# Geometry-Aware Maximum Likelihood Estimation of Intrinsic Dimension

arXiv: https://arxiv.org/abs/1904.06151 · HTML: https://mlsanity.com/paper/1904.06151/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Apr 2019.

## Built on by (most cited first, 15 of 15)

- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Scikit-dimension: a Python package for intrinsic dimension estimation](https://mlsanity.com/paper/2109.02596/) (2021)
- [Characterizing Truthfulness in Large Language Model Generations with Local Intrinsic Dimension](https://mlsanity.com/paper/2402.18048/) (2024)
- [The Intrinsic Manifolds of Radiological Images and their Role in Deep Learning](https://mlsanity.com/paper/2207.02797/) (2022)
- [Relating Regularization and Generalization through the Intrinsic Dimension of Activations](https://mlsanity.com/paper/2211.13239/) (2022)
- [Intrinsic Dimension for Large-Scale Geometric Learning](https://mlsanity.com/paper/2210.05301/) (2022)
- [Probabilistic and Semantic Descriptions of Image Manifolds and Their Applications](https://mlsanity.com/paper/2307.02881/) (2023)
- [Automatic Layer Selection for Hallucination Detection](https://mlsanity.com/paper/2605.26366/) (2026)
- [Autoencoders with Intrinsic Dimension Constraints for Learning Low Dimensional Image Representations](https://mlsanity.com/paper/2304.07686/) (2023)
- [Pre-processing and Compression: Understanding Hidden Representation Refinement Across Imaging Domains via Intrinsic Dimension](https://mlsanity.com/paper/2408.08381/) (2024)
- [Estimating Dataset Dimension via Singular Metrics under the Manifold Hypothesis: Application to Inverse Problems](https://mlsanity.com/paper/2507.07291/) (2025)
- [Surprisingly High Redundancy in Electronic Structure Data Across Materials Explained by Low Intrinsic Dimensionality](https://mlsanity.com/paper/2507.09001/) (2025)
- [DIT: Dimension Reduction View on Optimal NFT Rarity Meters](https://mlsanity.com/paper/2508.12671/) (2025)
- [A Survey and Comparative Evaluation of Intrinsic Dimension Estimators under the Manifold Hypothesis](https://mlsanity.com/paper/2509.15517/) (2025)
- [HomID : Benchmarking Intrinsic Dimension Estimators on Homogenous Manifolds with Anisotropic Embeddings](https://mlsanity.com/paper/2510.01335/) (2025)

## Abstract (arXiv)

The existing approaches to intrinsic dimension estimation usually are not reliable when the data are nonlinearly embedded in the high dimensional space. In this work, we show that the explicit accounting to geometric properties of unknown support leads to the polynomial correction to the standard maximum likelihood estimate of intrinsic dimension for flat manifolds. The proposed algorithm (GeoMLE) realizes the correction by regression of standard MLEs based on distances to nearest neighbors for different sizes of neighborhoods. Moreover, the proposed approach also efficiently handles the case of nonuniform sampling of the manifold. We perform numerous experiments on different synthetic and real-world datasets. The results show that our algorithm achieves state-of-the-art performance, while also being computationally efficient and robust to noise in the data.