CAREER: Statistical Advances in Topological Data Analysis with Applications to Astronomy

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

As data size and complexity increases, it is necessary to find ways to characterize features in the data for scientific studies. This research project concerns the development of statistical methodology in the field of topological data analysis (TDA). TDA offers a framework for quantifying shape-related features of data, such as holes with persistent homology (PH). The topological descriptors from PH (i.e., persistence diagrams) give a description of the different holes identified in data, which may be used in an analysis. This research project will advance the statistical foundations of PH and address major statistical challenges in astronomy. The research is motivated by two astronomical areas related to the large-scale structure (LSS) of the Universe and the detection of planets outside our Solar System. While PH gives informative descriptions of complex data, the computations are intensive, and the persistence diagrams are difficult objects to work with statistically. The project aims to improve the computational aspects of the PH algorithm and to develop a statistically sound framework for the analysis of complex data using PH. This research develops novel methodologies that contribute to the progress of science, especially astronomy. The project provides research training opportunities for graduate students, and the methodology, software, and materials produced will be available for researchers, instructors, and the broader public. This research project will advance the statistical foundations of PH for broad applicability, including addressing statistical challenges in astronomy. More specifically, both LSS and exoplanet data may be characterized by holes: LSS is a complex web of matter that includes clusters of matter, loops of filaments, and voids, while exoplanets induce periodic signals in starlight that may be construed as loops in certain embedding spaces. The three main objectives of this research are: (1) To develop TDA tools for big data with statistical guarantees. The project team will develop new algorithms for computing PH summaries to make them computable for large datasets, such as large cosmological simulations. (2) To build sound representations of holes for visualization and scientific discovery. The aim is to develop methods for estimating robust representations of homology group generators and assess their statistical properties. These representations identified in LSS have the potential to constrain the cosmological model. (3) To establish TDA methods for realistic time-series data. This will include state space reconstruction for time series data that are not uniformly spaced with noise, with an application to the detection of exoplanets in the presence of stellar variability. 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.