Morse Theory for Chromatic Delaunay Triangulations

Abstract: This talk is focused on new techniques for the topological data analysis (TDA) of labelled point cloud data. Well-established filtrations in TDA for a point cloud data include the Čech, Vietoris–Rips, and alpha filtrations. Bauer and Edelsbrunner [BE16] demonstrated that the Čech filtration can be simplicially collapsed onto the alpha filtration, showing that they are homotopy equivalent. Recent techniques in data collection across fields like cancer biology, geospatial analysis and ecology have produced chromatic (labeled) data that express interactions among points of different colors. In such cases, it is crucial to understand not only the overall spatial structure of the data but also the spatial relationships among subsets defined by their labels. The chromatic alpha filtration [Mon+24] is a generalization of the alpha filtration that captures these relationships, making it particularly useful for multi-species data in TDA. In this talk we introduce the chromatic Delaunay–Čech and chromatic Delaunay–Rips filtrations as computationally efficient alternatives to the chromatic alpha filtration. We use generalized discrete Morse theory to demonstrate that the Čech, chromatic Delaunay–Čech, and chromatic alpha filtrations are interconnected through simplicial collapses, extending Bauer and Edelsbrunner’s results from non-chromatic to chromatic contexts. Our findings offer theoretical support for the application of chromatic Delaunay–Čech and chromatic Delaunay–Rips filtrations, and we illustrate their computational advantages through numerical experiments. This is joint work with A. Natarajan, T. Chaplin and A. Brown from University of Oxford (arXiv:2405.19303). [BE16] Ulrich Bauer and Herbert Edelsbrunner. “The Morse theory of Cech and Delaunay complexes”. In: Transactions of the American Mathematical Society 369.5 (Dec. 27, 2016), pp. 3741–3762. DOI:10.1090/tran/6991. [Mon+24] Sebastiano Cultrera di Montesano et al. Chromatic Alpha Complexes. Feb. 7, 2024. arXiv: 2212.03128

geometric topology

Audience: researchers in the topic

GEOTOP-A seminar

Series comments: Web-seminar series on Applications of Geometry and Topology