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

Abstract

Let A$A$ be an n×n$n\times n$ real matrix, and let M$M$ be an n×n$n\times n$ random matrix whose entries are independent and identically distributed sub‐Gaussian random variables with mean 0 and variance 1. We make two contributions to the study of sn(A+M)$s_n(A+M)$, the smallest singular value of A+M$A+M$. (1)We show that for all ε⩾0$\epsilon \geqslant 0$, double-struckP[snfalse(A+Mfalse)⩽ε]=O(εn)+2e−normalΩfalse(nfalse),\begin{equation*} \mathbb {P}[s_n(A + M) \leqslant \epsilon ] = O(\epsilon \sqrt {n}) + 2e^{-\Omega (n)}, \end{equation*}provided only that A$A$ has normalΩfalse(nfalse)$\Omega (n)$ singular values which are Ofalse(nfalse)$O(\sqrt {n})$. This extends a well‐known result of Rudelson and Vershynin, which requires all singular values of A$A$ to be Ofalse(nfalse)$O(\sqrt {n})$. (2)We show that any bound of the form sup∥A∥⩽nC1double-struckP[snfalse(A+Mfalse)⩽n−C3]⩽n−C2\begin{equation*} \sup _{\Vert A\Vert \leqslant n^{C_1}}\mathbb {P}[s_n(A+M)\leqslant n^{-C_3}] \leqslant n^{-C_2} \end{equation*}must have C3=normalΩ(C1C2)$C_3 = \Omega (C_1 \sqrt {C_2})$. This complements a result of Tao and Vu, who proved such a bound with C3=O(C1C2+C1+1)$C_3 = O(C_1C_2 + C_1 + 1)$, and counters their speculation of possibly taking C3=O(C1+C2)$C_3 = O(C_1 + C_2)$.

Quantitative invertibility of non-Hermitian random matrices

“…In [40], it is shown that the above small ball probability bound can be significantly improved to match the average-case result of Rudelson and Vershynin [70], under the assumption that a positive fraction of the singular values of M are of order O( √ n). More specifically, for every c ∈ (0, 1) and C > 0, and any fixed matrix M with s n−⌊cn⌋ (M ) ≤ C√ n, one has…”

“…On the other hand, it was observed that for certain discrete random matrices, such as random sign (Bernoulli) matrices, no shift-independent small ball probability bounds for s min (A + M ) are possible [87,90,40]. In particular, it is shown in [40] that, assuming A has i.i.d entries taking values ±1 with probability 1/4 and zero with probability 1/2, every L ≥ 1, and every positive integer K, sup…”

“…where C, c > 0 may only depend on L and K. The smoothed analysis of the matrix condition number for discrete distributions was carried out in works [82,87,38,40] (see also references therein). The following result was proved in [87].…”