[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-120083-en":3,"doc-seo-120083-105":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},120083,2336464648322,"Aria","https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488",6,"Technology","Lecture Notes on Rough Paths and Applications to Machine Learning","These lecture notes develop a mathematical description of paths using iterated integrals and explain how the induced signature theory supports machine-learning and data-science tasks for streamed data. The notes connect the Chen-Fliess perspective on signatures with tree-like equivalence, then build toward the rough paths framework that enables pathwise, robust solutions for irregular differential equations. They further present signature transforms, signature kernels (including universal and characteristic cases), and learning formulations linking kernels to hypothesis testing and regression. The course extends the theory to rough differential equations and integrates numerical solvers with deep learning via neural ODEs, CDEs, RDEs, and neural SDE generative modeling, supported by exercises and learning objectives.","arXiv :2404 .06583v 1 [ cs .LG] 9 Apr 2024  \nLecture Notes on Rough Paths and Applications to  \nMachine Learning  \nThomas Cass and Cristopher Salvi  \nSpring 2024  \n2  \nContents  \nIntroduction 5  \n1 The signature transform 11  \n1.1 Controlled differential equations ...................... 11  \n1.1.1 Coordinate iterated integrals ................... 13  \n1.1.2 Algebras of tensors ......................... 15  \n1.1.3 Functions on the range of the signature transform ....... 18  \n1.2 Analytical properties ............................ 19  \n1.2.1 Sampling invariance ........................ 19  \n1.2.2 Fast decay in the magnitude of coefficients ............ 22  \n1.2.3 Uniqueness ............................. 22  \n1.3 Algebraic properties ............................ 24  \n1.3.1 Chen’s relation ........................... 24  \n1.3.2 The signature as a controlled differential equation ....... 27  \n1.3.3 The shuffle identity ......................... 29  \n1.4 Unparameterised paths ........................... 32  \n1.4.1 Functions on unparameterised paths ............... 33  \n1.4.2 Topology on unparameterised paths ............... 35  \n1.4.3 Compact sets in the product topology .............. 37  \n4 CONTENTS  \n1.4.4 Probability on unparameterised paths .............. 41  \n1.5 Learning with the signature transform .................. 43  \n1.6 Exercises .................................. 46  \n2 Signature kernels 53  \n2.1 From signatures to signature kernels ................... 54  \n2.1.1 Weighted inner products on tensor algebras ........... 54  \n2.1.2 Weighted signature kernels .................... 54  \n2.1.3 Reproducing signature kernel Hilbert spaces ........... 57  \n2.2 Universal and characteristic signature kernels .............. 61  \n2.2.1 Universality of a kernel ...................... 61  \n2.2.2 Characteristic kernels ....................... 64  \n2.2.3 Hypothesis testing ......................... 68  \n2.2.4 Distribution regression ....................... 69  \n2.3 Computing signature kernels ....................... 70  \n2.3.1 Truncated signature kernels .................... 71  \n2.3.2 Finite difference schemes for the signature kernel PDE ..... 72  \n2.3.3 Weighted signature kernels as averaged PDE solutions ..... 72  \n2.4 Learning with signature kernels ...................... 74  \n2.4.1 Representer theorems ....................... 74  \n2.5 Exercises .................................. 76  \n3 Rough paths and deep learning 81  \n3.1 Rough differential equations (RDEs) ................... 81  \n3.1.1 Rough paths ............................ 83  \n3.1.2 RDE solutions as limits of approximations ............ 85  \n3.1.3 Stratonovich SDEs as RDEs .................... 87  \nCONTENTS 5  \n3.2 Numerical RDE solvers .......................... 88  \n3.2.1 Euler scheme ............................ 88  \n3.2.2 The log-ODE method ....................... 89  \n3.3 Differential equations and deep learning ................. 92  \n3.3.1 Review of Deep Learning ..................... 92  \n3.3.2 Neural ODEs ............................ 94  \n3.3.3 Neural CDEs ............................ 96  \n3.3.4 Neural RDEs ............................ 100  \n3.3.5 Neural SDEs as generative models for time series ........ 101  \n3.4 Exercises .................................. 103  \n6 CONTENTS  \nIntroduction  \nThese lecture notes introduce a mathematical concept, that of describing a path through its iterated integrals, and explain how the resulting theory can be used in data science and machine learning tasks involving streamed data. The study of iterated integrals has deep-seated roots in both mathematics and physics, with the early mathematical work stretching back at least to a series of papers by K-T. Chen, see e.g. [12, 13, 14] . Chen’s contribution spanned various domains, with a central emphasis being on homotopy theory and the use of path space integration to illuminate the interplay between topology and analysis 1 . Without providing an exhaustive historical narrative, rela","cbCairLHqOCzGv8F","https://ap.wps.com/l/cbCairLHqOCzGv8F","pdf",3420920,1,111,"English","en",105,"# Introduction\n# The signature transform\n## Controlled differential equations\n## Analytical properties\n## Algebraic properties\n## Unparameterised paths\n## Learning with the signature transform\n# Signature kernels\n## From signatures to signature kernels\n## Universal and characteristic signature kernels\n## Computing signature kernels\n## Learning with signature kernels\n# Rough paths and deep learning\n## Rough differential equations (RDEs)\n## Numerical RDE solvers\n## Differential equations and deep learning","[{\"question\":\"What does the signature of a path represent in these notes?\",\"answer\":\"A path is described through its iterated integrals, forming the signature. The theory shows that signatures determine paths up to tree-like equivalence for continuous paths of finite 1-variation.\"},{\"question\":\"How do rough paths make differential equations well-posed for irregular inputs?\",\"answer\":\"Rough paths enhance the driving signal using iterated-integral terms so that differential equations of the form dyt = V(yt) dxt can be assigned a robust, pathwise notion of solution even when xt is highly irregular.\"},{\"question\":\"How are signature kernels used in machine learning tasks?\",\"answer\":\"Signature kernels convert signatures into kernel functions, enabling learning in kernel methods. The notes discuss universal/characteristic kernels for learning and connect them to applications such as hypothesis testing and distribution regression.\"}]","Lecture Notes on Rough Paths and Applications to Machine Learning | PDF",1785728063,280,{"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},"lecture-notes-on-rough-paths-and-applications-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/technology/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/lecture-notes-on-rough-paths-and-applications-to-machine-learning/120083/",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":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does the signature of a path represent in these notes?","Question",{"text":75,"@type":76},"A path is described through its iterated integrals, forming the signature. The theory shows that signatures determine paths up to tree-like equivalence for continuous paths of finite 1-variation.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do rough paths make differential equations well-posed for irregular inputs?",{"text":80,"@type":76},"Rough paths enhance the driving signal using iterated-integral terms so that differential equations of the form dyt = V(yt) dxt can be assigned a robust, pathwise notion of solution even when xt is highly irregular.",{"name":82,"@type":73,"acceptedAnswer":83},"How are signature kernels used in machine learning tasks?",{"text":84,"@type":76},"Signature kernels convert signatures into kernel functions, enabling learning in kernel methods. The notes discuss universal/characteristic kernels for learning and connect them to applications such as hypothesis testing and distribution regression.","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},[93,97,101,105,110,113,118,123,128,131,135],{"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":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":111,"slug":112},50,"technology",{"id":114,"doc_module":4,"doc_module_name":46,"category_name":115,"show_sort_weight":116,"slug":117},7,"Healthcare",40,"healthcare",{"id":119,"doc_module":4,"doc_module_name":46,"category_name":120,"show_sort_weight":121,"slug":122},8,"Research & Report",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"]