Dimensionality reduction based on persistence entropy

Abstract

[en] In this work we use different methods from algebraic topology, statistics, and data analysis to study a specific data set. This includes tools and analysis methods such as homology, simplicial complexes, persistent homology, bottleneck distance, Wasserstein distance, total persistence, persistence entropy, and directional hierar- chical analysis. Our aim is to study a database generated during a previous neuroscience experiment by Cos et al. (2021). This database is a high-dimensional electroencephalogram (EEG) data set of recordings from 11 participants in a decisionmaking experiment in which three motivational states were induced by manipulating social pressure onto participants. Our goal is to find out the intrinsic dimension of this database, that is, the number of latent variables, and look for subjects in the study population who are significantly different from the rest. This work was inspired by a paper by Ferrà et al. (2023), in which the authors present a new analytical approach using topological data analysis (TDA). Traditional dimensionality reduction methods determine how many dimensions should be retained attempting to preserve variance of the data, while topological data analysis estimates an optimal dimension by studying the data’s topology. While a TDA classifier was used by Ferrà et al., in this work we use directed hierarchical analysis combined with distances between persistence diagrams and persistent entropy to assess the amount of topological variation depending on the ambient dimension.