Persistent Homology & Topological Data Analysis
Multi-scale topological features of point clouds — and how to read a barcode.
Given a finite point cloud $X \subset \mathbb R^n$ and a scale $\varepsilon \geq 0$, form the Vietoris–Rips complex $R_\varepsilon(X)$: a simplex on a subset $S$ when all pairwise distances in $S$ are $\leq \varepsilon$. As $\varepsilon$ grows, complexes nest:
$$R_0 \subset R_{\varepsilon_1} \subset R_{\varepsilon_2} \subset \cdots$$Topological features (connected components, loops, voids) appear at some scale ($birth$) and may disappear at a larger scale ($death$). Persistent homology records these intervals $(b_i, d_i)$ in a barcode or persistence diagram.
Long bars represent robust topological features of the data; short bars are noise. The barcode is stable under perturbation (Cohen-Steiner–Edelsbrunner–Harer), making persistent homology a powerful multi-scale summary.
Applications: shape recognition, neural data analysis, materials science (porous structures), sensor networks, image analysis — anywhere "shape of data at multiple scales" matters.
Interactive: 0-dim persistence on a point cloud
Slide ε to grow the disks around points; connected components merge. Bars on the right show when each component dies (merges into another).