On the smoothed analysis of the smallest singular value with discrete noise

Let A $A$ be an n × $n\times n$ real matrix, and let M $M$ random matrix whose entries are independent identically distributed sub-Gaussian variables with mean 0 variance 1. We make two contributions to the study of s ( + ) $s_n(A+M)$ , smallest singular value $A+M$ . (1) show that for all ε ⩾ $\epsilon \geqslant 0$ P [ ⩽ ] = O 2 e − Ω \begin{equation*} \mathbb {P}[s_n(A M) \leqslant \epsilon O(\epsilon \sqrt {n}) 2e^{-\Omega (n)}, \end{equation*} provided only has $\Omega (n)$ values which $O(\sqrt {n})$ This extends a well-known result Rudelson Vershynin, requires (2) any bound form sup ∥ C 1 3 \sup _{\Vert A\Vert n^{C_1}}\mathbb {P}[s_n(A+M)\leqslant n^{-C_3}] n^{-C_2} must have $C_3 \Omega (C_1 {C_2})$ complements Tao Vu, who proved such O(C_1C_2 C_1 1)$ counters their speculation possibly taking O(C_1 C_2)$