Topological Data Analysis

Vietoris-Rips complex VR(X,ε): add a k-simplex whenever all pairwise distances ≤ ε.

As ε grows from 0 to ∞, topological features are born and die. A feature that persists long is real signal; short-lived features are noise.

H₀ (components): born at 0, die when merged.
H₁ (loops): born when loop forms, die when filled.

Stability theorem (Cohen-Steiner 2007): d_bottleneck(Dgm(f),Dgm(g)) ≤ ‖f-g‖_∞. Persistence is robust to perturbations.