CAREER: Reeb graph learning: Classification, Clustering, and Embedding of Graphical Signatures

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The constant increase in available data is a ubiquitous feature of modern life. This data comes in many forms such as points, images, and networks. Networks (equivalently called graphs in mathematical terminology) can themselves be data points in a large data set. For instance, one may wish to compare gene regulatory networks across many different species, or compare embedded graphs from different molecules to predict therapeutic properties. This research focuses on graphs with additional structure, which arise in the field of Topological Data Analysis (TDA), a modern take on data analysis that encodes shape and structure in data. The research objective of this project is to build the machine-learning theory necessary to utilize the rich structure of this kind of graph data which comes with both theoretical guarantees and practical algorithms. This research plan is combined with a fully integrated education plan including a series of workshops to stimulate interdisciplinary collaborations between starting researchers in TDA with domain scientists, as well as creation of open source code and educational materials to make the newly developed methods available to more interdisciplinary researchers. In more detail, this project focuses on graphical signatures, which are topological graphs equipped with a real-valued function further encoding some aspect of the topology or geometry of the structure represented. These include highly utilized constructions such as Reeb graphs, mapper graphs, merge trees, and contour trees. Recent work has developed metrics for comparing these objects, allowing for treating the space of, e.g., Reeb grahs as a metric space. The machine-learning interfaces developed for this project will utilize and respect the metrics available. While there is work aimed interfacing general graphs with machine learning, and separately for other topological signatures such as persistent homology, no tools yet exist for graphical signatures which take the additional function structure into account. The project team will further develop a theory of random Reeb graphs to generate data sets on which we can test the methods, as well as test the results in applications arising from the field of plant morphology.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.