# Persistent Homology and the Upper Box Dimension

arXiv: https://arxiv.org/abs/1802.00533 · HTML: https://mlsanity.com/paper/1802.00533/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Feb 2018.

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

- [Intrinsic Dimension Estimation for Robust Detection of AI-Generated Texts](https://mlsanity.com/paper/2306.04723/) (2023)
- [Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks](https://mlsanity.com/paper/2111.13171/) (2021)
- [The Shape of Learning: Anisotropy and Intrinsic Dimensions in Transformer-Based Models](https://mlsanity.com/paper/2311.05928/) (2023)
- [Generalization Bounds with Data-dependent Fractal Dimensions](https://mlsanity.com/paper/2302.02766/) (2023)
- [Metric Space Magnitude and Generalisation in Neural Networks](https://mlsanity.com/paper/2305.05611/) (2023)
- [Predicting the generalization gap in neural networks using topological data analysis](https://mlsanity.com/paper/2203.12330/) (2022)
- [On the Limitations of Fractal Dimension as a Measure of Generalization](https://mlsanity.com/paper/2406.02234/) (2024)
- [Topology Applied to Machine Learning: From Global to Local](https://mlsanity.com/paper/2103.05796/) (2021)
- [Memorization in Language Models through the Lens of Intrinsic Dimension](https://mlsanity.com/paper/2506.09591/) (2025)
- [T-REGS: Minimum Spanning Tree Regularization for Self-Supervised Learning](https://mlsanity.com/paper/2510.23484/) (2025)
- [When Personalization Tricks Detectors: The Feature-Inversion Trap in Machine-Generated Text Detection](https://mlsanity.com/paper/2510.12476/) (2025)
- [Unveiling Intrinsic Dimension of Texts: from Academic Abstract to Creative Story](https://mlsanity.com/paper/2511.15210/) (2025)
- [On topological data analysis for structural dynamics: an introduction to persistent homology](https://mlsanity.com/paper/2209.05134/) (2022)
- [What Makes a Good Layer? Assessing the Layer-Wise Intrinsic Properties of Music Foundation Models](https://mlsanity.com/paper/2608.14819/) (2026)
- [Linear Probing Provides Robust and Efficient Detection of Machine-Generated Text](https://mlsanity.com/paper/2608.24780/) (2026)

## Abstract (arXiv)

We introduce a fractal dimension for a metric space defined in terms of the persistent homology of extremal subsets of that space. We exhibit hypotheses under which this dimension is comparable to the upper box dimension; in particular, the dimensions coincide for subsets of ℝ2 whose upper box dimension exceeds 1.5. These results are related to extremal questions about the number of persistent homology intervals of a set of n points in a metric space.