[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-124594-en":3,"doc-seo-124594-105":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},124594,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","Persistent Homology in Machine Learning - Applied Sciences Review","Topological Data Analysis (TDA) strengthens data processing and Machine Learning (ML) by incorporating topology into feature extraction. Persistent homology (PH), a key part of the TDA toolkit, can be challenging to assess for newcomers because it differs from the mathematics commonly used in classical ML. This paper provides an overview of recent PH developments with an emphasis on applied sciences, covering physics, healthcare, material sciences, and related domains. It also summarizes current limitations and highlights potential future directions, supporting both experts and beginners.","Proceedings of the 11th International Conference on Applied Innovations in IT,(ICAIIT), March 2023  \nPersistent Homology in Machine Learning: Applied Sciences Review  \nOleksandr Yavorskyi 1, Andrii Asseko-Nkili 1 and Nataliia Kussul1,2,3  \n1Department of Mathematical Modeling and Data Analysis, Igor Sikorsky Kyiv Polytechnic Institute,  \nPeremohy Avenue 37, Kyiv, Ukraine  \n2Department of Space Information Technologies and System, Space Research Institute National Academy of Science of Ukraine an State Space Agency of Ukraine, Glushkov Avenue 40, Kyiv, Ukraine  \n3Anhalt University of Applied Sciences, Bernburger Str. 57, Köthen, Germany  \n[yaotianjiu@gmail.com](yaotianjiu@gmail.com), [a0494034@gmail.com](a0494034@gmail.com), [nataliia.kussul@gmail.com](nataliia.kussul@gmail.com)  \nKeywords: Algebraic Topology, Persistent Homology, Machine Learning, Physics, Healthcare, Topological Data Analysis, Chemistry, Biology, Material Sciences, Data Processing.  \nAbstract: Topological Data Analysis (‘TDA’) has become a vibrant and quickly developing field in recent years, providing topology-enhanced data processing and Machine Learning (‘ML’) applications. Due to the novelty of the field, as well as the dissimilarity between the mathematics behind the classical ML and TDA, it might be complicated for a field newcomer to assess the feasibility of the approaches proposed by TDA and the relevancy of the possible applications. The current paper aims to provide an overview of the recent developments that relate to persistent homology, a part of the mathematical machinery behind the TDA, with a particular focus on applied sciences. We consider multiple areas, such as physics, healthcare, material sciences, and others, examining the recent developments in the field. The resulting summary of this paper could be used by field experts to expand their knowledge on recent persistent homology applications, while field newcomers could assess the applicability of this TDA approach for their research. We also point out some of the current restrictions on the use of persistent homology, as well as potential development trajectories that might be useful to the whole field.  \n1 INTRODUCTION  \nArtificial Intelligence (‘AI’) is a fruitful and flourishing area that focuses on the development of algorithms that are capable of replicating human behavior. An important constituent of this area, which encompasses a variety of mathematical instruments developed for capturing, formalization, and optimization of methods that can help in the aims of AI, called Machine Learning (‘ML’)  \nLinear algebra, statistics, and probability theory, as well as functional analysis, comprise the list of the most widely-used mathematical instruments for Machine Learning. At the same time, more sophisticated mathematical machinery receives evergrowing attention from ML specialists [1] . Algebraic topology could be considered as one of the most important of such ‘mathematical newcomers’ to the field. The impact of topology on the current ML scene led to the emergence of a whole new area called Topological Data Analysis.  \nAlgebraic topology raises basic questions about the shape of the object and is especially interested in  \nthe shape features that are invariant under deformation. A hole in S2 sphere or torus is a typical example of such an invariant since it does not diminish up until we ‘cut’ the figure.  \n2 FUNDAMENTALS OF PERSISTENT HOMOLOGY  \nInformally, persistent homology (‘PH’) allows us to discover which features of the data set (called ‘point cloud’) are time-invariant. The latter notion gives researchers the right means to assess some constant geometrical (or, it is better to say, ‘topological’) features of the set. Below we give a more elaborate introduction to this concept.  \n2.1 Complexes, Filtrations and Persistence  \nA simplicial complex could be defined simply as a set consisting of points, lines, and n-order polytopes of  \nProceedings of the 11th International Confere","cbCaijyS1JfOYFR4","https://ap.wps.com/l/cbCaijyS1JfOYFR4","pdf",800886,1,6,"English","en",105,"# Introduction\n# Fundamentals of Persistent Homology\n## Complexes, Filtrations and Persistence\n## Persistent Homology as an Input","[{\"question\":\"What is Topological Data Analysis and why is it useful for Machine Learning?\",\"answer\":\"TDA is a fast-growing field that uses topology-enhanced data processing to support Machine Learning applications. It helps extract structure-aware features from data beyond standard approaches.\"},{\"question\":\"How does persistent homology work in the context of point clouds?\",\"answer\":\"Persistent homology identifies topological features of a point cloud and tracks how long they remain stable across a filtration. It summarizes features using tools such as persistence diagrams and barcodes.\"},{\"question\":\"Which application areas are emphasized for persistent homology?\",\"answer\":\"The paper focuses on applied sciences, including physics, healthcare, and material sciences, and discusses how PH developments relate to these domains.\"}]","Persistent Homology in Machine Learning - Applied Sciences Review | PDF",1785893220,15,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"persistent-homology-in-machine-learning-applied-sciences-review","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/persistent-homology-in-machine-learning-applied-sciences-review/124594/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-05",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is Topological Data Analysis and why is it useful for Machine Learning?","Question",{"text":75,"@type":76},"TDA is a fast-growing field that uses topology-enhanced data processing to support Machine Learning applications. It helps extract structure-aware features from data beyond standard approaches.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does persistent homology work in the context of point clouds?",{"text":80,"@type":76},"Persistent homology identifies topological features of a point cloud and tracks how long they remain stable across a filtration. It summarizes features using tools such as persistence diagrams and barcodes.",{"name":82,"@type":73,"acceptedAnswer":83},"Which application areas are emphasized for persistent homology?",{"text":84,"@type":76},"The paper focuses on applied sciences, including physics, healthcare, and material sciences, and discusses how PH developments relate to these domains.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":21,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]