[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-118086-en":3,"doc-seo-118086-105":30,"detail-sidebar-cat-0-en-105":83},{"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},118086,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","Random Convex Hulls and Kernel Quadrature - Thesis","Discretization of probability measures underpins classical numerical integration, data compression, and algorithmic acceleration in machine learning. This thesis studies random convex hulls and kernel quadrature starting from generalized Tchakaloff-type cubature. It analyzes the probability that a vector θ lies in the convex hull of independent copies of a random vector X, connecting sharp bounds with Tukey’s halfspace depth and deriving specialized results for θ = E[X]. It then develops convex kernel quadrature using spectral properties of the associated integral operator, with guarantees from eigenvalue decay, and proposes practical algorithmic variants. The work concludes with applications and future directions, including Bayesian numerical methods.","Random Convex Hulls and Kernel Quadrature  \nSatoshi Hayakawa St Catherine’s College University of Oxford  \nA thesis submitted for the degree of Doctor of Philosophy  \nTrinity 2023  \nAcknowledgements  \nFirst, I would like to express my deepest gratitude to my family, Saeko, Hisao, and Hisashi, for their invaluable support throughout my journey up to this point. Without their help, I could not have pursued my curiosity about mathematics from childhood.  \nI also deeply thank my colleagues in Oxford. Christina, Cris, James, and Patric greatly helped me regarding life in Oxford as a maths DPhil student. I am especially grateful to Jake, my first neighbor in Oxford, for cheering me up during the lockdown, without which I could not have gone through those depressing days.  \nI also thank all my academic collaborators during DPhil: Ken’ichiro Tanaka, Masaki Adachi, Saad Hamid, Xingchen Wan, Martin Jørgensen, Michael Osborne, Hayata Yamasaki, Sathyawageeswar Subramanian, Sho Sonoda, and my supervisors. I have to say a special thank you to Masaki for turning my mathematical ideas into something really useful.  \nThe university table tennis club is my best community in Oxford. I love all my teammates and especially thank Zhongyi and Haichuan for being my best friends during and not during the training. Thankyou to the men’s first team members: Kin for coaching and driving, Angus for being a good captain, and Alistair for that lovely two-handed backhands in our doubles. Finally, I am grateful to Taylor and Beverly for being my great rivals. I always enjoyed chopping your forehands.  \nI want to thank my dearest friends in Japan, including Catan-kai members. I appreciate your help and the time we spent together, without which completing this thesis would have been impossible.  \nI thank Toyota Riken Overseas Scholarship, Clarendon Scholarship, Oxford-Kobe Scholarship, Datasıg, and the CIMDA-Oxford for financial support on life in Oxford and academic activities.  \nI am deeply grateful to my examiners, Prof. Yuji Nakatsukasa and Prof. Chris Oates for assessing this thesis. Thank you to Benjamin Walker, Prof. Ken’ichiro Tanaka for proofreading the manuscript of the thesis. I am also grateful to Prof. Oliver Riordan, Prof. Raphael Hauser, and Prof. Massimiliano Gubinelli for assessing my transfer and confirmation theses. Thank you to all the anonymous reviewers of my papers for your strict comments, which improved the thesis a lot.  \nLast but not least, I am immensely grateful to my supervisors, Prof. Terry Lyons and Prof. Harald Oberhauser, for their help throughout my DPhil. They always patiently supported me and gave me countless ideas on life and mathematics.  \nAbstract  \nDiscretization of probability measures is ubiquitous in the field of applied mathematics, from classical numerical integration to data compression and algorithmic acceleration in machine learning. In this thesis, starting from generalized Tchakaloff-type cubature, we investigate random convex hulls and kernel quadrature.  \nIn the first two chapters after the introduction, we investigate the probability that a given vector θ is contained in the convex hull of independent copies of a random vector X . After deriving a sharp inequality that describes the relationship between the said probability and Tukey’s halfspace depth, we explore the case θ = E [X] by using moments of X and further the case when X enjoys some additional structure, which are of primary interest from the context of cubature.  \nIn the subsequent two chapters, we study kernel quadrature, which is numerical integration where integrands live in a reproducing kernel Hilbert space. By explicitly exploiting the spectral properties of the associated integral operator, we derive convex kernel quadrature with theoretical guarantees described by its eigenvalue decay. We further derive practical variants of the proposed algorithm and discuss their theoretical and computational aspects.  \nFinally, we briefly discuss the a","cbCaioUt6gXKrblq","https://ap.wps.com/l/cbCaioUt6gXKrblq","pdf",1378069,1,200,"English","en",105,"# Abstract\n## Random convex hull probabilities\n## Kernel quadrature via spectral properties\n## Applications and future work\n# Acknowledgements\n## Academic collaborators and supervision\n## Oxford community and scholarships\n# Publications\n## Related papers by chapter order\n# Breakdown of contributions\n## Author roles in each publication","[{\"question\":\"What applications and future directions are mentioned?\",\"answer\":\"The concluding chapter briefly discusses applications and future work, including Bayesian numerical methods.\"}]","Random Convex Hulls and Kernel Quadrature - Thesis | PDF",1785681462,504,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":28},"random-convex-hulls-and-kernel-quadrature-thesis","",{"@graph":36,"@context":77},[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/random-convex-hulls-and-kernel-quadrature-thesis/118086/",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],{"name":72,"@type":73,"acceptedAnswer":74},"What applications and future directions are mentioned?","Question",{"text":75,"@type":76},"The concluding chapter briefly discusses applications and future work, including Bayesian numerical methods.","Answer","https://schema.org",{"og:url":52,"og:type":79,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":81,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]