Title: Graph-based Methods in Topological Data Analysis

 

Abstract:

 

As collections of data grow in size, it is increasingly important to develop tools that can process and analyze data efficiently. Topological data analysis (TDA) uses the mathematical field of topology to study the underlying “shape" of a dataset to reveal novel insights that traditional data analysis methods miss. In this talk, I will explore the intimate connection between graphs and topological data analysis: not only are graph-based methods for analyzing point-cloud data a fundamental approach in TDA, but techniques from TDA are used to study graphs which naturally arise from real-world data. First, I will discuss one of the most popular tools in TDA, the mapper graph, which summarizes data, either discrete or continuous, into a graph. Mapper’s effectiveness depends on several tunable parameters, which can dramatically impact the resultant mapper graph. I will show how the mapper graph can dramatically vary based on input parameters, then present a method for automated cover selection which adds rigor and stability to the parameter selection process, making mapper viable for data analysis. Second, I will present an empirical study comparing classical, geometric, and topological graph distances on embedded graphs. The results of our experiments, which test computational efficiency and performance on classification tasks on real-world and synthetic datasets, validate a topological approach for comparing embedding graphs.