Weakening the sufficient extrapolation condition

Determine whether the sufficient condition \(\boldsymbol{b}\odot\boldsymbol{r}_p\in\ell^p_{\mathsf{M}}\) and \(\|\boldsymbol{b}\odot(\boldsymbol{r}_p-\boldsymbol{1})\|_1<\varepsilon\) can be weakened while retaining the stated out-of-distribution convergence guarantees for holomorphic functions.

Background

The paper uses the monotone ℓp\ell^p condition (b-r-cond-2-main) to construct estimators that work uniformly over unknown anisotropy parameters. The condition is stronger than the basic summability condition used for approximation and is needed in the analysis to control the relevant collection of polynomial subspaces.

Numerical experiments show convergence beyond the domains certified by this condition, suggesting that it may not be sharp. The authors explicitly leave open whether the condition can be relaxed.

References

It is currently unknown whether \ef{b-r-cond-2-main} can be weakened.

— Into the danger zone: stable extrapolation in high-dimensional function and operator learning  (2609.36709 - Adcock et al., 29 Sep 2026) in Remark “Condition (b-r-cond-main) versus (b-r-cond-2-main),” Section 4.1; Section 7, subsection “Weaker conditions and sharper rates”

This suggests the main condition \ef{b-r-cond-2-main} may not be sharp -- improving it is an open problem.

— Into the danger zone: stable extrapolation in high-dimensional function and operator learning  (2609.36709 - Adcock et al., 29 Sep 2026) in Remark “Differences between Theorem ... and the experiments,” Section 5.1.1