Positivity-compatible off-diagonal packing for matching minimax lower bounds

Construct a positivity-compatible packing of covariance perturbations supported on h^{-2}/q off-diagonal cells with amplitude h^\beta, thereby restoring the matching minimax exponent \beta/(\beta+1) for sparse functional covariance estimation under a cyclic symmetry of order q.

Background

The paper establishes a lower bound for the invariant covariance class using a positivity-preserving construction based on orbit sums of rank-one perturbations. This construction carries only diagonal information and yields a lower-bound exponent smaller than the estimator's upper-bound exponent by one factor of the bandwidth h.

The unresolved issue is whether one can retain the full count of h^{-2}/q off-diagonal cells while preserving positive semidefiniteness, the Hölder constraint, and adequate likelihood control. Solving this problem would close the polynomial gap between the upper and lower minimax rates below saturation and would make the transition result sharp.