound the minimum singular values of the sum of independent positive semidefinite self-adjoint matrices and the linear combination of independent random matrices with independent Gaussian coefficients. Both are under a global small-ball probability assumption. As a major application, we prove a better singular value bound for the Krylov space matrix, which leads to a faster and simpler algorithm for solving sparse linear systems. Our algorithm runs in $\tilde{O}\left(n^{\frac{3\omega-4}{\omega-1}}\right)=O(n^{2.2716})$ time where $\omega<2.37286$ is the matrix multiplication exponent, improving on the previous fastest one in $\tilde{O}\left(n^{\frac{5\omega-4}{\omega+1}}\right)=O(n^{2.33165})$ time by Peng and Vempala.

翻译：$165=m=mm 随机基质 $M$的最小单值上,我们提供一个多元下限 165=mm 随机基质 $M$ 与 Gausian 联合条目,在矩阵规范的多元基质约束下,并提供一个全球小球概率以$\inf ⁇ xx,ym-1=mathb{P ⁇ left{left{left}M\x ⁇ M y\r\\\\\\\\m\\\%1}O(1)\\\right\\\\\\\\\\%2}$美元为最低单值,加上一个额外的假设,即$M$(美元)是自调整的,全球小球概率的值可以用一个较弱的版本来取代。我们设置两个矩阵的反浓缩不平等性基质基质基质,这个基质基质的最小值是独立的正正正半确定基质自联合基质基质基质基质总和独立高斯系数的最小值。 两种基质概率假设的主要应用,我们证明Krylov=_____________________________%______}l=xxxl=xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx