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

2025/05/22 by Hannah Sansford, Sansford, Hannah, Nick Whiteley +3 · 1 citation
Computer Science · Mathematics · #Curse of dimensionality #Dimension (graph theory) #Dimensionality reduction #FOS: Computer and information sciences #Function (biology) #Intrinsic dimension #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Manifold (fluid mechanics) #Morphological variations and asymmetry #Nonlinear dimensionality reduction #Point processes and geometric inequalities #Representation (politics) #Topological and Geometric Data Analysis #Topological data analysis

paper · pdf · doi:10.48550/arxiv.2505.16879

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2025/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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 p_int, 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 p_int≫ 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.

Citations

• Hierarchical clustering with dot products recovers hidden tree structure
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