[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-123168-en":3,"doc-seo-123168-105":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":20,"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},123168,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Non-Euclidean Monotone Operator Theory and Applications","While monotone operator theory is often developed on Hilbert spaces, many machine learning and optimization problems naturally live in finite-dimensional vector spaces equipped with non-Euclidean norms. This paper generalizes monotone operator theory to such settings using weak pairings and logarithmic norms. It proves analogous properties for the resolvent and reflected resolvent of non-Euclidean monotone mappings, and establishes convergence of classical iterative and splitting methods. Applications include equilibrium computation and tighter Lipschitz constant bounds for recurrent neural networks via forward-backward splitting.","POLITECNICO DI TORINO Repository ISTITUZIONALE  \nNon-Euclidean Monotone Operator Theory and Applications  \nOriginal  \nNon-Euclidean Monotone Operator Theory and Applications / Davydov, Alexander; Jafarpour, Saber; Proskurnikov, Anton V. ; Bullo, Francesco. -In: JOURNAL OF MACHINE LEARNING RESEARCH. -ISSN 1532-4435. -2024:25(2024), pp. 1-33.  \nAvailability:  \nThis version is available at: 11583/2994482 since: 2024-11-16T22:09:44Z  \nPublisher: MIT Press  \nPublished DOI:  \nTerms of use:  \nThis article is made available under terms and conditions as specified in the corresponding bibliographic description in the repository  \nPublisher copyright  \n(Article begins on next page)  \n27 March 2025  \nNon-Euclidean Monotone Operator Theory and Applications  \nAlexander Davydov 􀀃 [davydov@ucsb.edu](davydov@ucsb.edu)  \nCenter for Control, Dynamical Systems, and Computation University of California, Santa Barbara  \nSanta Barbara, CA 93106-5070, USA  \nSaber Jafarpour􀀃 [saber.jafarpour@colorado.edu](saber.jafarpour@colorado.edu)  \nDepartment of Electrical, Computer, and Energy Engineering University of Colorado, Boulder  \nBoulder, CO 80309-0020, USA  \nAnton V. Proskurnikov [anton.p.1982@ieee.org](anton.p.1982@ieee.org)  \nDepartment of Electronics and Telecommunications Politecnico di Torino  \nTurin, Italy  \nFrancesco Bullo [bullo@ucsb.edu](bullo@ucsb.edu)  \nCenter for Control, Dynamical Systems, and Computation University of California, Santa Barbara  \nSanta Barbara, CA 93106-5070, USA  \nEditor: Silvia Villa  \nAbstract  \nWhile monotone operator theory is often studied on Hilbert spaces, many interesting problems in machine learning and optimization arise naturally in 􀀌nite-dimensional vector spaces endowed with non-Euclidean norms, such as diagonally-weighted ` 1 or `1 norms. This paper provides a natural generalization of monotone operator theory to 􀀌nitedimensional non-Euclidean spaces. The key tools are weak pairings and logarithmic norms.  \nWe show that the resolvent and re􀀍ected resolvent operators of non-Euclidean monotone mappings exhibit similar properties to their counterparts in Hilbert spaces. Furthermore, classical iterative methods and splitting methods for 􀀌nding zeros of monotone operators are shown to converge in the non-Euclidean case. We apply our theory to equilibrium computation and Lipschitz constant estimation of recurrent neural networks, obtaining novel iterations and tighter upper bounds via forward-backward splitting.  \nKeywords: non-Euclidean norms, monotone operator theory, 􀀌xed point equations, nonexpansive maps  \n1. Introduction  \nProblem description and motivation: Monotone operator theory is a fertile 􀀌eld of nonlinear functional analysis that extends the notion of monotone functions on R to mappings on Hilbert spaces. Monotone operator methods are widely used to solve problems in machine learning (Combettes and Pesquet, 2020b; Winston and Kolter, 2020), data science (Combettes and Pesquet, 2021), optimization and control (Simonetto, 2017; Bernstein et al. ,  \n􀀃 . The 􀀌rst two authors contributed equally  \n􀀍c2024 Alexander Davydov, Saber Jafarpour, Anton V. Proskurnikov, and Francesco Bullo.  \nLicense: CC-BY 4.0, see [https://creativecommons.org/licenses/by/4.0/](https://creativecommons.org/licenses/by/4.0/. Attribution)[. Attribution](https://creativecommons.org/licenses/by/4.0/. Attribution) requirements are provided  \nat [http://jmlr.org/papers/v25/23-0805.html](http://jmlr.org/papers/v25/23-0805.html).  \nDavydov, Jafarpour, Proskurnikov, and Bullo  \n2019), game theory (Pavel, 2020), and systems analysis (Cha􀀋ey et al., 2021) . A crucial part of this theory is the design of algorithms for computing zeros of monotone operators. This problem is central in convex optimization since (i) the subdi􀀋erential of any convex function is monotone and (ii) minimizing a convex function is equivalent to 􀀌nding a zero of its subdi􀀋erential. To this end, there has been extensive research in the last decade in applying monoton","cbCaie5DylQ2iaqs","https://ap.wps.com/l/cbCaie5DylQ2iaqs","pdf",685877,1,34,"English","en",105,"# Introduction\n## Problem description and motivation\n## Literature review\n# Abstract and main contributions\n## Weak pairings and logarithmic norms\n## Resolvent and reflected resolvent properties\n## Convergence of iterative and splitting methods\n# Applications\n## Equilibrium computation\n## Lipschitz constant estimation for recurrent neural networks\n# Keywords","[{\"question\":\"What problem does non-Euclidean monotone operator theory address?\",\"answer\":\"It extends monotone operator methods from Hilbert spaces to finite-dimensional spaces endowed with non-Euclidean norms, which arise naturally in machine learning and optimization.\"},{\"question\":\"Which key mathematical tools are introduced in the paper?\",\"answer\":\"The paper relies on weak pairings and logarithmic norms to build the generalized theory in non-Euclidean settings.\"},{\"question\":\"Do iterative and splitting algorithms still converge in the non-Euclidean case?\",\"answer\":\"Yes. Classical iterative methods and splitting methods for finding zeros of monotone operators are shown to converge for the non-Euclidean framework.\"}]","Non-Euclidean Monotone Operator Theory and Applications | PDF",1785815012,86,{"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},"non-euclidean-monotone-operator-theory-and-applications","",{"@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/non-euclidean-monotone-operator-theory-and-applications/123168/",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":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does non-Euclidean monotone operator theory address?","Question",{"text":75,"@type":76},"It extends monotone operator methods from Hilbert spaces to finite-dimensional spaces endowed with non-Euclidean norms, which arise naturally in machine learning and optimization.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which key mathematical tools are introduced in the paper?",{"text":80,"@type":76},"The paper relies on weak pairings and logarithmic norms to build the generalized theory in non-Euclidean settings.",{"name":82,"@type":73,"acceptedAnswer":83},"Do iterative and splitting algorithms still converge in the non-Euclidean case?",{"text":84,"@type":76},"Yes. Classical iterative methods and splitting methods for finding zeros of monotone operators are shown to converge for the non-Euclidean framework.","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"]