Manifold Dimension Estimation: An Empirical Study
This work provides practical guidance for researchers and practitioners in machine learning and data science, but it is incremental as it synthesizes and evaluates existing methods rather than introducing new ones.
The paper tackles the fragmented state of manifold dimension estimation by conducting a comprehensive survey and empirical evaluation of eight representative estimators, analyzing factors like noise and curvature, and finding that simpler methods often perform better in practice.
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.