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

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

URL: http://arxiv.org/abs/2505.16879v1
Date: Thu, 22 May 2025 16:34:15 GMT
Title: How high is `high'? Rethinking the roles of dimensionality in topological data analysis and manifold learning
Authors: Hannah Sansford, Nick Whiteley, Patrick Rubin-Delanchy,
Abstract summary: We present a generalised Hanson-Wright inequality and use it to establish new statistical insights into the geometry of data point-clouds.<n>We revisit the ground-breaking neuroscience discovery of isometric toroidal structure in grid-cell activity made by Gardner et al.<n>Our findings reveal, for the first time, evidence that this structure is in fact to physical space, meaning that grid cell activity conveys a geometrically faithful representation of the real world.
Score: 8.397730500554047
License: http://creativecommons.org/licenses/by/4.0/
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.

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