How High is ‘High’? Rethinking the Roles of Dimensionality in Topological Data Analysis and Manifold Learning

Hannah Sansford, Nick Whiteley, Patrick Rubin-Delanchy
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:107580-107612, 2026.

Abstract

High-dimensionality of data is often regarded as a fundamental statistical impediment in Machine Learning and AI. The purpose of this paper is to clarify, on the contrary, when and how high-dimensionality may be beneficial. In the setting of a general random function model of data we delineate between three notions of dimensionality: effective dimension $p_{\mathrm{eff}}$, measuring total variability across feature directions; correlation rank $r$, measuring functional complexity across samples; and latent intrinsic dimension $d$ of manifold structure hidden in data. Via a generalized Hanson-Wright inequality, we show that increasing $p_{\mathrm{eff}}$ drives a blessing of dimensionality phenomenon, whereby data dot-products concentrate about their expectations. In turn, we show that, under mild continuity assumptions (ensuring that features bring additional information as dimension grows), persistence diagrams recover latent homology when $p_{\mathrm{eff}} \in \omega (\log n)$ as $n\to\infty$. Informed by our theory, we revisit the ground-breaking neuroscience discovery of toroidal structure in grid-cell activity made by Gardner et al. (2022): our findings provide the first empirical evidence that this structure is isometric to a flat torus model of physical space, suggesting that grid cell activity conveys a geometrically faithful representation of the real world.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-sansford26a, title = {How High is ‘{H}igh’? {R}ethinking the Roles of Dimensionality in Topological Data Analysis and Manifold Learning}, author = {Sansford, Hannah and Whiteley, Nick and Rubin-Delanchy, Patrick}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {107580--107612}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/sansford26a/sansford26a.pdf}, url = {https://proceedings.mlr.press/v306/sansford26a.html}, abstract = {High-dimensionality of data is often regarded as a fundamental statistical impediment in Machine Learning and AI. The purpose of this paper is to clarify, on the contrary, when and how high-dimensionality may be beneficial. In the setting of a general random function model of data we delineate between three notions of dimensionality: effective dimension $p_{\mathrm{eff}}$, measuring total variability across feature directions; correlation rank $r$, measuring functional complexity across samples; and latent intrinsic dimension $d$ of manifold structure hidden in data. Via a generalized Hanson-Wright inequality, we show that increasing $p_{\mathrm{eff}}$ drives a blessing of dimensionality phenomenon, whereby data dot-products concentrate about their expectations. In turn, we show that, under mild continuity assumptions (ensuring that features bring additional information as dimension grows), persistence diagrams recover latent homology when $p_{\mathrm{eff}} \in \omega (\log n)$ as $n\to\infty$. Informed by our theory, we revisit the ground-breaking neuroscience discovery of toroidal structure in grid-cell activity made by Gardner et al. (2022): our findings provide the first empirical evidence that this structure is isometric to a flat torus model of physical space, suggesting that grid cell activity conveys a geometrically faithful representation of the real world.} }
Endnote
%0 Conference Paper %T How High is ‘High’? Rethinking the Roles of Dimensionality in Topological Data Analysis and Manifold Learning %A Hannah Sansford %A Nick Whiteley %A Patrick Rubin-Delanchy %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-sansford26a %I PMLR %P 107580--107612 %U https://proceedings.mlr.press/v306/sansford26a.html %V 306 %X High-dimensionality of data is often regarded as a fundamental statistical impediment in Machine Learning and AI. The purpose of this paper is to clarify, on the contrary, when and how high-dimensionality may be beneficial. In the setting of a general random function model of data we delineate between three notions of dimensionality: effective dimension $p_{\mathrm{eff}}$, measuring total variability across feature directions; correlation rank $r$, measuring functional complexity across samples; and latent intrinsic dimension $d$ of manifold structure hidden in data. Via a generalized Hanson-Wright inequality, we show that increasing $p_{\mathrm{eff}}$ drives a blessing of dimensionality phenomenon, whereby data dot-products concentrate about their expectations. In turn, we show that, under mild continuity assumptions (ensuring that features bring additional information as dimension grows), persistence diagrams recover latent homology when $p_{\mathrm{eff}} \in \omega (\log n)$ as $n\to\infty$. Informed by our theory, we revisit the ground-breaking neuroscience discovery of toroidal structure in grid-cell activity made by Gardner et al. (2022): our findings provide the first empirical evidence that this structure is isometric to a flat torus model of physical space, suggesting that grid cell activity conveys a geometrically faithful representation of the real world.
APA
Sansford, H., Whiteley, N. & Rubin-Delanchy, P.. (2026). How High is ‘High’? Rethinking the Roles of Dimensionality in Topological Data Analysis and Manifold Learning. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:107580-107612 Available from https://proceedings.mlr.press/v306/sansford26a.html.

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