Refine search
Actions for selected content:
1 results
EXTREME LEAST SINGULAR VALUES OF GAUSSIAN ROW SUBMATRICES AND A PHASE RETRIEVAL STABILITY PROBLEM
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 14 October 2026, pp. 1-9
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
-
Let $\mathbb {F}$
be either $\mathbb {R}$
or $\mathbb {C}$
, and put $d_{\mathbb {F}}=\dim _{\mathbb {R}}\mathbb {F}$
. Let $A_m\in \mathbb {F}^{N_m\times m}$
have independent standard Gaussian entries and assume $N_m/m\to \gamma>1$
. We determine the exponential scale of the smallest least singular value among all square row submatrices: $$ \begin{align*} M_m^{\mathbb{F}}=\min_{T\subset [N_m],\, |T|=m}\sigma_{\mathrm{min}}(A_{m,T}). \end{align*} $$
With $c_\gamma =\gamma ^\gamma /(\gamma -1)^{\gamma -1}$
, we prove $m^{-1}\log M_m^{\mathbb {F}}\xrightarrow {P}-d_{\mathbb {F}}^{-1}\log c_\gamma $
and, when $N_m=\gamma m+O(1)$
, the convergence holds at rate $O(m^{-1})$
. As an application, for real phase retrieval at the critical number $N=2m-1$
of measurements, the Balan–Wang stability parameter satisfies $\omega (A_m)=4^{-m+o_P(m)}$
, so the sharp exponential base in the Gaussian Balan–Wang problem is $1/4$
. The lower estimate extends to all bounded-density ensembles and continuity is necessary.
