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

Abstract

Let $ξ$ 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 $ξ$, where $N_m/m\toγ>1$. For each set $I\subset[N_m]$ with $|I|=m$, let $(A_m)_I$ denote the row submatrix indexed by $I$, and define $M_m(A_m):=\min_{I\subset[N_m],\,|I|=m}σ_{\min}((A_m)_I)$. We determine its exponential scale: $\frac{1}{m}\log M_m(A_m)\xrightarrow{\mathbb{P}}-h(γ)$, where $h(γ):=γ\logγ-(γ-1)\log(γ-1)$. 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 $\varepsilon\in(0,h(γ))$, the probability of an $\varepsilon$-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.