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topology-data-analysis

Topological data analysis: persistent homology, Mapper, and TDA tools

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# Topological Data Analysis

A skill for applying topological data analysis (TDA) methods to research data. Covers persistent homology, Vietoris-Rips complexes, persistence diagrams, the Mapper algorithm, and vectorization methods for integrating topological features into machine learning pipelines.

## Core Concepts

### Simplicial Complexes from Data

TDA extracts topological features (connected components, loops, voids) from data by building simplicial complexes at multiple scales:

| Complex | Construction | Computational Cost |
|---------|-------------|-------------------|
| Vietoris-Rips | Edge if distance < epsilon | O(n^d) for d-simplices |
| Cech | Ball intersection (exact) | Computationally expensive |