# Minimax Lower Bound for Estimating Diffusion-based Local Intrinsic Dimension

arXiv: https://arxiv.org/abs/2609.04822 · HTML: https://mlsanity.com/paper/2609.04822/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 19 references that are in the map.
- First posted on arXiv 4 Sep 2026.

## Builds on (most cited first, 8 of 8)

- [Characterizing Adversarial Subspaces Using Local Intrinsic Dimensionality](https://mlsanity.com/paper/1801.02613/) (2018)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models](https://mlsanity.com/paper/2406.03537/) (2024)
- [LIDL: Local Intrinsic Dimension Estimation Using Approximate Likelihood](https://mlsanity.com/paper/2206.14882/) (2022)
- [Minimax Rates for Estimating the Dimension of a Manifold](https://mlsanity.com/paper/1605.01011/) (2016)
- [A Deep Generative Approach to Stratified Learning](https://mlsanity.com/paper/2604.10650/) (2026)
- [Local Hessian Spectral Filtering for Robust Intrinsic Dimension Estimation](https://mlsanity.com/paper/2605.01221/) (2026)

## Abstract (arXiv)

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 σd, 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 (nσd)-1 for estimating this finite-scale field from n observations, for n-1/(2α+d)lesssimσ≤σ0. At the smallest scale covered by our lower-bound construction, the bound becomes the nonparametric rate n-2α/(2α+d).