{"id":"https://openalex.org/W7166850450","doi":"https://doi.org/10.1137/25m1764955","title":"Log-PDE Methods for Rough Signature Kernels","display_name":"Log-PDE Methods for Rough Signature Kernels","publication_year":2026,"publication_date":"2026-07-01","ids":{"openalex":"https://openalex.org/W7166850450","doi":"https://doi.org/10.1137/25m1764955"},"language":"en","primary_location":{"id":"doi:10.1137/25m1764955","is_oa":false,"landing_page_url":"https://doi.org/10.1137/25m1764955","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Numerical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5055402057","display_name":"Maud Lemercier","orcid":null},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Maud Lemercier","raw_affiliation_strings":["Mathematical Institute, University of Oxford, Oxford, UK"],"raw_orcid":"https://orcid.org/0009-0000-0735-829X","affiliations":[{"raw_affiliation_string":"Mathematical Institute, University of Oxford, Oxford, UK","institution_ids":["https://openalex.org/I40120149"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5139843612","display_name":"Terry Lyons","orcid":null},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Terry Lyons","raw_affiliation_strings":["Mathematical Institute, University of Oxford, Oxford, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematical Institute, University of Oxford, Oxford, UK","institution_ids":["https://openalex.org/I40120149"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5046539767","display_name":"Cristopher Salvi","orcid":"https://orcid.org/0000-0002-4185-5908"},"institutions":[{"id":"https://openalex.org/I47508984","display_name":"Imperial College London","ror":"https://ror.org/041kmwe10","country_code":"GB","type":"education","lineage":["https://openalex.org/I47508984"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Cristopher Salvi","raw_affiliation_strings":["Department of Mathematics, Imperial College London, London, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Imperial College London, London, UK","institution_ids":["https://openalex.org/I47508984"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.87041037,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"64","issue":"4","first_page":"1095","last_page":"1125"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10775","display_name":"Generative Adversarial Networks and Image Synthesis","score":0.23829999566078186,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10775","display_name":"Generative Adversarial Networks and Image Synthesis","score":0.23829999566078186,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.227400004863739,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":0.07129999995231628,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/signature","display_name":"Signature (topology)","score":0.682699978351593},{"id":"https://openalex.org/keywords/iterated-function","display_name":"Iterated function","score":0.6392999887466431},{"id":"https://openalex.org/keywords/kernel","display_name":"Kernel (algebra)","score":0.6173999905586243},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.6164000034332275},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.4740000069141388},{"id":"https://openalex.org/keywords/path","display_name":"Path (computing)","score":0.42239999771118164},{"id":"https://openalex.org/keywords/scale","display_name":"Scale (ratio)","score":0.41929998993873596},{"id":"https://openalex.org/keywords/bounded-variation","display_name":"Bounded variation","score":0.35920000076293945}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7986000180244446},{"id":"https://openalex.org/C2779696439","wikidata":"https://www.wikidata.org/wiki/Q7512811","display_name":"Signature (topology)","level":2,"score":0.682699978351593},{"id":"https://openalex.org/C140479938","wikidata":"https://www.wikidata.org/wiki/Q5254619","display_name":"Iterated function","level":2,"score":0.6392999887466431},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.6173999905586243},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6164000034332275},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.49540001153945923},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.4740000069141388},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4277999997138977},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.42239999771118164},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.41929998993873596},{"id":"https://openalex.org/C27851653","wikidata":"https://www.wikidata.org/wiki/Q829653","display_name":"Bounded variation","level":2,"score":0.35920000076293945},{"id":"https://openalex.org/C161584116","wikidata":"https://www.wikidata.org/wiki/Q1952580","display_name":"Multivariate