The smallest singular value of products of random combinatorial and deterministic matrices

Abstract

Let $p\in(0,1/2]$ be fixed, and let $d,n$ be positive integers satisfying $d=pn$. Let $Q_n$ be an $n\times n$ random combinatorial matrix whose rows are independent and uniformly distributed over the vectors in $\{0,1\}^n$ having exactly $d$ ones. For an arbitrary deterministic invertible $n\times n$ matrix $M$, we establish two-sided bounds for the smallest singular value of $Q_nM$. More precisely, for every $\varepsilon\in(0,1)$, \[ \mathbb P\!\left( \varepsilon\le s_n(Q_nM)\cdot \|M^{-1}\|_{\mathrm{HS}} \le \varepsilon^{-2} \right) \ge 1-C\left( \varepsilon+d^{-1/2}+e^{-c\,\operatorname{sr}(M^{-1})} \right)-2e^{-cn}, \] where positive constants $C, c$ depend only on $p$ and $\operatorname{sr}$ denotes the stable rank. Consequently, for any sequence of deterministic invertible matrices $M$ satisfying $\operatorname{sr}(M^{-1})\to\infty$, $s_n(Q_nM)$ is typically of order $\|M^{-1}\|_{\mathrm{HS}}^{-1}$. In particular, taking $M=\mathbf I_n$ recovers the known $n^{-1/2}$ scale for $s_n(Q_n)$.