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

We conjecture that the resulting convergence rate for this simplified model is minimax optimal.

— Nonparametric Estimation of the Diffusion-Interaction Function in Particle Systems  (2609.24374 - Amorino et al., 21 Sep 2026) in Section 4, subsection “Understanding the convergence rate”

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”