Comprehensive Exam of CMSE Ishika Gosh

4/14/2025, 10:30 AM @ EB 1502 & 1503

Zoom link:  https://msu.zoom.us/j/92223647520

Passcode: 978106

 

By: Ishika Ghosh

Abstract:

 

Topological Data Analysis (TDA) offers powerful tools for extracting shape-based insights from complex, high-dimensional data. Sitting at the crossroads of math, computer science, and data analysis, TDA focuses on using topological features to analyze and visualize data. One key tool is the mapper graph, which gives a simplified view of data by preserving the connected components under a chosen function and cover. Mapper graphs are widely used in fields like biology and machine learning to explore intrinsic data structures.

Given their popularity, comparing mapper graphs in a meaningful, quantitative way is crucial. However, defining a practical and computable distance between them remains a challenge. One promising idea is interleaving distance, a stable metric grounded in category theory. Recent work has adapted this concept from Reeb graphs to mapper graphs but noted its NP-hardness. To work around this, they proposed a polynomial-time loss function that upper-bounds the interleaving distance. Our work builds on this by designing an algorithm to minimize that loss using integer linear programming (ILP). This creates a robust pipeline for mapper graph comparison, which we plan to apply in machine learning contexts. For future projects, we aim to further refine our mapper comparison pipeline and demonstrate its effectiveness

This work lays the foundation for the practical use of interleaving distances in TDA, with implications for machine learning and exploratory data science. We also would make our work accessible to a broader audience by publishing an accompanying open-source implementation. Through this study, we aim to bridge the gap between combinatorial optimization and topological methods, demonstrating how ILP can be effectively used to refine Mapper Graph representations.

 

 

Committee:

Dr. Elizabeth Munch

Dr. Ekaterina A. Rapinchuk

Dr. Teena Gerhardt

Dr. Yang Yang