Generality of finite-dimensional data-driven error underestimation

Determine whether the observed tendency of the finite-dimensional estimator \(\ell^{(p)}/\theta_j^{(p)}\) to underestimate the true finite-dimensional out-of-subspace error \(\sin^2\angle_p(h_j,\mathcal B)\) is generic beyond the reported simulation experiment.

Background

The paper proves that the data-driven ratio $\ell^{(p)}/\theta_j^{(p)}$ and the true out-of-subspace error have the same limit as the number of variables tends to infinity. The finite-dimensional values need not coincide, however.

In the reported simulation, the estimator tended to underestimate the realized out-of-subspace error for moderate values of $p$. The authors explicitly state that they have not determined whether this behavior is generic, leaving its generality unresolved.