Minimax Lower Bound for Estimating Diffusion-based Local Intrinsic Dimension

• URL: http://arxiv.org/abs/2609.04822v1
• Date: Fri, 04 Sep 2026 07:18:53 GMT
• Title: Minimax Lower Bound for Estimating Diffusion-based Local Intrinsic Dimension
• Abstract summary: We study estimation of the finite-scale population functional underlying FLIPD.<n>We show uniformly over the model class that the finite-scale field differs from the manifold dimension $d$ by at most $O(2)$.<n>We then establish a minimax lower bound of order $(nd)-1$ for estimating this finite-scale field from $n$ observations.
• Score: 2.5556158622876195
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $σ^d$, 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 $O(σ^2)$. We then establish a minimax lower bound of order $(nσ^d)^{-1}$ for estimating this finite-scale field from $n$ observations, for $n^{-1/(2α+d)}\lesssimσ\leσ_0$. At the smallest scale covered by our lower-bound construction, the bound becomes the nonparametric rate $n^{-2α/(2α+d)}$.

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