[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117835-en":3,"doc-seo-117835-105":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},117835,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Topological data analysis and machine learning","Topological data analysis provides methods to compute robust, abstract “shapes” from complex datasets, based on quantities invariant under continuous deformations. The work reviews applications of TDA in physics and physics-driven machine learning, emphasizing tasks such as unsupervised detection of phase transitions. It also introduces core techniques, surveys how TDA can reveal order parameters and novel phases, and highlights future research directions supported by emerging extensions like zigzag persistence.","arXiv :2206 . 15075v3 [ cond-mat .mes-hall ] 25 Jul 2023  \nTopological data analysis and machine learning  \nDaniel Leykam 1 and Dimitris G. Angelakis 1, 2, 3  \n1 Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543  \n2 School of Electrical and Computer Engineering, Technical University of Crete, Chania, Greece 73100  \n3 AngelQ Quantum Computing, 531A Upper Cross Street, \\#04-95 Hong Lim Complex, Singapore 051531 (Dated: July 26, 2023)  \nTopological data analysis refers to approaches for systematically and reliably computing abstract \\shapes\" of complex data sets. There are various applications of topological data analysis in life and data sciences, with growing interest among physicists. We present a concise review of applications of topological data analysis to physics and machine learning problems in physics including the unsupervised detection of phase transitions. We 􀀌nish with a preview of anticipated directions for future research.  \nI. INTRODUCTION  \nTopological quantities are invariant under continuous deformations; an often-cited example is that a doughnut can be continuously transformed into co􀀋ee mug-both are topologically equivalent to a torus. The robustness of topological quantities to perturbations is inspiring physicists in many 􀀌elds, including condensed matter, photonics, acoustics, and mechanical systems [1{4] . In all these areas topology has enabled the prediction and explanation of surprisingly robust physical e􀀋ects. Most famously, the extremely precise quantisation of the Hall conductivity observed in twodimensional electronic systems since the 1980s was explained as a novel topological phase of matter, the quantum Hall phase [5] . In this and many other examples from physics, we deal with smooth deformations in some parameter space, such as the energy bands of solid state electronic systems.  \nPhysics is however an outlier among 􀀌elds of science in that idealised continuous models and functions can explain a wide variety of observed phenomena. Other 􀀌elds do not have the luxury of continuity and have to make do out of sparse data and limited observations in high dimensional parameter spaces. Despite this very di􀀋erent setting, topological approaches remain powerful.  \nA suite of computational topological techniques known as topological data analysis (TDA) has been developed over the past twenty years to systematically de􀀌ne and study the \\shape\" of complex discrete data in high dimensional spaces. TDA is attracting growing interest among physicists, particularly those working on topological materials or the application of machine learning techniques to physics [6{10] .  \nAt this time we are aware of two existing reviews on TDA aimed at the physics audience. The 􀀌rst by Carlsson, one of the founders of the 􀀌eld, gave a broad survey of di􀀋erent techniques of TDA and their applications in various areas of science [11] . The second review, by Murugan and Robertson, provided a detailed pedagogical and physicist-friendly introduction to two important techniques, persistent homology and the Mapper algorithm, applying them to the example of an astronomical dataset [12] .  \nSince publication of these two reviews there has been growing interest in applying TDA methods to physics, including the incorporation of TDA into physics-targeted machine learning, with applications including the unsupervised detection of phase transitions. Moreover, the 􀀌eld of TDA has continued to evolve with new generalisations and techniques being actively studied.  \nThe aim of this article is to review cutting edge applications of TDA to physics. We will provide a gentle introduction to the basic techniques, survey how TDA shows promise for the detection of novel phases of matter, and speculate on what we believe to be important directions for future research, including opportunities o􀀋ered by newer TDA methods such as zigzag persistence.  \nThe structure of this article is as follows: S","cbCaiupFZlhqLmuD","https://ap.wps.com/l/cbCaiupFZlhqLmuD","pdf",1268631,1,15,"English","en",105,"# Introduction\n## Topological invariants and robustness\n## Motivation for TDA in physics\n## Prior reviews and scope of this article\n# Topological data analysis\n## From point clouds to persistence diagrams\n## Persistence diagrams and key representations","[{\"question\":\"What is the main idea of topological data analysis (TDA) in this article?\",\"answer\":\"TDA systematically computes abstract “shapes” of complex data using topological quantities that remain invariant under continuous deformations, making them robust to perturbations.\"},{\"question\":\"How does TDA connect to machine learning in physics?\",\"answer\":\"The article reviews how TDA features can be integrated into physics-targeted machine learning pipelines, including unsupervised detection of phase transitions.\"},{\"question\":\"What future directions does the article discuss for TDA research in physics?\",\"answer\":\"It outlines anticipated research directions and mentions opportunities from newer TDA methods such as zigzag persistence.\"}]","Topological data analysis and machine learning | 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is the main idea of topological data analysis (TDA) in this article?","Question",{"text":75,"@type":76},"TDA systematically computes abstract “shapes” of complex data using topological quantities that remain invariant under continuous deformations, making them robust to perturbations.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does TDA connect to machine learning in physics?",{"text":80,"@type":76},"The article reviews how TDA features can be integrated into physics-targeted machine learning pipelines, including unsupervised detection of phase transitions.",{"name":82,"@type":73,"acceptedAnswer":83},"What future directions does the article discuss for TDA research in physics?",{"text":84,"@type":76},"It outlines anticipated research directions and mentions opportunities from newer TDA methods such as zigzag 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