# HomID : Benchmarking Intrinsic Dimension Estimators on Homogenous Manifolds with Anisotropic Embeddings

arXiv: https://arxiv.org/abs/2510.01335 · HTML: https://mlsanity.com/paper/2510.01335/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 76 references that are in the map.
- First posted on arXiv 1 Oct 2025.

## Builds on (most cited first, 19 of 19)

- [How Contextual are Contextualized Word Representations? Comparing the Geometry of BERT, ELMo, and GPT-2 Embeddings](https://mlsanity.com/paper/1909.00512/) (2019)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks](https://mlsanity.com/paper/2111.13171/) (2021)
- [Scikit-dimension: a Python package for intrinsic dimension estimation](https://mlsanity.com/paper/2109.02596/) (2021)
- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [Blessing of dimensionality: mathematical foundations of the statistical physics of data](https://mlsanity.com/paper/1801.03421/) (2018)
- [IsoScore: Measuring the Uniformity of Embedding Space Utilization](https://mlsanity.com/paper/2108.07344/) (2021)
- [A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models](https://mlsanity.com/paper/2406.03537/) (2024)
- [Topological Deep Learning: A Review of an Emerging Paradigm](https://mlsanity.com/paper/2302.03836/) (2023)
- [LIDL: Local Intrinsic Dimension Estimation Using Approximate Likelihood](https://mlsanity.com/paper/2206.14882/) (2022)
- [Intrinsic Dimension Estimation Using Wasserstein Distances](https://mlsanity.com/paper/2106.04018/) (2021)
- [Learning Manifold Dimensions with Conditional Variational Autoencoders](https://mlsanity.com/paper/2302.11756/) (2023)
- [Grassmannian Learning: Embedding Geometry Awareness in Shallow and Deep Learning](https://mlsanity.com/paper/1808.02229/) (2018)
- [Hardness of Learning Neural Networks under the Manifold Hypothesis](https://mlsanity.com/paper/2406.01461/) (2024)
- [Geometry-Aware Maximum Likelihood Estimation of Intrinsic Dimension](https://mlsanity.com/paper/1904.06151/) (2019)
- [Relating Regularization and Generalization through the Intrinsic Dimension of Activations](https://mlsanity.com/paper/2211.13239/) (2022)
- [Robust estimation of the intrinsic dimension of data sets with quantum cognition machine learning](https://mlsanity.com/paper/2409.12805/) (2024)

## Abstract (arXiv)

The manifold hypothesis suggests that data lies on manifolds with smaller intrinsic dimension (ID) than their ambient dimension. However there is no empirical agreement on the estimates for ID from different estimators for realistic datasets. Thus it is important to test ID estimators (IDEs) with targeted stressors. In this work, we consider the role of anisotropy. To this end, we propose HomID, a collection of homogeneous spaces with anisotropic embedding, for benchmarking ID estimators. We observe that methods that perform well on standard benchmarks systematically degrade on HomID under identical resource allocation. We further observe that anisotropic distortion of such benchmarks also results in performance degradation. Finally, we demonstrate that controlled anisotropic distortions induce systematic shifts in the distributions on which these methods rely, providing a concrete mechanism for the resulting estimation errors in two particular IDEs.