EXTREME LEAST SINGULAR VALUES OF GAUSSIAN ROW SUBMATRICES AND A PHASE RETRIEVAL STABILITY PROBLEM

Abstract Let F $\mathbb {F}$ double struck upper F be either R $\mathbb {R}$ double struck upper R or C $\mathbb {C}$ double struck upper C , and put d F = dim R ⁡ F $d_{\mathbb {F}}=\dim _{\mathbb {R}}\mathbb {F}$ 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 $A_m\in \mathbb {F}^{N_m\times m}$ 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 $N_m/m\to \gamma>1$ 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: