In these times, scientists are not the only ones struggling to grasp the overall picture from a flood of specific, detailed, high-dimensional data. Understanding a high-dimensional data set often means understanding a lower-dimensional space that generates it, as, say, the important part of the image of someone playing a body-controlled video game is the few angles of their arm and leg joints, and not the many pixels coming from a camera and depth sensor. Topology is the branch of mathematics that looks at connections rather than specific points or values; two popular tools in topological data analysis, hierarchical clustering and Reeb graphs, are based on path-connectivity. This project generalizes both, developing generalizations to parameterized hierarchical clustering and Reeb spaces. There is an exciting interplay between the practical questions of data analysis and the theoretical and mathematical tools to answer them -- each deepens the other. This project will train students through this interplay.
The PI will develop algorithmic foundations for Reeb spaces and theoretical foundations for parameterized hierarchical clustering. The Reeb space of a continuous map summarizes the path-connectivity of its fibers. The Reeb graph, which is the Reeb space of a map to the real line and for which there are fast algorithms to compute, is now a commonly used tool in topological data analysis. The PI will focus on developing practical algorithms for computing the Reeb space of a map to higher dimensional Euclidean spaces. When the domain of the continuous map is equipped with a metric, the path-connected components of each fiber assembles into a (Vietoris-Rips) hierarchical cluster. Thus the map gives rise to a family of hierarchical clusters parameterized by the co-domain of the map. The PI will use sheaf theory to develop a theoretical foundation for parameterized hierarchical clustering.
