How high is `high'? Rethinking the roles of dimensionality in topological data analysis and manifold learning
This work clarifies fundamental statistical limits in data geometry for researchers in topological data analysis and neuroscience, offering a theoretical framework to interpret high-dimensional data structures.
The authors tackled the problem of understanding how different notions of dimensionality affect topological data analysis and manifold learning, showing that persistence diagrams can reveal latent homology when the ambient intrinsic dimension is much larger than the logarithm of the sample size. They applied this theory to neuroscience data, providing the first evidence that grid-cell activity forms an isometric representation of physical space.
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.