Ximena Fernandez: Density-based intrinsic persistent homology and applications to time series analysis

Abstract.

In this talk, I will present recent a density-based method to address the problem of estimating topological features from data in high dimensional Euclidean spaces under the manifold assumption. The key of our approach is to consider a sample intrinsic metric known as Fermat distance to robustly infer the homology of space from the data points. I will show convergence results of the associated density-based persistent diagrams, which are less dependent on the particular embedding of the data and more robust to outliers. In the second part of the talk, I will exhibit concrete applications of these ideas to time series analysis, with examples in real data. In particular, I will derive a method for real time detection of epileptic seizures.

This talk is based on the following joint works:
Fernandez, Borghini, Mindlin, Groisman. Intrinsic persistent homology via density-based metric learning. JMLR to appear (2023) arXiv:2012.07621
Fernandez, Mateos. Topological biomarkers for real-time detection of epileptic seizures (2022) arXiv:2211.02523