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

2025-05-22

AI Summary

This work investigates the influence of three critical dimensional notions—ambient intrinsic dimension, correlation rank, and latent intrinsic dimension—on the statistical characterization of point-cloud geometry in topological data analysis (TDA) and manifold learning. Methodologically, it establishes a generalized Hanson–Wright inequality, constructs a stochastic functional model, and integrates TDA with statistical geometric analysis. The contributions are threefold: (i) it provides the first systematic theoretical delineation of the distinct roles these dimensions play in uncovering hidden manifold structure and persistent homology; (ii) it rigorously proves that TDA and manifold learning achieve theoretical validity when the latent intrinsic dimension satisfies $p_{ ext{int}} gg log n$; and (iii) it delivers the first isometricity proof for the toroidal encoding structure of grid cells reported in *Nature* (2022), thereby verifying its geometric fidelity in representing physical space.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Data transparency and provenance
• Web Mining and Content Analysis: Models for Web evolution

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 $p_{mathrm{int}}$, 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 $p_{mathrm{int}}gg log n$, 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.

Problem

• Clarifying roles of three dimensionality notions in data geometry
• Establishing conditions for persistence diagrams to reveal latent homology
• Identifying isometric structure in grid-cell activity to physical space

Innovation

• Generalized Hanson-Wright inequality for data geometry
• Three dimensionality notions clarify manifold structure
• Isometric toroidal structure in grid-cell activity