# Intrinsic Dimension of Geometric Data Sets

arXiv: https://arxiv.org/abs/1801.07985 · HTML: https://mlsanity.com/paper/1801.07985/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jan 2018.

## Built on by (most cited first, 7 of 7)

- [Intrinsic Dimension for Large-Scale Geometric Learning](https://mlsanity.com/paper/2210.05301/) (2022)
- [Scaling Dimension](https://mlsanity.com/paper/2302.09101/) (2023)
- [Selecting Features by their Resilience to the Curse of Dimensionality](https://mlsanity.com/paper/2304.02455/) (2023)
- [Towards Ordinal Data Science](https://mlsanity.com/paper/2307.09477/) (2023)
- [Formal concept analysis for evaluating intrinsic dimension of a natural language](https://mlsanity.com/paper/2311.10862/) (2023)
- [What is the textitintrinsic dimension of your binary data? -- and how to compute it quickly](https://mlsanity.com/paper/2404.06326/) (2024)
- [Reproducibility and Geometric Intrinsic Dimensionality: An Investigation on Graph Neural Network Research](https://mlsanity.com/paper/2403.08438/) (2024)

## Abstract (arXiv)

The curse of dimensionality is a phenomenon frequently observed in machine learning (ML) and knowledge discovery (KD). There is a large body of literature investigating its origin and impact, using methods from mathematics as well as from computer science. Among the mathematical insights into data dimensionality, there is an intimate link between the dimension curse and the phenomenon of measure concentration, which makes the former accessible to methods of geometric analysis. The present work provides a comprehensive study of the intrinsic geometry of a data set, based on Gromov's metric measure geometry and Pestov's axiomatic approach to intrinsic dimension. In detail, we define a concept of geometric data set and introduce a metric as well as a partial order on the set of isomorphism classes of such data sets. Based on these objects, we propose and investigate an axiomatic approach to the intrinsic dimension of geometric data sets and establish a concrete dimension function with the desired properties. Our model for data sets and their intrinsic dimension is computationally feasible and, moreover, adaptable to specific ML/KD-algorithms, as illustrated by various experiments.