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. Nearest-neighbor estimators infer it from how the probability of finding a neighbor grows around each sampled point. Because the neighbor distances r decrease as the sample grows, the estimated dimension can depend strongly on the number N of available points, which in molecular dynamics is set by trajectory length and saving frequency. 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 local uniformity produce corrections in powers of N−2/d. Isotropic noise adds a correction proportional to N+2/d; for a five-dimensional hyperplane in 100 noisy features, the extended expansion recovers the dimension within a few percent when noise is moderate. We test the scaling on ten independent 100 μs simulations of alanine dipeptide, represented by the pairwise distances among its ten heavy atoms, whose geometric dimension is 3nat−6=24. Over the investigated range, the estimate does not depend systematically on the temporal spacing between configurations, but increases from $7.6$ to $15.6$ as N grows from 102 to 2×105. Extrapolations that retain corrections through r2, r4, and r6 give limiting dimensions of $24.81$, $23.47$, and $25.30$, respectively. All three estimates lie within 6% of the known dimension and bracket it, and their spread measures the uncertainty due to truncation. The derived scaling provides a practical route from finite-sample estimates to the geometric dimension.
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BibTeX
@misc{capelli2026derivation,
title = {{Derivation of the Sample-Size Scaling of TWO-NN Intrinsic-Dimension Estimates from Molecular Dynamics Trajectories}},
author = {Riccardo Capelli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.04914},
eprint = {2609.04914},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04914},
}
RIS
TY - UNPB
TI - Derivation of the Sample-Size Scaling of TWO-NN Intrinsic-Dimension Estimates from Molecular Dynamics Trajectories
AU - Riccardo Capelli
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2609.04914
UR - https://arxiv.org/abs/2609.04914
AB - The intrinsic dimension of a dataset is the number of independent directions needed to describe the space occupied by its data. Nearest-neighbor estimators infer it from how the probability of finding a neighbor grows around each sampled point. Because the neighbor distances r decrease as the sample grows, the estimated dimension can depend strongly on the number N of available points, which in molecular dynamics is set by trajectory length and saving frequency. 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 local uniformity produce corrections in powers of N^−2/d. Isotropic noise adds a correction proportional to N^+2/d; for a five-dimensional hyperplane in 100 noisy features, the extended expansion recovers the dimension within a few percent when noise is moderate. We test the scaling on ten independent 100 μs simulations of alanine dipeptide, represented by the pairwise distances among its ten heavy atoms, whose geometric dimension is 3n_at−6=24. Over the investigated range, the estimate does not depend systematically on the temporal spacing between configurations, but increases from $7.6$ to $15.6$ as N grows from 10^2 to 2×10^5. Extrapolations that retain corrections through r^2, r^4, and r^6 give limiting dimensions of $24.81$, $23.47$, and $25.30$, respectively. All three estimates lie within 6% of the known dimension and bracket it, and their spread measures the uncertainty due to truncation. The derived scaling provides a practical route from finite-sample estimates to the geometric dimension.
ER -