Sharp minimax theory for Sobolev covariance classes with diagonal bias

Derive a sharp minimax theory for covariance estimation over Sobolev classes with integer regularity, accounting for the diagonal-band contribution to the integrated squared bias, and determine whether diagonal-corrected smoothers recover the h^{2\beta} bias rate.

Background

The paper notes that Fourier-decay covariance classes with integer regularity can fail to be Hölder classes of the same order because of a logarithmic singularity along the diagonal. For these Sobolev-type covariances, the integrated squared bias is observed to scale as h{2\beta-1}, rather than h{2\beta}.

The diagonal band therefore determines the bias behavior and may alter minimax rates even though the threshold exponent associated with symmetry is unchanged. A complete minimax treatment and the potential correction of this effect by specialized smoothers remain unresolved.

References

A sharp minimax theory over the Sobolev scale --- where the diagonal band, not the bulk, sets the bias --- would quantify what integer-regularity classes cost, and whether diagonal-corrected smoothers recover h{2\beta}.

— How far can symmetry help? Phase transitions and symmetry selection in sparse functional data analysis  (2608.27055 - Nembe, 27 Aug 2026) in Section 5, Discussion

Whether the rate is attainable uniformly over all of \mathcal Ws(B) is left open.

— Sieve Estimation of Optimal Transport Maps from Paired Data in Gaussian Spaces  (2609.09089 - Jin et al., 8 Sep 2026) in Section 4, subsection 4.2, paragraph immediately preceding Definition 4.1