CSE - Topological Data Analysis and Shape Transforms

Topological data analysis (TDA) studies the "shape" of data, using tools from topology and geometry to extract robust, quantitative descriptors of structure in point clouds, images, and other complex datasets. These descriptors are increasingly used across a range of applications areas like biology, materials science, and machine learning to capture features that classical statistics can miss, such as loops, voids, and other multi-scale patterns.

This project will center on a family of tools called topological transforms, including the persistent homology transform (PHT) and the radial transform, which represent a shape by tracking how its topology changes as it is viewed from many directions or scales. These transforms have appealing theoretical properties (e.g., some are provably injective, meaning they can reconstruct the original shape) as well as growing practical use in shape comparison and classification.

We'll begin with foundational reading on simplicial complexes, persistent homology, and persistence diagrams, then move into papers on topological transforms specifically. From there, the project can develop in either a mathematical or a more practical direction, ranging from invertibility or stability on the math side to implementing these objects on a variety of data sets on the practical. Specific goals:

• Build a working understanding of simplicial homology and persistent homology from first principles.
• Read and present key papers establishing the radial transform and/or persistent homology transform.
• Either: (a) investigate open questions about algorithmic complexity or reconstruction guarantees for these transforms, or (b) implement a transform and evaluate it on a dataset of interest.
• Produce a final writeup or presentation suitable as a capstone project or springboard to further research.