[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-118629-en":3,"doc-seo-118629-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},118629,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",6,"Technology","Topological Methods in Machine Learning - A Tutorial for Practitioners","Topological Methods in Machine Learning (TML) applies algebraic topology tools to complex data to reveal structure that conventional machine learning may miss. The tutorial introduces two core techniques—persistent homology and the Mapper algorithm—with a practical, data-centric focus. Readers get step-by-step explanations, implementations, hands-on examples, and case studies showing how multi-scale topological features can be translated into interpretable graph summaries and vectors for real-world tasks, with guidance for incorporating these methods into research workflows. Tutorial code is provided.","arXiv :2409 .0290 1v 1 [ cs .LG] 4 Sep 2024  \nTopological Methods in Machine Learning: A Tutorial for Practitioners  \nBARIS COSKUNUZER, University of Texas at Dallas, USA  \nCÜNEYT GÜRCAN AKÇORA, University of Central Florida, USA  \nTopological 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](https://github.com/cakcora/TopologyForML).  \nCCS Concepts: • Mathematics of computing → Algebraic topology; • Computing methodologies → Learning paradigms; Machine learning algorithms; Machine learning applications; • Applied computing → Life and medical sciences; Physical sciences and engineering.  \nAdditional Key Words and Phrases: Topological Data Analysis, Machine Learning, Persistent Homology, Mapper, Multiparameter Persistence  \nACM Reference Format:  \nBaris Coskunuzer and Cüneyt Gürcan Akçora. 2023. Topological Methods in Machine Learning: A Tutorial for Practitioners. 1, 1  \n(September 2023), 54 pages. [https://doi.org/10.1145/XXXXXXX.XXXXXXX](https://doi.org/10.1145/XXXXXXX.XXXXXXX)  \nContents  \nAbstract 1  \nContents 1  \n1 Introduction 3  \n1.1 Roadmap for the Tutorial 4  \n2 Background 5  \n2.1 A Crash Course on Topology 5  \n2.1.1 Topological Space 5  \n2.1.2 Topological Equivalence 7  \n2.1.3 Topological Invariant 8  \n2.1.4 Geometry vs. Topology 8  \nAuthors’ addresses: Baris Coskunuzer, [coskunuz@utdallas.edu](coskunuz@utdallas.edu), University of Texas at Dallas, Richardson, TX, USA; Cüneyt Gürcan Akçora, cuney. [akcora@ucf.edu](akcora@ucf.edu), University of Central Florida, Orlando, FL, USA.  \nPermission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, [requires prior specific permission and/or a fee. Request permissions from permissions@acm.org](requires prior specific permission and/or a fee. Request permissions from permissions@acm.org).  \n© 2023 Copyright held by the owner/author(s) . Publication rights licensed to ACM.  \nManuscript submitted to ACM  \nManuscript submitted to ACM 1  \n2 Coskunuzer-Akcora  \n2.2 Homology 9  \n2.2.1 TLDR 9  \n2.2.2 Computation of Homology 11  \n3 Persistent Homology 14  \n3.1 Constructing Filtrations 14  \n3.1.1 Filtrations for Point Clouds 15  \n3.1.2 Filtrations for Images 17  \n3.1.3 Filtrations for Graphs 18  \n3.1.4 Choosing Thresholds 21  \n3.2 Persistence Diagrams 22  \n3.2.1 Interpretation of Persistence Diagrams 23  \n3.2.2 Wasserstein Distance 24  \n3.3 Integrating PDs to ML tasks 1: Vectorizations 25  \n3.3.1 Stability 29  \n3.3.2 Choice of Vectorization and Hyperparameters 29  \n3.4 Integrating PDs to ML task","cbCaipC2LYBT5Bxh","https://ap.wps.com/l/cbCaipC2LYBT5Bxh","pdf",7043804,1,54,"English","en",105,"# Contents\n## 1 Introduction\n## 2 Background\n## 3 Persistent Homology\n## 4 Multiparameter Persistence\n## 5 Mapper\n## 6 Applications\n## 7 Future Directions\n## 8 Conclusion\n## A Notation Table\n## B Dataset Resources","[{\"question\":\"What are the two main techniques covered in the tutorial?\",\"answer\":\"The tutorial focuses on persistent homology and the Mapper algorithm as the two key approaches in topological machine learning (TML).\"},{\"question\":\"How does persistent homology contribute to machine learning tasks?\",\"answer\":\"Persistent homology captures multi-scale topological features and is integrated into ML via representations such as vectorizations and learning with neural networks, including stability considerations and complexity aspects.\"},{\"question\":\"What types of applications are demonstrated?\",\"answer\":\"Applications include shape recognition from point clouds, anomaly forecasting for graphs, cancer diagnosis from histopathological images, drug discovery with multiparameter persistence, and cancer genotyping from RNA sequencing using Mapper.\"}]","Topological Methods in Machine Learning - 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