Source-linked AI summary
Estimating the intrinsic dimension of datasets by a minimal neighborhood information
Elena Facco, Maria d'Errico, Alex Rodriguez, Alessandro Laio
TL;DR
Estimating intrinsic dimension is challenging for curved manifolds and non-uniform point densities. The paper introduces TWO-NN, which uses only the first two nearest neighbors and combines it with block analysis to resolve scale-dependent dimensions. The method is theoretically density-independent under local uniformity and is applied to noisy, image, and molecular-dynamics datasets.
Problem
Estimating the minimal number of variables needed to describe relevant dataset features is difficult when data lie on curved manifolds with non-uniform density.
Method
TWO-NN estimates intrinsic dimension from the distances to each point’s first two nearest neighbors and uses block analysis to examine scale dependence.
Results
TWO-NN provides consistent measures in general and supports intrinsic-dimension estimation for manifolds perturbed by high-dimensional noise, image data, and molecular-dynamics configurations.
Takeaways & Limitations
Using a minimal neighborhood reduces the effects of density inhomogeneities and curvature while avoiding mixing dataset features across scales.
Takeaways & Limitations
The theoretical derivation assumes density is constant around each point, while the algorithm requires local uniformity over the second-neighbor range.
Abstract
from arXiv · showhide
Analyzing large volumes of high-dimensional data is an issue of fundamental importance in data science, molecular simulations and beyond. Several approaches work on the assumption that the important content of a dataset belongs to a manifold whose Intrinsic Dimension (ID) is much lower than the crude large number of coordinates. Such manifold is generally twisted and curved, in addition points on it will be non-uniformly distributed: two factors that make the identification of the ID and its exploitation really hard. Here we propose a new ID estimator using only the distance of the first and the second nearest neighbor of each point in the sample. This extreme minimality enables us to reduce the effects of curvature, of density variation, and the resulting computational cost. The ID estimator is theoretically exact in uniformly distributed datasets, and provides consistent measures in general. When used in combination with block analysis, it allows discriminating the relevant dimensions as a function of the block size. This allows estimating the ID even when the data lie on a manifold perturbed by a high-dimensional noise, a situation often encountered in real world data sets. We demonstrate the usefulness of the approach on molecular simulations and image analysis.
1 SCIentIfIC REPOrTS | 7: 12140 | DOI:10.1038/s41598-017-11873-y
The paper frames intrinsic-dimension estimation as difficult for curved, non-uniform manifolds and introduces TWO-NN, which uses only the first two nearest-neighbor distances. It further examines asymptotic convergence, density and curvature effects, scale-dependent dimensions, image datasets, and molecular dynamics.
- Motivation: Existing intrinsic-dimension methods include projection, distance-preservation, local-connectivity, and fractal approaches.Fractal methods can require exponentially many points as dimension increases.
- Method: TWO-NN estimates intrinsic dimension using only each point’s first and second nearest-neighbor distances.This minimal neighborhood reduces the influence of density inhomogeneities.
- Method: The estimator derives a distribution of the ratio of second-to-first neighbor distances that depends on intrinsic dimension but not local density.The derivation assumes density is approximately constant over the second-neighbor length scale.
- Scope: The study evaluates TWO-NN under non-uniform density and curvature, tests multiscaling for noisy data, and applies it to image and molecular-dynamics datasets.The applications include the Isomap face dataset, MNIST, and molecular configurational space.
Results
TWO-NN fits an intrinsic dimension from the empirical distribution of ratios between second- and first-neighbor distances. Its local construction reduces sensitivity to density variation and supports scale-dependent analysis, with tests spanning synthetic, image, and molecular-dynamics data.
- Estimator construction: For each point, the sorted neighbor distances r1, r2, …, rk define r1 as the nearest-neighbor distance and r2 as the second-neighbor distance.The convention sets r0 = 0.
- Theoretical basis: Under locally constant density, successive neighbor shells have independent exponential volume increments with rate ρ.This assumption provides the theoretical basis for the nearest-neighbor ratio distribution.
- Theoretical basis: The functions f and F depend explicitly on intrinsic dimension d but not on local density.The resulting dimension equation therefore removes ρ from the estimator.
- Algorithm: TWO-NN estimates d by fitting a line through the origin to points with coordinates (log(μi), −log(1 − Femp(μi))).The empirical cumulative distribution Femp is obtained by sorting the μ values.
- Assumptions: TWO-NN requires only local density uniformity over the second-neighbor range, rather than at larger competing-method neighborhood scales.This condition is theoretically satisfied as N tends to infinity and is empirically reasonably insensitive to finite-sample density variation.
- Synthetic benchmarks: The estimates appear to converge to the real dimension across hypercube, Gaussian, Cauchy, and hypersphere datasets as sample size increases.Convergence is faster at lower dimensions, while Cauchy samples retain high variance with few points because of outliers.
- Scale-dependent dimension: At N around 1000, noisy two-dimensional planes show a plateau at dimension 2 before additional noisy directions increase the estimated dimension.The same dimension-versus-sample-size behavior appears for a Gaussian distribution wrapped around a Swiss Roll.
Discussion
TWO-NN estimates intrinsic dimension from only two nearest neighbors, reducing sensitivity to density inhomogeneity and curvature while remaining numerically consistent in general settings. Combined with subsampling analysis, it can identify relevant dimensions in noisy real-world datasets.
- TWO-NN estimates intrinsic dimension using only the first two nearest-neighbor distances for each dataset point.
- Minimal neighborhoods reduce the effects of density inhomogeneities and curvature and avoid mixing dataset features across scales.
- For locally uniform point distributions, TWO-NN has a robust theoretical framework, while in general cases it provides numerically consistent estimates.
- Subsample analysis identifies relevant dimensions through a plateau in estimated ID as sample size increases, indicating a noise-insensitive region.
- The method detects relevant directions despite curvature, density variation, and high-dimensional Gaussian noise, including in image and molecular-dynamics datasets.
Additional Information
The paper is accompanied by supplementary information and includes standard declarations concerning competing interests, publisher neutrality, and open-access licensing.
- Supplementary information accompanies the paper.
- The authors declare that they have no competing interests.
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- The article is distributed under a Creative Commons Attribution 4.0 International License with attribution and change-notification requirements.