{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/127381"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/127381","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Nonlinear and geometric control methods in deep learning theory","abstract":"Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2026-12-01","abstract_html":"Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2026-12-01","abstract_has_math":false,"creators":["Hanson, Joshua McKinley"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Electrical & Computer Engr","degree_department":null,"school":null,"contributors":["Raginsky, Maxim","Baryshnikov, Yuliy","Belabbas, Mohamed Ali","Liberzon, Daniel"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-12-03","date_published":"2024-12-03","updated_at":"2026-07-22T22:25:04Z","subjects":["Neural Networks","Statistical Learning Theory","Rademacher Complexity","Deep Learning","Nonlinear Control","Geometric Control"],"languages":["en","eng"],"rights":["Copyright 2024 Joshua Hanson"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/127381","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Raginsky, Maxim","Baryshnikov, Yuliy","Belabbas, Mohamed Ali","Liberzon, Daniel"]},{"key":"dc:creator","label":"Author","values":["Hanson, Joshua McKinley"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2024-12-03","2024-12"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Electrical & Computer Engr"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Neural Networks","Statistical Learning Theory","Rademacher Complexity","Deep Learning","Nonlinear Control","Geometric Control"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2024 Joshua Hanson"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/127381"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2026-12-01","The student, Joshua Hanson, accepted the attached license on 2024-12-02 at 13:55.","The student, Joshua Hanson, submitted this Dissertation for approval on 2024-12-02 at 14:53.","This Dissertation was approved for publication on 2024-12-03 at 14:21.","DSpace SAF Submission Ingestion Package generated from Vireo submission #21449 on 2025-03-28 at 14:43:36","The practical success and popularity of deep neural networks for solving modern machine learning problems motivates us to develop a rigorous theoretical understanding of how depth affects model expressivity and generalizability. Interpreting deep learning architectures as control systems unlocks versatile tools from nonlinear systems theory to study these models and associated statistical learning problems. In this dissertation, we present three main technical works taking advantage of this perspective, which are summarized as follows: The first work describes an encoder-decoder architecture for learning immersed submanifolds from data inspired by the structure of the group action in Sussmann’s orbit theorem, which is built from composing forward- and backward-in-time flow maps. We proceed to develop generalization bounds for this model class and apply these results to a handful of illustrative examples. The second work investigates a technique for proving generalization bounds for neural ordinary differential equations based on transforming the model into an equivalent infinite-dimensional kernel machine through the use of the Chen–Fliess expansion, which expresses the model output as an infinite series in terms of signature integrals of the control and iterated Lie derivatives of the output map. This technique differs from strategies based on bounding the covering number of the model class by propagating a parameter perturbation through the flow map, and instead takes advantage of standard tools applicable to kernel machines. The third work focuses on deriving quantitative approximation error bounds for neural ordinary differential equations having at most quadratic nonlinearities in the dynamics. The simple dynamics of this model form demonstrates how expressivity can be derived primarily from iteratively composing many basic elementary operations, versus from the complexity of those elementary operations themselves. These results contribute to our understanding of what depth imparts to the capabilities of deep learning architectures."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Nonlinear and geometric control methods in deep learning theory"]}]}],"canonical_facts":{"dc:contributor":["Raginsky, Maxim","Baryshnikov, Yuliy","Belabbas, Mohamed Ali","Liberzon, Daniel"],"dc:creator":["Hanson, Joshua McKinley"],"dc:date":["2024-12-03","2024-12"],"dc:description":["Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2026-12-01","The student, Joshua Hanson, accepted the attached license on 2024-12-02 at 13:55.","The student, Joshua Hanson, submitted this Dissertation for approval on 2024-12-02 at 14:53.","This Dissertation was approved for publication on 2024-12-03 at 14:21.","DSpace SAF Submission Ingestion Package generated from Vireo submission #21449 on 2025-03-28 at 14:43:36","The practical success and popularity of deep neural networks for solving modern machine learning problems motivates us to develop a rigorous theoretical understanding of how depth affects model expressivity and generalizability. Interpreting deep learning architectures as control systems unlocks versatile tools from nonlinear systems theory to study these models and associated statistical learning problems. In this dissertation, we present three main technical works taking advantage of this perspective, which are summarized as follows: The first work describes an encoder-decoder architecture for learning immersed submanifolds from data inspired by the structure of the group action in Sussmann’s orbit theorem, which is built from composing forward- and backward-in-time flow maps. We proceed to develop generalization bounds for this model class and apply these results to a handful of illustrative examples. The second work investigates a technique for proving generalization bounds for neural ordinary differential equations based on transforming the model into an equivalent infinite-dimensional kernel machine through the use of the Chen–Fliess expansion, which expresses the model output as an infinite series in terms of signature integrals of the control and iterated Lie derivatives of the output map. This technique differs from strategies based on bounding the covering number of the model class by propagating a parameter perturbation through the flow map, and instead takes advantage of standard tools applicable to kernel machines. The third work focuses on deriving quantitative approximation error bounds for neural ordinary differential equations having at most quadratic nonlinearities in the dynamics. The simple dynamics of this model form demonstrates how expressivity can be derived primarily from iteratively composing many basic elementary operations, versus from the complexity of those elementary operations themselves. These results contribute to our understanding of what depth imparts to the capabilities of deep learning architectures."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/127381"],"dc:language":["en","eng"],"dc:rights":["Copyright 2024 Joshua Hanson"],"dc:subject":["Neural Networks","Statistical Learning Theory","Rademacher Complexity","Deep Learning","Nonlinear Control","Geometric Control"],"dc:title":["Nonlinear and geometric control methods in deep learning theory"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Electrical & Computer Engr"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:04Z"}