从分子动力学轨迹推导TWO-NN本征维数估计的样本量标度

中文摘要 AI 辅助

数据集的本征维数是描述数据点所占据空间所需的独立方向数量。基于近邻的估计器通过每个采样点周围找到近邻点的概率增长情况来推断该维数。由于近邻点之间的距离r会随样本量增大而减小，估计的维数会强烈依赖于可用点的数量。本文推导了从平滑d维空间抽取数据的TWO-NN估计器的大样本行为：典型近邻距离按N^{-1/d}标度，局部均匀分布的平滑偏差会产生与N^{-2/d}成正比的逐次修正项。我们使用丙氨酸二肽的10次独立100μs模拟轨迹验证该结果，构型由10个重原子间的所有两两距离表示，该表示的已知几何维数为3n_at-6=24。在研究范围内，TWO-NN估计值对构型的时间间隔无系统性依赖，但会随样本量从10²增至2×10⁵而从约7.5升至15.6。保留r²、r⁴、r⁶阶修正的外推结果分别给出极限维数25.23、22.89、27.00，三个估计值均接近已知维数且共同将其包含在内，支持所提出的标度关系；它们的离散程度可直接估计因截断产生的系统不确定性。因此，推导的标度关系解释了TWO-NN估计值对样本量的强烈依赖，并提供了从有限样本估计值到潜在几何维数的实用途径。

英文摘要

The intrinsic dimension of a dataset is the number of independent directions needed to describe the space occupied by its data. Estimators based on nearest neighbors infer this number from how the probability to find a neighbor point grows around each sampled point. Because the distances $r$ between neighbor points decrease as the sample grows, the estimated dimension can depend strongly on the number of available points. Here, we derive the large-sample behavior of the TWO-NN estimator for data drawn from a smooth $d$-dimensional space. The typical nearest-neighbor distance scales as $N^{-1/d}$, and smooth deviations from a locally uniform distribution produce successive corrections proportional to $N^{-2/d}$. We test this result using the trajectories coming from ten independent $100~μ$s simulations of alanine dipeptide. Configurations are represented by all pairwise distances among the ten heavy atoms. This representation has a known geometric dimension of $3n_{\mathrm{at}}-6=24$. Over the investigated range, the TWO-NN estimate shows no systematic dependence on the temporal spacing between configurations, but increases from approximately $7.5$ to $15.6$ as the sample size grows from $10^2$ to $2\times10^5$. Extrapolations that retain corrections through $r^2$, $r^4$, and $r^6$ give limiting dimensions of $25.23$, $22.89$, and $27.00$, respectively. All three estimates lie close to the known dimension and collectively bracket it, supporting the proposed scaling. Their spread provides a direct estimate of the systematic uncertainty associated with the truncation. The derived scaling therefore explains the strong sample-size dependence of TWO-NN and provides a practical route from finite sample estimates to the underlying geometric dimension.

发表机构

• Università degli Studi di Milano（米兰大学）

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