Persistent Homology
Measuring the shape of data across scales.
Motivation
Topological data analysis asks a question orthogonal to most of machine learning: what is the shape of a dataset? Persistent homology answers by tracking topological features — connected components, loops, voids — as a scale parameter grows.
Method
A filtration builds a nested family of simplicial complexes over the data. Homology is computed at every scale, and features are summarized by persistence — how long each feature survives — visualized as a barcode or persistence diagram. Long-lived features are signal; short-lived ones are noise.
Mathematical core
The stability of persistence diagrams makes topology a robust feature for learning. This project anchors my broader interest in geometric and topological structure — including using topological invariants to understand latent spaces and continuous-depth models.