[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-125887-en":3,"doc-seo-125887-105":31,"detail-sidebar-cat-0-en-105":93},{"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},125887,1099523885336,"Violet","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","The Signature Transform in Numerics and Machine Learning - Dissertation","This dissertation studies the signature transform from the viewpoint of applied and numerical mathematics. It introduces the signature as a map from continuous paths of bounded variation to ordered tensor algebras, then develops approximation results and computational considerations with explicit examples. Key properties are highlighted for non-linear approximation, dimension reduction, and probabilistic extensions. The work then uses the theory to design numerical experiments for data science and machine learning, including time series classification, clustering, correlation detection, and artificial sample generation.","The Signature Transform in Numerics and Machine Learning  \nDissertation  \nzur  \nErlangung des Doktorgrades (Dr. rer. nat.) der  \nMathematisch-Naturwissenschaftlichen Fakultät  \nder  \nRheinischen Friedrich-Wilhelms-Universität Bonn  \nvorgelegt von  \nBiagio Paparella  \naus  \nTerlizzi (Italien)  \nBonn 2024  \nAngefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn  \nBetreuer: Prof. Dr. Michael Griebel  \nGutachter: Prof. Dr. Jochen Garcke  \nTag der Promotion: 19.04.2024  \nErscheinungsjahr: 2024  \nAbstract  \nIn this work we study the signature transform from the viewpoint of applied and numerical mathematics.  \nThe theoretical background is established in the 􀀌rst part, where the signature is de􀀌ned as a map going from continuous paths of bounded variations to ordered tensor algebras. Approximation theorems and computational considerations are clari􀀌ed, together with explicit and well commented examples. Only selected essential properties are pointed out, useful for non-linear approximation of functionals, dimension reduction and extension to the probabilistic setting.  \nIn the second part we use all the previously introduced theory to design numerical experiments of interest in data science and machine learning, targeting problems like time series classi􀀌cation, clustering, correlation detection and generation of arti􀀌cial samples. A small section on agents classi􀀌cation for reinforcement learning is also included.  \nFinally, the reader is given a list of possible connections to other areas of mathematics like PDE, kernel theory, jump processes and even algebraic geometry. We did our best to keep the exposition clear and compact.  \nContents  \nI Foundation 9  \n1 Background 10  \n1.1 Tensor algebras ............................ 10  \n1.1.1 Tensor products: algebraic de􀀌nition ............ 10  \n1.1.2 Connection to words and alphabets ............. 11  \n1.1.3 Connection to non-commutative polynomials ....... 11  \n1.1.4 On the tensor notation .................... 12  \n1.1.5 Vectorization, P-notation and scalar products ....... 12  \n1.1.6 The tensor algebras T1 (E), TN (E) ............ 13  \n1.1.7 The Hilbert space T (E) ................... 14  \n1.1.8 Tensors in computer science ................. 15  \n1.2 Elementary path integrals ...................... 16  \n1.2.1 Path integrals for smooth curves .............. 16  \n1.2.2 Invariances .......................... 16  \n1.2.3 Bounded variation functions ................. 18  \n1.2.4 Linear piecewise approximation ............... 22  \n1.2.5 Riemann-Stieltjes integrals ................. 24  \n1.2.6 Recap of the section ..................... 26  \n1.2.7 An additional remark on generic p variation ........ 26  \n2 The signature transform: de􀀌nition 27  \n2.1 De􀀌ning the signature transform .................. 27  \n2.1.1 Motivation .......................... 27  \n2.1.2 The signature coe􀀎cients .................. 28  \n2.1.3 The signature transform ................... 29  \n2.1.4 Notation in other papers ................... 30  \n2.1.5 The signature truncation .................. 31  \n2.1.6 The signature decay ..................... 32  \n2.1.7 Practical conclusion ..................... 33  \n2.2 Computing the signature transform ................. 33  \n2.2.1 Working with the segment on [0 ; 1] ............. 33  \n2.2.2 The factorization property .................. 34  \n2.2.3 Working on the canonical interval ............. 35  \n2.2.4 All paths can start from the origin ............. 37  \n2.2.5 Recipe for the signature computation ........... 37  \n2.2.6 Studying a one dimensional case .............. 37  \n2.2.7 The Signatory library .................... 38  \n2.2.8 A small and complete numerical example ......... 40  \n2.2.9 Monte Carlo and other integration strategies ....... 42  \n3 The signature transform: key properties 43  \n3.1 Four properties of relevance ..................... 43  \n3.1.1 Comments about surjectivity ................ 43  \n3.1.2","cbCaike3KQugRWhi","https://ap.wps.com/l/cbCaike3KQugRWhi","pdf",1733981,7,1,107,"English","en",105,"# Contents\n## Foundation\n### Background\n## The signature transform: deffinition\n## The signature transform: key properties\n## Applications\n### Clustering and visualization\n## Approximating nonlinear functionals","[{\"question\":\"How is the signature transform defined in this work?\",\"answer\":\"The signature is defined as a map from continuous paths of bounded variation into ordered tensor algebras.\"},{\"question\":\"What approximation and computational aspects are covered?\",\"answer\":\"The dissertation clarifies approximation theorems and computational considerations, including practical guidance and explicit numerical examples.\"},{\"question\":\"Which machine learning tasks are targeted by the numerical experiments?\",\"answer\":\"Experiments target problems such as time series classification, clustering, correlation detection, and generation of artificial samples.\"}]","The Signature Transform in Numerics and Machine Learning - 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