statistics","level":2,"score":0.33660000562667847},{"id":"https://openalex.org/C122280245","wikidata":"https://www.wikidata.org/wiki/Q620622","display_name":"Kernel method","level":3,"score":0.3188000023365021},{"id":"https://openalex.org/C122383733","wikidata":"https://www.wikidata.org/wiki/Q865920","display_name":"Approximation error","level":2,"score":0.2985000014305115},{"id":"https://openalex.org/C145242015","wikidata":"https://www.wikidata.org/wiki/Q774123","display_name":"Approximation theory","level":2,"score":0.2957000136375427},{"id":"https://openalex.org/C193386753","wikidata":"https://www.wikidata.org/wiki/Q1130396","display_name":"Approximations of \u03c0","level":2,"score":0.2955999970436096},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.29409998655319214},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.2597000002861023},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.25519999861717224},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.25060001015663147}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/25m1764955","is_oa":false,"landing_page_url":"https://doi.org/10.1137/25m1764955","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Numerical Analysis","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4500485203","display_name":"The Alan Turing Institute","funder_award_id":"EP/N510129/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G6618037410","display_name":null,"funder_award_id":"G0095","funder_id":"https://openalex.org/F4320310833","funder_display_name":"Lloyd's Register Foundation"},{"id":"https://openalex.org/G8593039737","display_name":"Unparameterised multi-modal data, high order signatures, and the mathematics of data science","funder_award_id":"EP/S026347/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320310833","display_name":"Lloyd's Register Foundation","ror":"https://ror.org/057q4mw47"},{"id":"https://openalex.org/F4320314707","display_name":"Government of the United Kingdom","ror":"https://ror.org/05wnh3t63"},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":23,"referenced_works":["https://openalex.org/W617501864","https://openalex.org/W1604692157","https://openalex.org/W2004095444","https://openalex.org/W2092946348","https://openalex.org/W2132870739","https://openalex.org/W2321579306","https://openalex.org/W3005953992","https://openalex.org/W3138784139","https://openalex.org/W3141455185","https://openalex.org/W3155745390","https://openalex.org/W3196640496","https://openalex.org/W3198653666","https://openalex.org/W3204752081","https://openalex.org/W3214779242","https://openalex.org/W4240805545","https://openalex.org/W4385487120","https://openalex.org/W4387667062","https://openalex.org/W4389390888","https://openalex.org/W4391333058","https://openalex.org/W4415795263","https://openalex.org/W4415797998","https://openalex.org/W4417226790","https://openalex.org/W7133239061"],"related_works":[],"abstract_inverted_index":{"Abstract.":[0],"Signature":[1],"kernels,":[2],"inner":[3],"products":[4],"of":[5,97,114,122,132],"path":[6],"signatures,":[7],"underpin":[8],"several":[9],"machine":[10],"learning":[11],"algorithms":[12],"for":[13,53],"multivariate":[14],"time":[15,78],"series":[16],"analysis.":[17],"For":[18],"bounded":[19],"variation":[20],"paths,":[21,57,100],"the":[22,87,95,102,123],"signature":[23,73,105,149],"kernel":[24],"was":[25],"recently":[26],"shown":[27],"to":[28,44,61,69,89,147],"solve":[29],"a":[30,65,111,135,144],"Goursat":[31],"PDE.":[32],"However,":[33],"existing":[34],"PDE":[35],"solvers":[36],"only":[37],"use":[38],"increments":[39],"as":[40,58],"input":[41,56,124],"data,":[42],"leading":[43],"first-order":[45],"approximation":[46,146],"errors.":[47],"These":[48],"approaches":[49],"become":[50],"computationally":[51],"intractable":[52],"highly":[54],"oscillatory":[55],"they":[59],"have":[60],"be":[62,108],"resolved":[63],"at":[64],"fine":[66],"enough":[67],"scale":[68],"accurately":[70],"recover":[71],"their":[72],"kernel,":[74],"resulting":[75,103],"in":[76],"significant":[77],"and":[79,92,138],"memory":[80],"complexities.":[81],"In":[82],"this":[83,130],"paper,":[84],"we":[85],"extend":[86],"analysis":[88],"rough":[90,99,104,125,148],"paths":[91],"show,":[93],"leveraging":[94],"framework":[96],"smooth":[98],"that":[101,129],"kernels":[106],"can":[107],"approximated":[109],"by":[110],"novel":[112],"system":[113,131],"PDEs":[115,133],"whose":[116],"coefficients":[117],"involve":[118],"higher-order":[119,145],"iterated":[120],"integrals":[121],"paths.":[126],"We":[127],"show":[128],"admits":[134],"unique":[136],"solution":[137],"establish":[139],"quantitative":[140],"error":[141],"bounds":[142],"yielding":[143],"kernels.":[150]},"counts_by_year":[],"updated_date":"2026-07-02T06:18:51.028212","created_date":"2026-07-02T00:00:00"}
