Persistence Barcode And Betti Curves

Persistent homology produces intervals. ML models usually need fixed-width vectors. This page shows the bridge: barcode intervals become Betti curves, persistence images, or small signature dictionaries.

timeline
  title Filtration To Features
  radius 0.00 : points are isolated : beta0 is high
  radius 0.25 : close components merge : short H0 bars die
  radius 0.75 : loop appears : beta1 rises
  radius 1.25 : loop fills : beta1 falls

Active API

import topoml

diagram = topoml.persistent_homology(points, max_dim=1, max_radius=2.0)
curve = topoml.BettiCurve(radii=[0.0, 0.25, 0.75, 1.25]).fit_transform([diagram])
image = topoml.PersistenceImage(width=16, height=16).fit_transform([diagram])

Reading The Graph

The barcode answers "how long did a feature live?" The Betti curve answers "how many features are alive at this radius?" The persistence image answers "where are stable features concentrated in birth-persistence space?"

flowchart TB
  A["Persistence pairs (birth, death, dim)"] --> B["Barcode for inspection"]
  A --> C["BettiCurve for tabular models"]
  A --> D["PersistenceImage for image-like vectors"]
  C --> E["PHFeaturizer"]
  D --> E

Claim Boundary

These are active feature encoders. They are not universal best encodings. A training claim must state the radii, homology dimensions, image resolution, baseline feature set, split policy, and whether topology improved held-out metrics.