[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-120597-en":3,"doc-seo-120597-105":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":20,"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},120597,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","OPERATOR-VALUED KERNELS - MACHINE LEARNING, AND DYNAMICAL SYSTEMS - Abstract and Key Results","This paper studies kernel optimization using general positive operator-valued kernels and proves a result that yields new factorizations and realizations. The work extends these implications to Hilbert space-valued Gaussian processes, linking operator-valued kernels with applications in dynamics and machine learning where uncertainty-aware probabilistic models are needed. Additional applications are developed in non-commutative probability, culminating in a new non-commutative Radon–Nikodym theorem, with further connections to induced scalar counterparts and quantum channels.","arXiv :2405 .09315v2 [math .OA] 11 Oct 2024  \nOPERATOR-VALUED KERNELS, MACHINE LEARNING, AND  \nDYNAMICAL SYSTEMS  \nPALLE E.T. JORGENSEN AND JAMES TIAN  \nAbstract. In the context of kernel optimization, we prove a result that yields new factorizations and realizations. Our initial context is that of general positive operator-valued kernels. We further present implications for Hilbert spacevalued Gaussian processes, as they arise in applications to dynamics and to machine learning. Further applications are given in non-commutative probability theory, including a new non-commutative Radon–Nikodym theorem.  \nContents  \n1. Introduction 1  \n2. L (H)-valued kernels and H-valued Gaussian processes 3  \n3. Dynamics and a non-commutative Radon􀀕Nikodym theorem 5  \n4. Completely positive maps, quantum channels 8  \nReferences 11  \n1. Introduction  \nMachine learning, particularly kernel methods, heavily relies on the concept of positive de􀀜nite (p.d.) kernels. These kernels allow the transformation of data into higher-dimensional spaces where linear separability is possible, leading to e􀀛ective classi􀀜cation and regression models. The positive operator-valued kernels discussed in our paper extend this idea to more complex structures, such as Hilbert spacevalued Gaussian processes. These processes enable the construction of probabilistic models that can handle uncertainty and provide predictive distributions.  \nFor instance, in support vector machines (SVMs) and Gaussian process regression (GPR), the choice and optimization of kernels are crucial for model performance. Our factorization results provide new ways to construct and optimize kernels, potentially leading to more e􀀞cient algorithms and improved generalization in machine learning models. This is mainly due to the induced scalar-valued counterparts of these operator-valued kernels. More speci􀀜cally, for a 􀀜xed L (H)-valued kernel K (s, t), (s, t) ∈ S × S, there is a family of scalar-valued kernels K􀀚 (s, t) = Trace (􀀚K (s, t)), indexed by positive trace class operators 􀀚 in the  \n2020 Mathematics Subject Classi􀀜cation. 46E22, 46L53, 47A20, 60G15, 68T07, 81P15, 81P47 .  \nKey words and phrases. Positive definite functions, Gaussian processes, covariance, dilation, non-commutative Radon-Nikodym derivatives, completely positive maps, measurement, quantum states, quantum gates, kernel method.  \n2  \nHilbert space H. This, in turn, yields the following kernel optimization model:  \nmf,in􀀚 (Xi |f (si) − ci | 2 + 􀀌 kf kHK 􀀚 : f ∈ HK 􀀚 , 􀀚 > 0) (1.1)  \nin which HK􀀚 is the reproducing Hilbert space (RKHS) of K􀀚 .  \nIn a Bayesian interpretation of (1.1), we impose a prior belief on the structure of the kernel function through the constraint 􀀚 . This positive operator 􀀚 can be seen as de􀀜ning a distribution over the ambient space of kernel functions (assuming Trace (􀀚) = 1), re􀀝ecting prior knowledge or assumptions about the functions we aim to model.  \nThe data 􀀜t term in (1.1) plays the role of the likelihood function. It assesses how well a particular kernel function, determined by 􀀚, explains the observed data. Essentially, this term measures the likelihood of the data given the chosen kernel. By combining the prior and likelihood, we can derive a posterior distribution over the space of kernel functions. This posterior represents our updated belief about the kernel after observing the data. The solution to our optimization problem (1.1) can then be interpreted as a Maximum a Posteriori (MAP) estimate, meaning that a solution corresponds to the kernel function that is most probable under the posterior distribution, given the observed data and our prior assumptions.  \nMain results. In this paper, motivated by the model (1.1), we present a result that o􀀛ers a canonical link between a general setting for operator-valued completely positive maps and their induced scalar-valued counterparts. We further provide applications to a variety of neighboring areas, including the following six closely interre","cbCaidqGb5Z93Xhg","https://ap.wps.com/l/cbCaidqGb5Z93Xhg","pdf",232893,1,13,"English","en",105,"# Introduction\n# L (H)-valued kernels and H-valued Gaussian processes\n# Dynamics and a non-commutative Radon–Nikodym theorem\n# Completely positive maps, quantum channels\n# References","[{\"question\":\"What is the main contribution of the paper in kernel optimization?\",\"answer\":\"It proves a result for general positive operator-valued kernels that yields new factorizations and realizations, enabling induced scalar-valued counterparts useful for kernel construction and optimization.\"},{\"question\":\"How do operator-valued kernels connect to Gaussian processes and machine learning?\",\"answer\":\"The paper presents implications for Hilbert space-valued Gaussian processes and discusses how kernel choice and optimization affect probabilistic models used in dynamics and machine learning tasks such as classification and regression.\"},{\"question\":\"What role does non-commutative probability play in the results?\",\"answer\":\"It provides further applications in non-commutative probability theory, including a new non-commutative Radon–Nikodym theorem, and relates operator-valued structures to quantum-related concepts.\"}]","OPERATOR-VALUED KERNELS - MACHINE LEARNING, AND DYNAMICAL SYSTEMS - Abstract and Key Results | PDF",1785730825,33,{"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},"operator-valued-kernels-machine-learning-and-dynamical-systems-abstract-and-key-results","",{"@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/operator-valued-kernels-machine-learning-and-dynamical-systems-abstract-and-key-results/120597/",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","2026-08-03",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the main contribution of the paper in kernel optimization?","Question",{"text":75,"@type":76},"It proves a result for general positive operator-valued kernels that yields new factorizations and realizations, enabling induced scalar-valued counterparts useful for kernel construction and optimization.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do operator-valued kernels connect to Gaussian processes and machine learning?",{"text":80,"@type":76},"The paper presents implications for Hilbert space-valued Gaussian processes and discusses how kernel choice and optimization affect probabilistic models used in dynamics and machine learning tasks such as classification and regression.",{"name":82,"@type":73,"acceptedAnswer":83},"What role does non-commutative probability play in the results?",{"text":84,"@type":76},"It provides further applications in non-commutative probability theory, including a new non-commutative Radon–Nikodym theorem, and relates operator-valued structures to quantum-related concepts.","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"]