Minimax Lower Bound for Estimating Diffusion-based Local Intrinsic Dimension

Abstract: While diffusion-based methods have recently emerged as effective tools for probing the intrinsic geometry of high-dimensional data, their statistical difficulty remains largely unexplored. We study estimation of the finite-scale population functional underlying FLIPD (Kamkari et al., 2024; arXiv:2406.03537), a diffusion-based local intrinsic dimension (LID) quantity defined through the logarithmic scale derivative of a Gaussian-smoothed density. Intuitively, Gaussian smoothing turns local dimension into a scale law: near a $d$-dimensional manifold, the kernel mass grows like $σ<sup>d$, so differentiating with respect to the noise scale reveals the intrinsic exponent. Under a regular manifold model, we show uniformly over the model class that the finite-scale field differs from the manifold dimension $d$ by at most $O(σ<sup>2)$. We then establish a minimax lower bound of order $(nσ<sup>d)<sup>{-1}$ for estimating this finite-scale field from $n$ observations, for $n<sup>{-1/(2α+d)}\lesssimσ\leσ_0$. At the smallest scale covered by our lower-bound construction, the bound becomes the nonparametric rate $n<sup>{-2α/(2α+d)}$.