Dissertation Defense of CMSE Elena Wang
Department of Computational Mathematics, Science & Engineering
Michigan State University
Dissertation Defense Notice
March 18th, 3pm
EB 1502
Meeting ID: 94410134089
Passcode: 763779
Topological data analysis based distances between directional transform representations
of graphs
By Elena Wang
Abstract:
Shape analysis is important in fields like computational geometry, biology, and machine
learning, where understanding differences in structure and tracking changes over time
is useful. Topological Data Analysis (TDA) provides tools to study shape in a way
that is resistant to noise and captures both fine and large-scale features. This dissertation
focuses on directional transforms, a method that encodes shape by looking at its structure
from different directions. Then, we can evaluate the output using various topological
signatures from TDA, such as persistence diagrams and merge trees. In particular,
we focus on creating and computing distances between the resulting objects in order
to compare the input graphs in applications.
We introduce the Labeled Merge Tree Transform (LMTT), a new way to represent embedded graphs by combining merge trees with directional transforms, and utilize the labeled merge tree distance to compare the outputs. We test this method on real-world datasets and show that it works better for classification than existing distance measures in some empirical settings. We also develop a kinetic data structure (KDS) for the bottleneck distance between persistence diagrams, which allows us to update shape comparisons efficiently when the data changes over time. We apply this method to the Persistent Homology Transform (PHT) and show that it reduces computation time while keeping accurate results. These contributions improve the use of topology in studying dynamic shapes and open new research possibilities in both theory and practical applications.
We introduce the Labeled Merge Tree Transform (LMTT), a new way to represent embedded graphs by combining merge trees with directional transforms, and utilize the labeled merge tree distance to compare the outputs. We test this method on real-world datasets and show that it works better for classification than existing distance measures in some empirical settings. We also develop a kinetic data structure (KDS) for the bottleneck distance between persistence diagrams, which allows us to update shape comparisons efficiently when the data changes over time. We apply this method to the Persistent Homology Transform (PHT) and show that it reduces computation time while keeping accurate results. These contributions improve the use of topology in studying dynamic shapes and open new research possibilities in both theory and practical applications.
Committee Members:
Elizabeth Munch (chair)
Erin Chambers
Teena Gerhardt
Longxiu Huang