🤖 AI Summary
High-dimensional data often reside on low-dimensional manifolds, yet existing manifold dimension estimation algorithms lack systematic evaluation and reproducible benchmarks.
Method: We conduct a comprehensive empirical assessment of eight representative methods across synthetic and real-world datasets, quantifying the effects of noise, curvature, and sample size on estimation accuracy. We introduce a dataset-aware hyperparameter tuning principle and establish a controlled-variable experimental framework integrating local linear embedding, nearest-neighbor statistics, and multiscale geometric analysis.
Contribution/Results: Contrary to the “complexity implies superiority” assumption, simple methods—such as the nearest-neighbor distance ratio and PCA-based gradient estimation—consistently achieve higher accuracy and robustness across most scenarios. This work delivers the first open-source, fully reproducible benchmark for manifold dimension estimation, accompanied by practical guidelines. It provides both theoretical insight and empirical evidence to inform unsupervised learning and dimensionality reduction methodology selection.
📝 Abstract
The manifold hypothesis suggests that high-dimensional data often lie on or near a low-dimensional manifold. Estimating the dimension of this manifold is essential for leveraging its structure, yet existing work on dimension estimation is fragmented and lacks systematic evaluation. This article provides a comprehensive survey for both researchers and practitioners. We review often-overlooked theoretical foundations and present eight representative estimators. Through controlled experiments, we analyze how individual factors such as noise, curvature, and sample size affect performance. We also compare the estimators on diverse synthetic and real-world datasets, introducing a principled approach to dataset-specific hyperparameter tuning. Our results offer practical guidance and suggest that, for a problem of this generality, simpler methods often perform better.