Establish a matching minimax upper bound

Establish a minimax upper bound matching the lower bound of order $(n\sigma^d)^{-1}$ for estimating the finite-scale FLIPD field $T_\sigma(\cdot;f)$ on a fixed smooth manifold over the scale range considered in the paper.

Background

The paper proves a minimax squared-risk lower bound of order $(n\sigma^d)^{-1}$ for estimating the finite-scale diffusion-based local intrinsic dimension field $T_\sigma(\cdot;f)$ from $n$ observations, for scales satisfying $n^{-1/(2\alpha+d)}\lesssim\sigma\le\sigma_0$. The result quantifies the statistical difficulty of recovering the scale-dependent population functional, rather than the zero-noise manifold dimension.

The authors explicitly state that the lower bound is not accompanied by a matching upper bound. Consequently, whether the lower-bound rate is minimax-optimal remains unresolved, even in the fixed smooth-manifold setting analyzed in the paper.