TDA — Vietoris-Rips Filtration & Betti Numbers
As the radius ε grows, edges and triangles fill in. Betti numbers β₀ (components) and β₁ (loops) track the topology of the evolving simplicial complex
Persistent homology: Build Vietoris-Rips complex: include edge (i,j) when d(pᵢ,pⱼ) ≤ ε; include triangle {i,j,k} when all pairwise distances ≤ ε. Track β₀ = #(connected components) and β₁ = #(independent cycles). As ε increases from 0: β₀ decreases (components merge), β₁ first rises (loops form) then falls (holes fill in). Loops that persist over a wide ε-range are topologically significant (encoded in persistence barcode). The barcode shows birth/death of each feature — long bars = real structure, short bars = noise.