Module tda_vr 

Topological Data Analysis (TDA) and Persistent Homology

This module provides implementations of persistent homology for analyzing topological features of data. Key concepts include:

• Simplicial complexes: Geometric objects built from vertices, edges, triangles, etc.
• Filtrations: Nested sequences of simplicial complexes parameterized by a scale
• Persistence diagrams: Collections of (birth, death) pairs representing topological features
• Barcodes: Interval representations of persistent homology
• Persistence images: Stable vectorizations of persistence diagrams

Algorithms

• Vietoris-Rips complex construction via distance-based filtration
• Boundary matrix reduction for computing persistent homology
• Bottleneck distance between persistence diagrams
• Wasserstein distance between persistence diagrams
• Persistence image vectorization

References

• Edelsbrunner, H., Letscher, D., & Zomorodian, A. (2002). Topological persistence and simplification.
• Zomorodian, A., & Carlsson, G. (2005). Computing persistent homology.
• Adams, H., et al. (2017). Persistence images: A stable vector representation of persistent homology.

Structs

Barcode

Persistence barcode: a multi-set of intervals representing topological features.

BarcodeInterval

An interval [birth, death) in a persistence barcode

PersistenceDiagram

Persistence diagram: a collection of (birth, death) pairs per dimension.

PersistenceImage

Persistence image: a stable vector representation of persistence diagrams.

PersistenceLandscape

Persistence landscape: a functional summary of persistence diagrams.

PersistencePoint

A single persistence point (birth, death) pair in a persistence diagram. The dimension indicates which homological dimension this feature belongs to.

VietorisRips

Vietoris-Rips complex for computing persistent homology

Enums

PersistenceWeight

Weight function for persistence image computation

Functions

bottleneck_distance

Compute the bottleneck distance between two persistence diagrams.

bottleneck_distance_dim

Compute the bottleneck distance between two persistence diagrams for a specific dimension.

wasserstein_distance

Compute the Wasserstein-p distance between two persistence diagrams.