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Topology and geometry in data science - mklabgr - 09-15-2026
The article “On the shape that matters – topology and geometry in data science” by Paweł Dłotko explores how topological data analysis (TDA) can capture the shape and structure of datasets in ways that traditional statistics often cannot. While conventional statistical methods reduce complex, high-dimensional data to numerical summaries, datasets with almost identical statistics can have radically different geometric shapes.
TDA addresses this by constructing mathematical objects such as simplicial complexes from point clouds and studying their topological properties—connected components, holes, and cavities—as the scale of observation changes. The article introduces constructions such as Vietoris–Rips and alpha complexes, which turn raw data into multiscale structures known as filtrations.
A central tool discussed is the Euler characteristic curve (ECC), which summarizes how the topology of a dataset evolves across different scales. The ECC can provide robust descriptions of high-dimensional data because it is invariant under transformations such as permutations of coordinate axes and affine transformations. The article particularly highlights its use in statistical goodness-of-fit testing: the resulting “TopoTests” use topological information as an alternative to the traditional cumulative distribution function and can outperform established methods, even for relatively small datasets. Overall, the article presents topology and geometry not merely as abstract mathematical theories, but as practical tools for visualising, comparing, classifying and statistically analysing complex data, with applications ranging from data science to cosmology.
ARTICLE [PDF]