A way to do statistics on data tables that have no built-in geometry: run UMAP to obtain a local, neighbourhood-based distance for every pair of rows, and treat those distances as the metric of a Riemannian manifold carrying the data. With that metric in place, the Euclidean mean and Pearson correlation are replaced by a Frechet-style Riemannian mean (the minimiser of summed squared local distances) and a Riemannian correlation built from a rescaled subtraction, which restores properties such as correlation-circle coordinates staying inside the unit circle. Students learn why applying Euclidean statistical indices to manifold-valued data is an error and how a learned local metric repairs it.
This 2024 arxiv paper presents a comprehensive framework for applying statistical methods to data lying on Riemannian manifolds (non-Euclidean spaces with a smoothly varying metric). It extends funda…