Module homology 

Persistent Homology and Topological Data Analysis

Topological methods for analyzing shape and structure in data.

Key Capabilities

• Persistent Homology: Track topological features (components, loops, voids)
• Betti Numbers: Count topological features at each scale
• Persistence Diagrams: Visualize feature lifetimes
• Bottleneck/Wasserstein Distance: Compare topological signatures

Integration with Mincut

TDA complements mincut by providing:

• Long-term drift detection (shape changes over time)
• Coherence monitoring (are attention patterns stable?)
• Anomaly detection (topological outliers)

Mathematical Background

Given a filtration of simplicial complexes K_0 ⊆ K_1 ⊆ … ⊆ K_n, persistent homology tracks when features are born and die.

Birth-death pairs form the persistence diagram.

Structs

AlphaComplex

Alpha complex filtration (more efficient than Rips for low dimensions)

BettiNumbers

Betti numbers at a given scale

BirthDeathPair

Birth-death pair in persistence diagram

BottleneckDistance

Bottleneck distance between persistence diagrams

Filtration

Filtration: sequence of simplicial complexes

PersistenceDiagram

Persistence diagram: collection of birth-death pairs

PersistentHomology

Persistent homology computation

Point

Point in Euclidean space

PointCloud

Point cloud for TDA

Simplex

A simplex (k-simplex has k+1 vertices)

SimplicialComplex

Simplicial complex (collection of simplices)

VietorisRips

Vietoris-Rips filtration

WassersteinDistance

Wasserstein distance between persistence diagrams