Optimality for higher-order Hermite interactions

Establish whether the rate N^{-s/(q+2s)} attained by the orthogonal Hermite sieve is minimax-optimal for the bounded-chaos interaction class \mathcal C_{q,m}^{s}(B) when the interaction order satisfies q\ge2.

Background

The structured Hermite sieve has parameter dimension of order dq when each potential term involves at most q coordinates. Balancing estimation and truncation errors yields the rate N{-s/(q+2s)}.

The paper proves matching minimax upper and lower bounds only in the additive case q=1. For higher-order interactions, the derived rate is an attained upper bound, but no corresponding lower bound is established.

References

The estimator attains s/(q+2s) at every q; for q\ge2 its optimality is open.

— Sieve Estimation of Optimal Transport Maps from Paired Data in Gaussian Spaces  (2609.09089 - Jin et al., 8 Sep 2026) in Section 5, subsection 5.2, final paragraph before Remark 5.3

No matching lower bound is available for \mathcal C_{q,m}{s} with q\ge2, or for \mathcal K_q{s,t,\varpi} at every q including q=1, so the exponents there are attained upper bounds whose optimality is open.

— Sieve Estimation of Optimal Transport Maps from Paired Data in Gaussian Spaces  (2609.09089 - Jin et al., 8 Sep 2026) in Section 7, Discussion, paragraph beginning “No matching lower bound”