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.

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“ps as potential financial competing interests. Declaration of generative AI and AI-assisted technologies in the manuscript preparation process During the preparation of this work, the authors used generative AI tools to accelerate drafting and revision. All mathematical claims, proofs, numerical interpretations, and bibliographic information were subsequently reviewed and edited by the authors, who take full responsibility for the content of the”

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2026-08-23T05:00:09.461721Z

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