[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81588-en":3,"doc-seo-81588-105":30,"detail-sidebar-cat-0-en-105":95},{"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":13,"seo_description":14,"update_tm":28,"read_time":29},81588,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Principles of Lipschitz continuity in neural networks","Deep learning delivers strong performance across many domains, yet robustness to small input perturbations and generalization to out-of-distribution data remain difficult. The thesis develops a principled understanding of how Lipschitz continuity governs robustness and generalization, focusing on worst-case output sensitivity. It argues for reliable, resilient, and trustworthy systems, especially in safety-critical contexts. Two complementary views are used: training dynamics over time and feature-wise modulation, including frequency signal propagation; three research questions structure the inquiry.","arXiv :2602 .04078v2 [ cs .LG] 10 Jul 2026  \nPrinciples of Lipschitz continuity in neural networks  \nby  \nRóisín Luo  \nA thesis presented in fulfillment of the requirements for the degree of  \nDoctor of Philosophy  \nat  \nOllscoil na Gaillimhe  \nUniversity of Galway  \nIreland  \nand  \nTaighde Éireann – Research Ireland Irish National Centre for Research Training in AI (CRT-AI)  \nDoctoral Supervisors: Dr. Colm O’Riordan  \nDr. James McDermott  \nScoil na Ríomheolaíochta  \nSchool of Computer Science  \nSeptember, 2025 (v3)  \nCite as:  \nRóisín Luo. Principles of Lipschitz continuity in neural networks. Doctoral Thesis, Research Ireland – Centre for Research Training in Artificial Intelligence (CRT-AI), University of Galway, Ireland, September, 2025 (v3) .  \nTo my beloved grandparents.  \nVoyagers: Edge of the Bubble (NASA/JPL-Caltech) 1. Two tiny spacecraft — the Voyagers—manifest our spirit of exploration to an extraordinary level, a spirit that defines us as human beings. They have been sailing outward into vast interstellar space, reaching the frontier of our solar system, where the star winds meet the solar winds from our world, forming a vast bubble. Within this bubble lies every story, every life, every dream we have ever known—yet the Voyagers press onward, carrying our boldness and wonder into the vast, eternal, and spectacular sea between the stars, destined for an eternal journey, a testament to our instinct to go beyond the horizon where no human has gone before.  \nI am only stardust—learning to think, to chase dreams.  \n1 Image use follows the NASA Images and Media Usage Guidelines.  \n“Nunquam praescriptos transibunt sidera fines—Nothing exceeds the limits of the stars.”  \nJules Henri Poincaré, 1890  \nR E S EA R C H F U N D I N G  \nThis Ph.D. training program is a part of the Centre for Research Training in Artificial Intelligence (CRT-AI), funded by Taighde Éireann – Research Ireland under Grant Number 18/CRT/6223 .  \nA B ST RA CT  \nDeep learning has achieved remarkable success across a wide range of domains, significantly expanding the frontiers of what is achievable in artificial intelligence. Yet, despite these advances, critical challenges remain—most notably, ensuring robustness to small input perturbations and generalization to out-of-distribution data. These critical challenges underscore the need to understand the underlying fundamental principles that govern robustness and generalization. This understanding is indispensable for establishing deep learning systems that are: reliable—performing consistently under expected conditions on in-distribution data; resilient—capable of recovering from unexpected conditions such as noise or adversarial attacks; and trustworthy—behaving transparently, ethically, in alignment with intended use, and technically robust, particularly in safety-critical applications.  \nAmong the theoretical tools available, Lipschitz continuity plays a pivotal role in governing the fundamental properties of neural networks related to robustness and generalization. It quantifies the worst-case sensitivity of network’s outputs to small input perturbations. While its importance is widely acknowledged, prior research has predominantly focused on empirical regularization approaches based on Lipschitz constraints, leaving the underlying principles less explored. This thesis seeks to advance a principled understanding of the principles of Lipschitz continuity in neural networks within the paradigm of machine learning, examined from two complementary perspectives: an internal perspective—focusing on the temporal evolution of Lipschitz continuity in neural networks during training (i.e., training dynamics); and an external perspective—investigating how Lipschitz continuity modulates the behavior of neural networks with respect to features in the input data, particularly its role in governing frequency signal propagation (i.e., modulation of frequency signal propagation) .  \nGuided by these perspectives, the","cbCaibKVoxdPWtrO","https://ap.wps.com/l/cbCaibKVoxdPWtrO","pdf",12264633,3,1,289,"English","en",105,"# Abstract\n## Training dynamics and feature-wise modulation\n## Research questions (RQ1–RQ3)","[{\"question\":\"What problem does the thesis address in deep learning?\",\"answer\":\"It targets two key challenges: robustness to small input perturbations and generalization to out-of-distribution data.\"},{\"question\":\"How does Lipschitz continuity relate to neural network robustness?\",\"answer\":\"Lipschitz continuity quantifies the worst-case sensitivity of a network’s outputs to small input perturbations.\"},{\"question\":\"What two perspectives does the thesis use to study Lipschitz continuity?\",\"answer\":\"It studies training dynamics over time during training (internal perspective) and how Lipschitz continuity modulates neural network behavior with respect to input features, especially frequency signal propagation (external perspective).\"},{\"question\":\"What are the three main research questions?\",\"answer\":\"RQ1 asks the state of knowledge; RQ2 asks how Lipschitz continuity evolves during training; RQ3 asks how Lipschitz continuity modulates frequency signal propagation.\"}]",1784174536,728,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":90,"head_meta":92,"extra_data":94,"updated_unix":28},"principles-of-lipschitz-continuity-in-neural-networks","",{"@graph":36,"@context":89},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/principles-of-lipschitz-continuity-in-neural-networks/81588/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81,85],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the thesis address in deep learning?","Question",{"text":75,"@type":76},"It targets two key challenges: robustness to small input perturbations and generalization to out-of-distribution data.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does Lipschitz continuity relate to neural network robustness?",{"text":80,"@type":76},"Lipschitz continuity quantifies the worst-case sensitivity of a network’s outputs to small input perturbations.",{"name":82,"@type":73,"acceptedAnswer":83},"What two perspectives does the thesis use to study Lipschitz continuity?",{"text":84,"@type":76},"It studies training dynamics over time during training (internal perspective) and how Lipschitz continuity modulates neural network behavior with respect to input features, especially frequency signal propagation (external perspective).",{"name":86,"@type":73,"acceptedAnswer":87},"What are the three main research questions?",{"text":88,"@type":76},"RQ1 asks the state of knowledge; RQ2 asks how Lipschitz continuity evolves during training; RQ3 asks how Lipschitz continuity modulates frequency signal propagation.","https://schema.org",{"og:url":51,"og:type":91,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":93,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":96},[97,101,105,109,114,119,124,127,132,135,139],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":106,"show_sort_weight":107,"slug":108},"Exam",70,"exam",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},5,"Comic",60,"comic",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},6,"Technology",50,"technology",{"id":120,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":122,"slug":123},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":125,"slug":126},30,"research-report",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":130,"slug":131},9,"Religion & Spirituality",20,"religion-spirituality",{"id":130,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":130,"slug":134},"World Cup","world-cup",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":136,"slug":138},10,"Lifestyle","lifestyle",{"id":140,"doc_module":4,"doc_module_name":46,"category_name":141,"show_sort_weight":110,"slug":142},19,"General","general"]