# A Survey and Comparative Evaluation of Intrinsic Dimension Estimators under the Manifold Hypothesis

arXiv: https://arxiv.org/abs/2509.15517 · HTML: https://mlsanity.com/paper/2509.15517/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 70 references that are in the map.
- First posted on arXiv 19 Sep 2025.

## Builds on (most cited first, 15 of 15)

- [Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning](https://mlsanity.com/paper/2012.13255/) (2020)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [Scikit-dimension: a Python package for intrinsic dimension estimation](https://mlsanity.com/paper/2109.02596/) (2021)
- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Data-driven discovery of intrinsic dynamics](https://mlsanity.com/paper/2108.05928/) (2021)
- [Intrinsic Dimensionality Estimation within Tight Localities: A Theoretical and Experimental Analysis](https://mlsanity.com/paper/2209.14475/) (2022)
- [Intrinsic Dimension Estimation Using Wasserstein Distances](https://mlsanity.com/paper/2106.04018/) (2021)
- [Manifold Learning by Mixture Models of VAEs for Inverse Problems](https://mlsanity.com/paper/2303.15244/) (2023)
- [Geometry-Aware Maximum Likelihood Estimation of Intrinsic Dimension](https://mlsanity.com/paper/1904.06151/) (2019)
- [A Survey of Dimension Estimation Methods](https://mlsanity.com/paper/2507.13887/) (2025)
- [Minimax Rates for Estimating the Dimension of a Manifold](https://mlsanity.com/paper/1605.01011/) (2016)
- [Tangent Space and Dimension Estimation with the Wasserstein Distance](https://mlsanity.com/paper/2110.06357/) (2021)
- [Rdimtools: An R package for Dimension Reduction and Intrinsic Dimension Estimation](https://mlsanity.com/paper/2005.11107/) (2020)
- [CA-PCA: Manifold Dimension Estimation, Adapted for Curvature](https://mlsanity.com/paper/2309.13478/) (2023)
- [Intrinsic Dimension Estimation via Nearest Constrained Subspace Classifier](https://mlsanity.com/paper/2002.03228/) (2020)

## Abstract (arXiv)

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 for estimator selection and yield insights that will inform future estimator design.