EVENT DOSSIER
On September 7, 2026, arXiv cs.LG published the paper “Minimax Lower Bound for Estimating Diffusion-based Local Intrinsic Dimension”, which proposed a method for estimating the minimum lower bound of the local intrinsic dimension based on diffusion models. This research aims to address the theoretical limitations of diffusion models in evaluating the local structural complexity of their generated data, providing a new theoretical basis and analytical tools for understanding the intrinsic dimension of such models through the establishment of a mathematical framework.
本文研究了基于扩散模型的局部内在维度（LID）估计的统计难度。针对 FLIPD 定义的通过高斯平滑密度对数尺度导数得到的有限规模总体泛函，在正则流形模型下证明了该有限规模场与流形维度 $d$ 的差异不超过 $O(\sigma^2)$。研究建立了从 $n$ 个观测中估计该有限规模场的极小极大下界，其阶数为 $(n\sigma^d)^{-1}$，适用范围为 $n^{-1/(2\alpha+d)}\lesssim\sigma\le\sigma_0$。在构造下界覆盖的最小尺度处，该下界对应于非参数率 $n^{-2\alpha/(2\alpha+d)}$。