formalML

Persistent Homology

Tracking topological features across scales — the workhorse of topological data analysis

Persistent homology solves the fundamental problem of topological data analysis: at what scale should we analyze shape? By tracking how topological features — connected components, loops, voids — are born and die across a filtration of simplicial complexes, it produces a multiscale signature that is both theoretically grounded (via the Stability Theorem) and practically useful as input to machine learning pipelines.

1 prerequisite

Simplicial Complexes

The combinatorial scaffolding that turns point clouds into topology

Simplicial complexes are the bridge between raw point cloud data and topological invariants. We build them from scratch — starting with the geometric intuition of simplices as generalized triangles, then constructing the Vietoris-Rips complex that underlies most of modern topological data analysis.