Topological Machine Learning: Theoretical Foundations and a Custom Classifier on Artificial Data

Topological Machine Learning (TML) leverages tools from algebraic topology, especially persistent homology, to extract robust features from data that remain invariant under continuous deformations. This article presents a comprehensive overview of the theoretical underpinnings of TML, historical evolution, core equations, and a Python-based implementation of a topological handwritten digit classifier using artificial data. We avoid common libraries like sklearn and tensorflow, ensuring complete control and transparency of computation.