# How high is `high'? Rethinking the roles of dimensionality in topological data analysis and manifold learning

arXiv: https://arxiv.org/abs/2505.16879 · HTML: https://mlsanity.com/paper/2505.16879/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 53 references that are in the map.
- First posted on arXiv 22 May 2025.

## Builds on (most cited first, 6 of 6)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Statistical exploration of the Manifold Hypothesis](https://mlsanity.com/paper/2208.11665/) (2022)
- [Matrix factorisation and the interpretation of geodesic distance](https://mlsanity.com/paper/2106.01260/) (2021)
- [Hierarchical clustering with dot products recovers hidden tree structure](https://mlsanity.com/paper/2305.15022/) (2023)
- [Optimal rates of convergence for persistence diagrams in Topological Data Analysis](https://mlsanity.com/paper/1305.6239/) (2013)

## Abstract (arXiv)

We present a generalised Hanson-Wright inequality and use it to establish new statistical insights into the geometry of data point-clouds. In the setting of a general random function model of data, we clarify the roles played by three notions of dimensionality: ambient intrinsic dimension pint, which measures total variability across orthogonal feature directions; correlation rank, which measures functional complexity across samples; and latent intrinsic dimension, which is the dimension of manifold structure hidden in data. Our analysis shows that in order for persistence diagrams to reveal latent homology and for manifold structure to emerge it is sufficient that pintgg log n, where n is the sample size. Informed by these theoretical perspectives, we revisit the ground-breaking neuroscience discovery of toroidal structure in grid-cell activity made by Gardner et al. (Nature, 2022): our findings reveal, for the first time, evidence that this structure is in fact isometric to physical space, meaning that grid cell activity conveys a geometrically faithful representation of the real world.