Minimax Lower Bound for Estimating Diffusion-based Local Intrinsic Dimension

Diffusion-based tools have gained traction for probing the intrinsic geometry of high-dimensional data, yet their fundamental statistical limits have remained largely uncharacterized. Focusing on the finite-scale population functional underlying FLIPD, this work establishes a rigorous theoretical footing. The authors show that the finite-scale field approximates true manifold dimension with an $O(\sigma^2)$ bias, and derive a minimax lower bound of order $(n\sigma^d)^{-1}$ across noise scales $\sigma$. At the smallest resolvable scale, the rate coincides with the classical nonparametric lower bound $n^{-2\alpha/(2\alpha+d)}$, charting the exact statistical price of geometric extraction.