Existence of the limiting Hessian bound

Determine whether the limit used in the limiting Hessian lower-bound argument exists when taking the successive liminf limits in the metric-collapse and Calabi–Yau asymptotic regimes.

Background

The paper analyzes lower bounds for the covariant Hessian of the loss under metric collapse and in the Calabi–Yau regime. These bounds are expressed using successive liminf operations over parameters such as the minimum eigenvalue, the Fisher-type upper bound, the Hessian constant, and the network width. The authors explicitly note that their use of liminf is informal because they do not investigate whether the corresponding limit exists. Establishing existence would make the asymptotic lower-bound argument mathematically more rigorous.