[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117247-en":3,"doc-seo-117247-105":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},117247,2336464648322,"Aria","https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488",8,"Research & Report","C*-Algebraic Machine Learning - Moving in a New Direction","Machine learning has a long collaborative tradition with statistics, probability, and linear algebra, but its integration with operator-algebraic structures remains limited. The work proposes C*-algebraic machine learning as a cross-fertilization between C*-algebra and machine learning. C*-algebras generalize complex numbers and unify learning strategies, enabling new frameworks for information-rich data models. It explains why and how to use C*-algebras, focusing on kernel methods and neural networks, and discusses design considerations, open questions, and future applications.","C ∗-Algebraic Machine Learning: Moving in a New Direction  \nYuka Hashimoto 1 ,2 Masahiro Ikeda2 ,3 Hachem Kadri4  \n1. NTT Corporation, Tokyo, Japan  \n2. Center for Advanced Intelligence Project, RIKEN, Tokyo, Japan  \n3. Keio University, Yokohama, Japan  \n4. Aix-Marseille University, CNRS, LIS, Marseille, France  \narXiv :2402 .02637v2 [ cs .LG] 7 Jun 2024  \nAbstract  \nMachine learning has a long collaborative tradition with several fields of mathematics, such as statistics, probability and linear algebra. We propose a new direction for machine learning research: C ∗ -algebraic ML—a cross-fertilization between C ∗-algebra and machine learning. The mathematical concept of C ∗ -algebra is a natural generalization of the space of complex numbers. It enables us to unify existing learning strategies, and construct a new framework for more diverse and information-rich data models. We explain why and how to use C ∗-algebras in machine learning, and provide technical considerations that go into the design of C ∗-algebraic learning models in the contexts of kernel methods and neural networks. Furthermore, we discuss open questions and challenges in C ∗-algebraic ML and give our thoughts for future development and applications.  \n1 Introduction  \nMachine learning problems and methods are currently becoming more and more complicated. We have many types of structured data, such as time-series data, image data, and graph data. In addition, not only are the models large, but multiple models and tasks have to be considered in some situations.  \nTo address these situations, we propose C ∗ -algebraic machine learning: application of C ∗-algebra to machine learning methods. Typical examples of C ∗ -algebras are the space of continuous functions on a compact space and the space of bounded linear operators on a Hilbert space. C ∗-algebra was first proposed in quantum mechanics to model physical observablesand has been investigated in pure mathematics, math-  \nematical physics, and quantum mechanics. Whereas its rich mathematical and theoretical investigations, its main application is limited to quantum mechanics. In the current situation in machine learning, we believe that it is time to apply these rich investigations to machine learning methods. Since C ∗-algebras enable us to unify complex values, matrices, functions, and linear operators, we expect that the generalization of machine learning methods using C ∗-algebras allows us to unify existing methods and construct a framework for more complicated data and models. Figure 1 shows an overview of the C ∗-algebraic machine learning.  \nIn this paper, we mainly focus on two approaches: kernel methods and neural networks. For kernel methods, most of existing methods are realized using reproducing kernel Hilbert spaces (RKHSs) or vectorvalued RKHS (vvRKHS), which are constructed by positive definite kernels (Sch¨olkopf & Smola, 2001; Saitoh & Sawano, 2016) . The reproducing property enables us to evaluate the value of a function at a point using the inner product, which makes it easy for us to implement algorithms and analyze them theoretically. Moreover, we can apply kernel methods to probabilistic and statistical settings by embedding probability measures in an RKHS. This embedding is called the kernel mean embedding. However, since RKHSs (resp. vvRKHSs) are complex- (resp. vector-) valued function spaces, the output of the models is usually complex- or vector-valued. In addition, appropriate ways of the construction of positive definite kernels are not trivial. The generalization of RKHS by means of the C ∗-algebra enables us to output more general data, such as functions and operators (Hashimoto et al., 2021) . Moreover, C ∗-algebras give us a method to construct C ∗-algebra-valued pos-  \nitive definite kernels for structured data (Hasimoto et al., 2023a) . The noncommutative product structure in C ∗-algebras (ab  ba for elements a, b in the C ∗-algebra) enables us to construct an operation that ","cbCaidPAKISQ17au","https://ap.wps.com/l/cbCaidPAKISQ17au","pdf",885091,1,15,"English","en",105,"# Introduction\n## C*-Algebraic machine learning as a new direction\n## Kernel methods via C*-algebras\n## Neural networks and structured data challenges","[{\"question\":\"What is C*-algebraic machine learning?\",\"answer\":\"It applies C*-algebra concepts to machine learning methods, aiming to cross-fertilize operator-algebraic theory with learning models.\"},{\"question\":\"How do C*-algebras contribute to kernel methods?\",\"answer\":\"They generalize RKHS constructions and support building C*-algebra-valued positive definite kernels, enabling richer outputs such as functions and operators.\"},{\"question\":\"Which two main approaches does the paper focus on?\",\"answer\":\"The paper mainly focuses on kernel methods and neural networks, discussing how C*-algebras can fit into both frameworks.\"}]","C*-Algebraic Machine Learning - Moving in a New Direction | PDF",1785674653,38,{"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},"c-algebraic-machine-learning-moving-in-a-new-direction","",{"@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/c-algebraic-machine-learning-moving-in-a-new-direction/117247/",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","2026-08-02",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 C*-algebraic machine learning?","Question",{"text":75,"@type":76},"It applies C*-algebra concepts to machine learning methods, aiming to cross-fertilize operator-algebraic theory with learning models.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do C*-algebras contribute to kernel methods?",{"text":80,"@type":76},"They generalize RKHS constructions and support building C*-algebra-valued positive definite kernels, enabling richer outputs such as functions and operators.",{"name":82,"@type":73,"acceptedAnswer":83},"Which two main approaches does the paper focus on?",{"text":84,"@type":76},"The paper mainly focuses on kernel methods and neural networks, discussing how C*-algebras can fit into both frameworks.","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},[93,97,101,105,110,115,120,123,128,131,135],{"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":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]