Extreme least singular values of random row submatrices with bounded-density subgaussian entries

Abstract

Let $\xi$ be a centered real subgaussian random variable with positive variance and a bounded Lebesgue density, and let $A_m\in {\mathbb{R}}^{N_m\times m}$ have independent entries distributed as $\xi$, where $N_m/m\to \gamma >1$. For each set $I\subset [N_m]$ with $|I|=m$, let $(A_m{)}_I$ denote the row submatrix indexed by $I$, and define $M_m(A_m):={\min }_{I\subset [N_m],|I|=m}{\sigma }_{\min }((A_m{)}_I)$. We determine its exponential scale: $\frac{1}{m}\log M_m(A_m)\stackrel{\mathbb{P}}{\to }-h(\gamma )$, where $h(\gamma ):=\gamma \log \gamma -(\gamma -1)\log (\gamma -1)$. This extends the corresponding real Gaussian result. The main new ingredient is an upper-tail argument that avoids uniform control over exponentially many random hyperplanes. We combine a density-level local central limit theorem for delocalized directions, an averaged delocalization estimate for hyperplane normals, an exponential bound for nearly parallel pairs, and amplification using a linear number of independent probe rows. For every fixed $\epsilon \in (0,h(\gamma ))$, the probability of an $\epsilon$-deviation is at most $C\exp (-c\sqrt{m})$ for all sufficiently large $m$. Under the canonical coupling induced by a single infinite i.i.d. array, this summable deviation estimate yields a uniform almost-sure exponential law over every compact range of aspect ratios. In particular, at the real phase-retrieval threshold $N_m=2m-1$, the Balan--Wang stability parameter has exponential base $1/4$ in probability and, under this coupling, almost surely.