Stable, Discriminative Distances on Reeb Graphs and Merge Trees

Abstract

Topological Data Analysis provides us with low-dimensional, topological descriptions of various types of datasets. Recently, the graph-based descriptors -- specifically Reeb graphs and merge trees -- have garnered a lot of attention due to their low-dimensional yet rich representation of scalar fields. Researchers have constructed distances between these graph-based descriptors to 1) show that the Reeb graph is stable under perturbations of the underlying dataset and 2) to provide a measure for similarity between the underlying scalar fields. Chapter 3 of this dissertation is devoted to a thorough overview of the existing landscape of Reeb graph metrics. We provide an exhaustive list of properties which these distances exhibit as well as a set of detailed examples so that new researchers to this field may be able to better understand how these distance operate. To our discontent, the desirable properties of these distances make their computation complex. In contrast, other researchers have designed distances between these descriptors which loosen the restriction on stability; in turn constructing similarity measures between scalar fields which are computable. Chapter 4 is devoted to the construction of a novel distance on merge trees which we show experimentally exhibits stability and discriminativity to the well-studied bottleneck distance. We discuss the computational hurdles that we face in order to construct a distance which exhibits all the desired properties along with an accompanying algorithm of our distance. We show the application of our distance in settings where the number of vertices is small. This setting is made practical with the property that persistence simplification by $\varepsilon$ increases our distance by at most half of $\varepsilon$. Lastly, Chapter 5 is devoted to the construction of disjoint deformations -- transformations which we can apply to a source Reeb graph to construct a sequence of Reeb graphs where the pairwise distance between elements of the sequence is known.