topology-data-analysis
Installation
SKILL.md
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 |
| Alpha | Delaunay-based (exact in low dim) | Efficient in R^2, R^3 |
| Cubical | Grid-based (for images) | Linear in pixels |