Overparametrization: Hessian-rank bound

The linear loss of a QNN $U_\theta$ with observable $O$ and signed data operator $A=\sum_\mu c_\mu\,\rho_\mu$ (a signed combination of data states $\rho_\mu$) is the single expectation $$\ell(\theta)=\operatorname{Tr}\!\big[\,O\,U_\theta\,A\,U_\theta^{\dagger}\,\big],$$ linear in both $O$ and $A$. With $d$ the Hilbert-space dimension and $r=\min\{\operatorname{rank}A,\operatorname{rank}O\}$, assume (as named hypotheses, per Larocca) that at a minimizer $\theta_*$ the loss Hessian obeys both $\operatorname{rank}\nabla^2\ell(\theta_*)\le\dim\mathfrak g$ and $\operatorname{rank}\nabla^2\ell(\theta_*)\le 2dr-r^2-r$. Then $$\operatorname{rank}\nabla^2\ell(\theta_*)\le\min\{\dim\mathfrak g,\;2dr-r^2-r\}.$$