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Life sciences · Preprint
arXiv · September 4, 2026
Raises a question worth testing. It does not answer one.
This preprint establishes a minimax lower bound of order $(nσ^d)^{-1}$ for estimating a diffusion-based local intrinsic dimension (LID) quantity under a regular manifold model. The work characterizes the statistical difficulty of the estimation problem theoretically but does not provide empirical validation, algorithmic analysis, or demonstration that this bound is tight.
Theoretical analysis.
Under regular manifold model, finite-scale field differs from manifold dimension d by at most O(σ²) Minimax lower bound of order (nσ^d)^{-1} for n observations, valid for n^{-1/(2α+d)} ≤ σ ≤ σ₀ At smallest scale, lower bound becomes nonparametric rate n^{-2α/(2α+d)}
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This is a theoretical analysis establishing minimax lower bounds for a statistical estimation problem; it raises and partially answers a question about statistical difficulty but does not empirically validate the practical utility or achievability of these bounds.
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Quoted from the source exactly as published.
Graded across the dimensions that decide whether you should act, each from what the source actually supports. There is no single score, and where a dimension was not assessed it says so.
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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