Value of ω(A) for Gaussian matrices and relation to the Balan–Wang bound

Determine the typical magnitude (or asymptotic behavior) of ω(A) when A has i.i.d. standard Gaussian entries, and ascertain whether ω(A) obeys an exponential decay upper bound of the form ω(A) ≤ C (max_k ||A_k||) β^M; if so, identify the corresponding value of β.

Background

The functional ω(A) captures the minimal smallest singular value over complements of spanning subsets and is tightly connected to injectivity and stability of real phase retrieval. Understanding ω(A) in random ensembles such as Gaussian is central to average-case stability.

A positive resolution matching the conjectured exponential decay would quantify stability for Gaussian measurement designs and calibrate β.

References

This also motivates the following question (for which I do not know the state of the art, but a quick search did not reveal that the answer is already known). What is the value of $\omega(A)$ for $A$ with iid standard gaussian entries? Does it satisfy the bound in the conjecture? If so, what is $\beta$?

— Randomstrasse101: Open Problems of 2025  (2603.29571 - Bandeira et al., 31 Mar 2026) in Open Problem, Section “Injectivity and Stability of Phase Retrieval (ASB)” (Entry 10)