Topological Data Analysis (TDA) has grown tremendously as a rich toolkit for analyzing data sets where shape is an important feature. The success of TDA comes largely from its ability to take a complicated, potentially high-dimensional, data set and summarize it using topological descriptors known as merge trees, Reeb graphs, or persistence barcodes (also known as persistence diagrams). These descriptors are compact, lightweight summaries of a data set that are stable with respect to perturbations and certain types of noise. The passage from data to these summaries is sometimes called the ?TDA pipeline?. This project emerged from the observation that qualitatively different data sets can produce identical merge trees and barcodes, when passed through the TDA pipeline; see Figure 1 for a cartoon example. The goals of the project were to produce new enriched topological summaries (ETS) that stand upstream of traditional topological summaries (TS) that can distinguish these data sets as well data sets of practical interest, all without sacrificing stability or computability; see Figure 2. 

This project was successful in several regards. First, the PI and their collaborators were able to identify several enrichments of traditional TDA by first counting how many different data sets produced the same merge tree or barcode. The notion of ?data set? and ?different? was varied across a broad class of examples, including Morse functions and embeddings of shapes in three dimensions. The project advanced knowledge of the theory underlying TDA methods and produced new connections between TDA, (co)sheaf theory, combinatorics and geometric group theory. Additionally, one of the first combinatorial results that distinguishes merge trees from phylogenetic trees was proved as a result of this work, thereby opening the door to further avenues of exploration in the geometry of the space of all possible merge trees. Preliminary results on doing statistics on the space of merge trees and barcodes were also provided.

Perhaps most significantly, one of the main ETS developed by this project, the Decorated Merge Tree (DMT), was identified as a new lightweight and stable summary of data sets where features such as clusters and circles can be tracked in tandem. Both theoretical and computational definitions were provided and their implementation in Python is now freely available on GitHub. As an illustration of this new tool see Figure 3. In that figure, three synthetic point cloud examples are shown to the left, their persistence diagrams are shown in the middle---which are nearly identical for the three point clouds, and their decorated merge trees are shown on the right. Variations on this example are shown in Figure 4. The PI and their collaborators were able to define several distances on these DMTs of varying computability and have demonstrated their improved clustering power. Figure 5 shows how the six data sets from Figure 4 would be clustered using traditional methods, which are most clearly grouped using our new method. These methods also provide feature localization when studying time series, such as heart rate data (Figure 6), and in image data, such as in fMRI data.

Several graduate students received support from this grant to augment their mentorship as highly skilled mathematicians and programmers. At least one student has written a PhD thesis based on the work performed on this project and several postdocs at other institutions were able to augment their training through collaboration with the PI and their graduate students. The PI continues to teach data science courses at their host institution where these methods are brought into the classroom so that undergraduate and masters students are exposed to the latest methods in TDA.

Last Modified: 10/27/2021
Modified by: Justin M Curry