用拓扑方法发现数据中隐藏的形状结构，提升机器学习洞察力
Topological Methods in Machine Learning: A Tutorial for Practitioners
- 通过持久同调和Mapper算法捕捉数据的多尺度拓扑特征
- 可识别传统方法忽略的簇、环、空洞等复杂结构
- 适合想探索数据深层几何特性的研究人员与工程师
拓扑机器学习（TML）是新兴领域，利用代数拓扑技术分析传统机器学习难以捕捉的复杂数据结构。本教程系统介绍两种核心TML方法：持久同调与Mapper算法。持久同调可捕获数据在多尺度下的拓扑特征，如聚类、环路和空洞；Mapper算法则生成可解释的图结构以总结高维数据。为增强实用性，教程采用以数据为中心的方法，提供分步说明、代码实现、动手示例与案例研究，帮助读者在真实任务中应用这些工具。目标是让研究者与实践者掌握TML知识与资源，揭示常规方法难以察觉的数据隐含规律。相关代码已开源：https://github.com/cakcora/TopologyForML
原文摘要 · Abstract (English)
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
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