AI 中文总结
该研究针对经验度量-测度拉普拉斯算子的谱，在弱正则条件下建立了其与总体对应算子特征值的相对误差界，改进了现有技术，适用于度量图等多种空间的测度。
AI 中文摘要
非参数估计量的方差通常对被估计对象的正则性不敏感。我们针对带宽h>0固定时的图拉普拉斯矩阵的谱，建立了这类性质。具体而言，给定来自波兰度量空间上概率测度μ的n个独立同分布样本，在谱间隙条件下，我们将经验加权拉普拉斯算子Δ_μₙ^h的特征值与总体对应算子Δ_μ^h的特征值进行比较，对于阶小于h⁻²的特征值，将相对误差限制为1/√(nv_μ(h))，其中v_μ(h)是半径为h的球的最小质量。该界对μ的正则性条件要求极弱：若μ属于我们引入的粗PI测度类，则满足该条件。该类包含度量图上的测度、具有足够正则边界、角点或分支点的空间，以及这些空间在O(h)尺度下的离散化或加厚版本。即使是流形上具有正则性s>2的密度的测度（目前唯一已知的情况），我们的界也通过消除对数因子改进了现有技术。
英文摘要
The variance of nonparametric estimators is typically insensitive to the regularity of the object being estimated. We establish such a property for the spectra of graph Laplacian matrices at a fixed bandwidth $h>0$. Specifically, given $n$ i.i.d. samples from a probability measure $μ$ on a Polish metric space, we compare the eigenvalues of the empirical weighted Laplacian operator $Δ_{μ_n}^h$ to those of the population counterpart $Δ_μ^h$ under a spectral gap condition, bounding the relative error by $1/\sqrt{nv_μ(h)}$ for eigenvalues of order smaller than $h^{-2}$, where $v_μ(h)$ is the smallest mass of a ball of radius $h$. This bound requires very weak regularity conditions on $μ$: it is satisfied if $μ$ belongs to the class of coarse PI measures that we introduce. This class contains measures on metric graphs, spaces with sufficiently regular boundaries, corners, or branch points, together with discretizations or thickenings of these at scale $O(h)$. Even for measures having densities of regularity $s>2$ on manifolds (the only known case so far), our bound improves on the state-of-the-art by shaving off logarithmic factors.
