[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-121918-en":3,"doc-seo-121918-105":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},121918,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","Applications of stochastic analysis and algebra to machine learning","This thesis studies how tools from stochastic analysis and algebra can be used to build statistical methods for machine learning. The central framework is signature methods, where a path is mapped to a structured group representation, giving valuable invariances and an algebraically rich image. The work develops new signature cumulants for characterizing independence, kernelized cumulants related to reproducing kernel Hilbert spaces, and general feature map extensions into non-commutative algebras. It also proposes topology-aware feature maps for stochastic filtrations and scoring-rule families with divergences, entropies, and mutual informations for paths respecting group structure.","Applications of stochastic analysis and algebra to machine learning  \nPatric Ossian Mauritz Bonnier  \nSt John’s College University of Oxford  \nA thesis submitted for the degree of Doctor of Philosophy in Mathematics  \nAcknowledgements  \nI would like to express my gratitude to my supervisor Harald Oberhauser for his guidance and support. It has been a privilege to have been your student, and your guidance has been invaluable throughout.  \nI am very thankful to my examiners, Terry Lyons and Josef Teichmann for agreeing to asses this thesis, and to Zhongmin Qian, Ben Hambly, and Sam Cohen for assessing my transfer and confirmation of status. Your feedback and comments have been very helpful. The support, guidance and help from my collaborators has also been deeply helpful to me. Cristopher Salvi, Imanol Perez Arribas, Patrick Kidger, Csaba Toth, Chong Liu, Zoltán Szabó, and Yannic Vargas, thank you all for your help. I am also thankful to my colleagues and office mates Vlad Margarint, Christina Zou and Alexander Schell for your friendship and support.  \nI would like to express my gratitude to my family Claës, Yvonne, Isabelle, and Thérèce. Thank you for being there for me and for supporting me this whole time. You are all wonderful, and I am truly blessed to have you in my life. This gratitude gladly extends itself to my friends for supporting my through this journey. You mean more to me than you know, and I am very lucky to have such great people in my life. Lastly, my deepest gratitude goes to my girlfriend Jana and our dog Echo, who have been a constant source of love and support, even if the latter might not fully appreciate this thesis. I love you both.  \nAbstract  \nIn this thesis we consider the application of tools from stochastic analysis and algebra to statistics and machine learning. Most of these tools are different forms of what has become known as signature methods. The signature has been discovered and rediscovered in a few different areas of mathematics in the last 70 years. In short, it maps a path evolving in a vector space to a group enveloping that same space. The reason it is so useful for statistics is twofold: One, its set of invariances is highly desirable in many applications, and two, its image group is highly structured making it particularly amenable to mathematical study using algebraic tools.  \nThe primary aim here is to study how one may use the signature to express statistical properties of a given path, and how these properties can be applied to machine learning. This aim manifests itself as-among other things:  \n• a new type of cumulants for signatures that have unique combinatorial properties and can be used to characterise independence of paths,  \n• cumulants on reproducing kernel Hilbert spaces which are related to the signature cumulants, even though signatures are not used explicitly,  \n• A generalisation of the signature to other types of feature maps into non-commutative algebras,  \n• a feature map with an initial topology that captures properties of the filtration of stochastic processes,  \n• and a family of scoring rules with associated divergences, entropiesand mutual informations for paths that respect their group structure.  \nThese are divided into separate, self contained chapters that can be read independently of one another.  \nContents  \nIntroduction 1  \n1 Signature cumulants 6  \n1. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7  \n1.1. 1 Poset of Partitions, Moments and Cumulants . . . . . . . . . . 9  \n1.2 The Lattice of (Ordered) Partitions . . . . . . . . . . . . . . . . . . . 14  \n1.2. 1 Posets, Lattices, and Möbius inversion . . . . . . . . . . . . . 14  \n1.2.2 The lattice of ordered partitions . . . . . . . . . . . . . . . . . 16  \n1.3 The Signature Cumulant and its properties ............... 21  \n1.3.1 Geometric rough paths ...................... 21  \n1.3.2 Signature moments and cumulants................ 23  \n1.3.3 Cumulants, moments, and independe","cbCaipMT6cpmCYDa","https://ap.wps.com/l/cbCaipMT6cpmCYDa","pdf",2545346,1,235,"English","en",105,"# Introduction\n# Signature cumulants\n## Poset of Partitions, Moments and Cumulants\n## Lattice of (Ordered) Partitions\n## Signature Cumulant and its properties\n## Estimating Signature cumulants\n# Kernelized cumulants\n## Moments and Cumulants\n## Kernelized Cumulants\n## Experiments\n## Conclusion","[{\"question\":\"What is the core concept behind the thesis’s approach to statistics and machine learning?\",\"answer\":\"The thesis uses signature methods, which map a path evolving in a vector space to a structured group representation. This structure provides invariances and supports algebraic analysis for statistical applications.\"},{\"question\":\"How do signature cumulants contribute to understanding independence of paths?\",\"answer\":\"The thesis introduces a new type of cumulants for signatures with distinctive combinatorial properties. These cumulants can characterize independence of paths.\"},{\"question\":\"What additional techniques connect signatures to kernel methods and non-commutative feature maps?\",\"answer\":\"It develops cumulants in reproducing kernel Hilbert spaces related to signature cumulants, and generalizes signatures to other feature maps into non-commutative algebras. It also proposes a feature map that captures filtration properties via an initial topology.\"}]","Applications of stochastic analysis and algebra to machine learning | PDF",1785807731,592,{"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},"applications-of-stochastic-analysis-and-algebra-to-machine-learning","",{"@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/applications-of-stochastic-analysis-and-algebra-to-machine-learning/121918/",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","2026-08-04",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 the core concept behind the thesis’s approach to statistics and machine learning?","Question",{"text":75,"@type":76},"The thesis uses signature methods, which map a path evolving in a vector space to a structured group representation. This structure provides invariances and supports algebraic analysis for statistical applications.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do signature cumulants contribute to understanding independence of paths?",{"text":80,"@type":76},"The thesis introduces a new type of cumulants for signatures with distinctive combinatorial properties. These cumulants can characterize independence of paths.",{"name":82,"@type":73,"acceptedAnswer":83},"What additional techniques connect signatures to kernel methods and non-commutative feature maps?",{"text":84,"@type":76},"It develops cumulants in reproducing kernel Hilbert spaces related to signature cumulants, and generalizes signatures to other feature maps into non-commutative algebras. 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