The objective of topological data analysis is to extract information of topological nature (connected components, holes, voids...) from the data and use them as features to perform various machine-learning tasks. We start by building a nested sequence of simplicial complexes on top of the data and track the evolution of its topology along the sequence. Computing the Euler characteristic of each complex in the sequence yields a descriptor called the Euler characteristic curve that we use as a feature vector. We will demonstrate that this descriptor has a very good performance in terms of accuracy, strong explainability in terms of topology, and stability with respect to the input data while having a drastically reduced computational cost. We will also study integral transforms of this descriptor and show how this "topological signal processing" enables better performance, especially in an unsupervised setting. Joint work with Vadim Lebovici (Oxford)
Références utiles : Euler characteristic tools for topological data analysis, Hacquard and Lebovici (2023) An introduction to topological data analysis: fundamental and practical aspects for data scientists, Chazal and Michel (2021) Euler characteristic curves and profiles: a stable shape invariant for big data problems, Dlotko and Gurnari (2023)