Week 06 — Vietoris–Rips, Čech, Alpha, Witness Complexes

The decision that ends up mattering most in practice: which complex to build, and why.

Lecture

Vietoris–Rips · $\check{C}$ech · alpha complexes (Delaunay triangulation) · witness complexes for landmark-based reduction · the nerve theorem · what each complex costs and when each is appropriate.

Read before the lecture

• Edelsbrunner and Harer, *Computational Topology: An Introduction*, chapter III

Code lab

Comparing complex constructions

On the same point cloud, compute persistence with Rips, Čech, and Alpha complexes. Compare diagrams and computation time. Discuss when the differences matter.

Notebook: lab03-complex-comparison.ipynb  ·  Dataset: Public TDA benchmark suite (Anu Mall / scikit-tda).

Reference text for this week: chapter 06 of the bilingual notes — EN PDF · FR PDF.