Vietoris-Rips filtration on a point cloud: as ε grows, simplices appear and topological features (H₀ = components, H₁ = loops) are born and die. The persistence barcode records the lifetime of each feature — long bars are significant topology.
0.30
16
H₀ bars (components)
H₁ bars (loops)
Active edges
Stability: d_B(Dgm(f), Dgm(g)) ≤ ‖f − g‖∞ — small perturbations create small barcode changes.
Long bars = persistent topology; short bars ≈ noise. H₀: #bars = #components; H₁: #bars = #independent loops.