[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117186-en":3,"doc-seo-117186-105":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},117186,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","Randomized Signatures and Applications in Machine Learning - Bachelor Thesis","This bachelor thesis explores the intersection of rough path and signature theory with reservoir computing and machine learning, centered on the randomized signature. A randomized signature is formed by projecting the original signature representation into a lower-dimensional vector space via random projection. The thesis derives the projection maps and approximation error bounds using the Johnson–Lindenstrauss lemma. Results show that the randomized signature preserves key approximation properties while reducing computational cost, supported by numerical experiments. The work further connects controlled differential equations, rough paths, and signature theory to practical learning models, and validates randomized signature algorithms in machine learning and quantitative finance tasks.","Fakultät für  \nMathematik und Informatik  \nPascal Debus  \nRandomized Signatures and Applications in Machine Learning  \nLehrgebiet Angewandte Stochastik Bachelorarbeit  \nRandomized Signatures and Applications in  \nMachine Learning  \nBachelor Thesis  \nwritten by  \nPascal Debus  \nsupervised by Prof. Dr. Sebastian Riedel Chair of Applied Stochastics FernUniversität in Hagen  \nDecember 31, 2023  \nAbstract  \nThis thesis explores the intersection of rough path and signature theory, reservoir computing, and machine learning, focusing on the concept of the randomized signature. The randomized signature is a dimensionality-reduced path signature that is obtained by random projection from the original signature space to a lower dimensional vector space. The projections maps, as well as the error bounds of this approximation, are derived from the classical Johnson-Lindenstrauss lemma. It is shown that the randomized signature retains the favorable approximation properties of the original signature at a fraction of the computation cost, making it very attractive for applications in machine learning. The thesis provides a comprehensive theoretical foundation, touching upon controlled differential equations, rough path theory, and signature theory, as well as the connections to reservoir computing and machine learning, before delving into definitions and proofs of the randomized signature. It showcases the computational advantages and approximation capabilities of the randomized signature compared to its original counterpart. Numerical experiments validate the efficacy and efficiency of randomized signature algorithms across various tasks, illustrating their utility in machine learning and quantitative finance.  \nContents  \nList of Figures 2  \n1 Introduction 3  \n1.1 Motivation ........................................ 3  \n1.2 Objective and Problem Definition ............................ 3  \n1.3 Machine Learning and Reservoir Computing ...................... 4  \n1.4 Rough Path and Signature Theory ........................... 4  \n1.5 Structure of the Thesis .................................. 4  \n2 Controlled Differential Equations 5  \n2.1 Basic Definitions and Mathematical Setting ...................... 5  \n2.2 Picard Iteration ...................................... 6  \n2.3 Iterated Integrals and Signature ............................. 7  \n2.4 Relation to Rough Path Theory ............................. 7  \n3 Signature Theory 12  \n3.1 Basic Definition ...................................... 12  \n3.2 Properties ......................................... 14  \n3.3 Universal Approximation Theorem ........................... 19  \n4 Signatures and Machine Learning 21  \n4.1 Relevance for Machine Learning ............................. 21  \n4.2 Signatures from Discrete Data .............................. 22  \n4.3 Overview on Signature Methods in Machine Learning ................ 24  \n4.4 Reservoir Computing View on Signatures ....................... 27  \n5 Randomized Signature 31  \n5.1 Motivation ........................................ 31  \n5.2 The Johnson-Lindenstrauss Lemma ........................... 32  \n5.3 Approximation Theorems ................................ 37  \n5.4 Practical Implementation ................................ 40  \n6 Numerical Experiments 42  \n6.1 Approximation of the truncated Signature ....................... 42  \n6.2 Randomized Signature Autoencoder .......................... 43  \n6.3 Hurst Parameter Estimation ............................... 47  \n7 Conclusion and Outlook 48  \nA Additional Theorems and Results 50  \nList of Figures  \n2.1 Illustration of the Lévy area of (X1t, X2t) ........................ 8  \n3.1 Illustration of tree-like paths .............................. 18  \n4.1 An overview of different interpolation methods..................... 23  \n4.2 Plots of the lead-lag embedding. In (b) it becomes apparent that piecewise linear and rectilinear interpolation produce the same result.................. 24  \n4.3 Plots of th","cbCaitTwdsLmD3yt","https://ap.wps.com/l/cbCaitTwdsLmD3yt","pdf",4701795,1,58,"English","en",105,"# Introduction\n## Motivation\n## Objective and Problem Definition\n## Machine Learning and Reservoir Computing\n## Rough Path and Signature Theory\n## Structure of the Thesis\n# Controlled Differential Equations\n## Basic Definitions and Mathematical Setting\n## Picard Iteration\n## Iterated Integrals and Signature\n## Relation to Rough Path Theory\n# Signature Theory\n## Basic Definition\n## Properties\n## Universal Approximation Theorem\n# Signatures and Machine Learning\n## Relevance for Machine Learning\n## Signatures from Discrete Data\n## Overview on Signature Methods in Machine Learning\n## Reservoir Computing View on Signatures\n# Randomized Signature\n## Motivation\n## The Johnson-Lindenstrauss Lemma\n## Approximation Theorems\n## Practical Implementation\n# Numerical Experiments\n## Approximation of the truncated Signature\n## Randomized Signature Autoencoder\n## Hurst Parameter Estimation\n# Conclusion and Outlook\n## Additional Theorems and Results","[{\"question\":\"What is the randomized signature in this thesis?\",\"answer\":\"It is a dimensionality-reduced path signature obtained by applying random projection from the original signature space to a lower-dimensional vector space.\"},{\"question\":\"How are approximation error bounds for the randomized signature derived?\",\"answer\":\"The thesis derives both projection maps and error bounds using the classical Johnson–Lindenstrauss lemma.\"},{\"question\":\"What benefits do randomized signatures provide for machine learning applications?\",\"answer\":\"They retain favorable approximation properties of the original signature while substantially reducing computation cost, which is demonstrated through computational advantages and numerical experiments across tasks.\"}]","Randomized Signatures and Applications in Machine Learning - Bachelor Thesis | PDF",1785674277,146,{"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},"randomized-signatures-and-applications-in-machine-learning-bachelor-thesis","",{"@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/randomized-signatures-and-applications-in-machine-learning-bachelor-thesis/117186/",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 the randomized signature in this thesis?","Question",{"text":75,"@type":76},"It is a dimensionality-reduced path signature obtained by applying random projection from the original signature space to a lower-dimensional vector space.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are approximation error bounds for the randomized signature derived?",{"text":80,"@type":76},"The thesis derives both projection maps and error bounds using the classical Johnson–Lindenstrauss lemma.",{"name":82,"@type":73,"acceptedAnswer":83},"What benefits do randomized signatures provide for machine learning applications?",{"text":84,"@type":76},"They retain favorable approximation properties of the original signature while substantially reducing computation cost, which is demonstrated through computational advantages and numerical experiments across tasks.","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"]