Topological Data Analysis: what, why, how & when

Topological data analysis is a recent and fast growing field that approaches the analysis of (the shape of) data using techniques from algebraic topology. Its main tool, persistent homology, captures information about cycles in the data: its connected components, loops, voids, and so on. The first part of the thesis investigates the properties of persistent homology. Ho- mology captures cycles in the data, but what additional information is stored in the persistence (“the what”)? We show that it can capture the number of holes, but also important geometric notions of curvature and convexity. Stability theorems provide mathematically provable guarantees of desirable properties of persistent homology (“the why”), but how do these theoretical results translate to practice? We show that persistent homology does not always yield noise robust features in a classification task. The second part explores two applications of persistent homology. Firstly, we demonstrate how it can be used to study the preservation of topology and geometry (“the how”); here we focus on hyperdimensional computing that encodes the input data into a very high dimensional space. Secondly, we extract persistent homology features from EEG data during an audiovisual task in order to detect attention, providing some understanding when it can be useful in neuroscience applications (“the when”). In both parts, we depart from the dominant stream in the literature by highlight- ing the versatility of filtrations and signatures, the input and output of persistent homology, beyond the canonical choices. In this way, we showcase how persistent homology can be seen as a diverse family of rich descriptors of different aspects of shape.

Publication year:2024

Keywords:Computer. Automation

Accessibility:Open