Extreme Least Singular Values of Gaussian Row Submatrices and a Phase Retrieval Stability Problem

Let F F double struck upper F be either R R double struck upper R or C C double struck upper C , and put d F = dim R ⁡ F dF= RF d Subscript double struck upper F Baseline equals dimension Subscript double struck upper R Baseline double struck upper F . Let A m ∈ F N m × m Aₘ∈ FNₘ× m upper A Subscript m Baseline element of double struck upper F Superscript upper N Super Subscript m Superscript times m have independent standard Gaussian entries and assume N m / m → γ > 1 Nₘ/m→ γ>1 upper N Subscript m Baseline divided by m right arrow gamma greater than 1 . We determine the exponential scale of the smallest least singular value among all square row submatrices: