Topological Methods in Machine Learning: A Tutorial for Practitioners

TL;DR

This tutorial provides a comprehensive introduction to two key TML techniques, persistent homology and the Mapper algorithm, with an emphasis on practical applications.

Abstract

Topological Machine Learning (TML) is an emerging field that leverages techniques from algebraic topology to analyze complex data structures in ways that traditional machine learning methods may not capture. This tutorial provides a comprehensive introduction to two key TML techniques, persistent homology and the Mapper algorithm, with an emphasis on practical applications. Persistent homology captures multi-scale topological features such as clusters, loops, and voids, while the Mapper algorithm creates an interpretable graph summarizing high-dimensional data. To enhance accessibility, we adopt a data-centric approach, enabling readers to gain hands-on experience applying these techniques to relevant tasks. We provide step-by-step explanations, implementations, hands-on examples, and case studies to demonstrate how these tools can be applied to real-world problems. The goal is to equip researchers and practitioners with the knowledge and resources to incorporate TML into their work, revealing insights often hidden from conventional machine learning methods. The tutorial code is available at https://github.com/cakcora/TopologyForML

Figures (35)

• Figure 1: Simplicial Complexes. Among the complexes, only b and f fail to be simplicial complexes, as their simplices do not intersect at complete subsimplices. All others are valid simplicial complexes.
• Figure 2: The sphere and the cube are topologically equivalent, whereas the torus is different from both.
• Figure 3: Toy examples for homology. We present the ranks of the homology groups of various topological spaces, i.e., for $\mathcal{H}_0(\mathcal{X})=\mathbb{Z}^k$ we write only $\mathcal{H}_0=k$ for simplicity, representing the count of the $k$-dimensional holes in $\mathcal{X}$.
• Figure 4: Toy example for homology.
• Figure 5: $\partial_1:\mathcal{C}_1(\mathcal{X})\to\mathcal{C}_0(\mathcal{X})$ is represented as a $4\times 4$ binary matrix. Columns represent the edges in $\mathcal{C}_1(\mathcal{X})$, and rows correspond to the vertices in $\mathcal{C}_0(\mathcal{X})$. For example, $\partial e_3= v_2+v_3$ can be read from the third column