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^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(σ²). 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)≲σ≤σ₀. At the smallest scale covered by our lower-bound construction, the bound becomes the nonparametric rate n^{−2α/(2α+d)}.

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BibTeX

@misc{seo2026minimax,
  title = {{Minimax Lower Bound for Estimating Diffusion-based Local Intrinsic Dimension}},
  author = {Jaehee Seo and Wontae Jeong and Jisu Kim},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04822},
  eprint = {2609.04822},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04822},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Minimax Lower Bound for Estimating Diffusion-based Local Intrinsic Dimension
AU  - Jaehee Seo
AU  - Wontae Jeong
AU  - Jisu Kim
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.04822
UR  - https://arxiv.org/abs/2609.04822
AB  - 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)≲σ≤σ_0. At the smallest scale covered by our lower-bound construction, the bound becomes the nonparametric rate n^−2α/(2α+d).
ER  -

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