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 $N_m/m\to\gamma>1$, then \[ \min_{\substack{T\subset[N_m]\\ |T|=m}} \sigma_{\min}(A_{m,T}) = \left(\frac{\gamma^\gamma}{(\gamma-1)^{\gamma-1}}\right)^{-m/d_{\mathbb F}+o_P(m)} . \] If $N_m=\gamma m+O(1)$, the convergence of $m^{-1}\log M_m^{\mathbb F}$ has probability error $O(m^{-1})$. In particular, at the real phase-retrieval threshold $N=2m-1$, \[ \omega(A_m)=4^{-m+o_P(m)}, \] so the Gaussian Balan--Wang critical exponential base is $1/4$.