Extreme least singular values of Gaussian row submatrices and a phase retrieval stability problem

Abstract

Let $\mathbb F\in\{\mathbb R,\mathbb C\}$ and $d_{\mathbb F}=\dim_{\mathbb R}\mathbb F$. If $A_m\in\mathbb F^{N_m\times m}$ has independent standard Gaussian entries and $N_m/m\toγ>1$, then \[ \min_{\substack{T\subset[N_m]\\ |T|=m}} σ_{\min}(A_{m,T}) = \left(\frac{γ^γ}{(γ-1)^{γ-1}}\right)^{-m/d_{\mathbb F}+o_P(m)} . \] If $N_m=γm+O(1)$, the convergence of $m^{-1}\log M_m^{\mathbb F}$ has probability error $O(m^{-1})$. In particular, at the real phase-retrieval threshold $N=2m-1$, \[ ω(A_m)=4^{-m+o_P(m)}, \] so the Gaussian Balan--Wang critical exponential base is $1/4$.

Disclosure

“research and innovation programme (grant agreement No. 101041711), by the Simons Foundation, by Heights Labs, by the Israel Science Foundation (grant number 2258/19) and by the Israel Science Foundation (ISF Grant 4101/25). Declaration of generative AI and AI-assisted technologies in the manuscript preparation process During the preparation of this work the author used generative AI tools (large language model assistants) to assist with drafting, revision and bib- liographic organiza”

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2026-08-24T00:34:12.480Z

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