Topological data analysis (TDA) is a branch of data science that applies topology to study the shape of data, i.e., the coarse-scale, global, non-linear geometric features of data. Such features include clusters, loops, and tendrils in point cloud data and modes and ridges in functional data. While the history of TDA dates back to the 1990s, the field has advanced rapidly in recent years, leading to a rich theoretical foundation, highly efficient algorithms and software, and many applications. At the VU, Senja Barthel, Magnus Bakke Botnan, and Renee Hoekzema work on multiple facets of TDA, and TDA fits into a larger effort at the VU of analyzing higher-order interactions.
The most notable technique in TDA is persistent homology. Here, homology is a concept from algebraic topology, and persistent refers to the fact that topological properties are tracked as one or more parameters vary; the more persistent a feature, the less likely it is due to noise. For instance, one can form a one-parameter filtration by associating a function value to every vertex and every edge in a graph and then studying the connectivity properties of the sublevel sets of the function. For filtrations along a single parameter, the theory of persistent homology is well established, but for multiple filtration parameters, the situation is much more complicated. The branch of TDA that studies data via 'multifiltrations' is called multiparameter persistence. Such constructions arise naturally in many settings, e.g., data with outliers, time-varying data, functional data, and data equipped with a real-valued function (e.g., the partial charge function on the atom centers of a protein).
The authors hope that the article will serve as an invitation for others–mathematicians, computer scientists, statisticians, and application specialists—to join the effort to realize the potential of this approach to data analysis. The article assumes some familiarity with a few elementary mathematical topics that arise in studying 1-parameter persistence, such as basic category theory, homology, and abstract algebra.
The published paper is a chapter in the proceedings of the 2020 International Conference on Representations of Algebras, and is available here. The paper is also available via the arXiv.
Reference: Botnan, M. B., & Lesnick, M. (2023). An introduction to multiparameter persistence. EMS Series of Congress Reports, 77–150. https://doi.org/10.4171/ecr/19/4