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Module 6: Topology (Cubical, Rips, Alpha) and Features

  • Objectives
    • Compute persistence via cubical, Rips, Alpha backends
    • Extract feature vectors; understand persistence images/landscapes
  • Time: 60–90 minutes

6.1 Cubical persistence

import numpy as np
from mneme.core.topology import PersistentHomology

field = np.random.randn(64,64)
ph = PersistentHomology(max_dimension=2, filtration='sublevel', persistence_threshold=0.05)
diagrams = ph.compute_persistence(field)
features = ph.extract_features(diagrams)

6.2 Rips/Alpha via adapter

from mneme.core.topology import RipsComplex, AlphaComplex, field_to_point_cloud

pc = field_to_point_cloud(field, method='peaks', percentile=95.0, max_points=2000)
rc = RipsComplex(max_dimension=1)
rd = rc.compute_persistence(pc)
rf = rc.extract_features(rd)

6.3 Persistence images/landscapes (MVP)

  • Use compute_persistence_image/compute_persistence_landscape helpers for derived representations

Exercises

  1. Compare feature vectors across backends on the same field
  2. Increase percentile in field_to_point_cloud and see how point density affects Rips results
  3. Plot a persistence image for H1 and interpret visually

Run log (MVP)

  • Success: Cubical and Rips both produced 2 diagrams with 12-length feature vectors in a quick test.

Solutions (outline)

  • Different backends produce different sensitivities; features reflect scale and sampling
  • Higher threshold → fewer points → sparser complexes
  • Bright areas in persistence image correspond to long-lived features