Derivation of the Sample-Size Scaling of TWO-NN Intrinsic-Dimension Estimates from Molecular Dynamics Trajectories

Abstract

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 $100~\mu$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.