Efficient Computation of Generalized Persistence Diagrams

Topological data analysis (TDA) brings techniques from algebraic topology to the applied domains. It emphasizes methods that are stable, and as such resilient to noise in the data, as well as methods that are computationally efficient, and as such practical across a range of applications. Over the last two decades, TDA techniques have found applications in many domains, including biochemistry, cosmology, materials science, neuroscience, climate research, and many others. One significant limitation that practitioners encounter is that one of the main methods in TDA, persistent homology, is limited to studying one-parameter data, i.e., measurements of a single quantity. The focus of this project is that in practice, multiple measurements are available, and it is precisely the correlation between them that will be used to reveal important features of the problem. Software for practitioners using this methodology will be developed and the project will include graduate student training in topological data analysis. Recently, generalized persistent homology was introduced. It interprets persistence as a Moebius inversion of a certain function derived from the changes in topology of the data across parameters. The construction generalizes all the properties of 1-parameter persistence needed in applications, including stability and the particular structure of the diagrams used in machine learning and statistical pipelines. This project will develop a software implementation for computing generalized persistence diagrams, a crucial gap in this research program and the missing bridge between theory and applications. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.