{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/132799"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/132799","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Scalable second-order Riemannian optimization for K-means clustering","abstract":"Clustering is a fundamental problem in unsupervised learning. The classical K-means formulation for clustering is a worst-case NP-hard discrete optimization problem. Despite being NP-hard, the SDP relaxation of the discrete formulation is guaranteed to recover the true cluster whenever it is statistically solvable. In this thesis, we propose to solve the relaxed K-means problem as an unconstrained optimization problem on a smooth manifold. The proposed manifold can be parametrized by a product manifold with simple structures, allowing the application of second-order Riemannian algorithms. We show how to efficiently implement the cubic-regularized Riemannian Newton method by exploiting the structure of the Hessian. Numerical results show that our proposed algorithm converges faster while achieving similar accuracy compared with existing methods.","abstract_html":"Clustering is a fundamental problem in unsupervised learning. The classical K-means formulation for clustering is a worst-case NP-hard discrete optimization problem. Despite being NP-hard, the SDP relaxation of the discrete formulation is guaranteed to recover the true cluster whenever it is statistically solvable. In this thesis, we propose to solve the relaxed K-means problem as an unconstrained optimization problem on a smooth manifold. The proposed manifold can be parametrized by a product manifold with simple structures, allowing the application of second-order Riemannian algorithms. We show how to efficiently implement the cubic-regularized Riemannian Newton method by exploiting the structure of the Hessian. Numerical results show that our proposed algorithm converges faster while achieving similar accuracy compared with existing methods.","abstract_has_math":false,"creators":["Hou, Chun Ying"],"institution":"University of Illinois Urbana-Champaign","degree_name":"M.S.","degree_level":"Thesis","degree_discipline":"Electrical & Computer Engr","degree_department":null,"school":null,"contributors":["Zhang, Richard Y"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025-12","date_published":"2025-12","updated_at":"2026-07-22T22:25:07Z","subjects":["K-means clustering","manifold optimization"],"languages":["en"],"rights":["Copyright 2025 Chun Ying Hou"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/132799","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Zhang, Richard Y"]},{"key":"dc:creator","label":"Author","values":["Hou, Chun Ying"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2025-12","2025-12-10"]},{"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":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M.S."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["K-means clustering","manifold optimization"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2025 Chun Ying Hou"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/132799"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Clustering is a fundamental problem in unsupervised learning. The classical K-means formulation for clustering is a worst-case NP-hard discrete optimization problem. Despite being NP-hard, the SDP relaxation of the discrete formulation is guaranteed to recover the true cluster whenever it is statistically solvable. In this thesis, we propose to solve the relaxed K-means problem as an unconstrained optimization problem on a smooth manifold. The proposed manifold can be parametrized by a product manifold with simple structures, allowing the application of second-order Riemannian algorithms. We show how to efficiently implement the cubic-regularized Riemannian Newton method by exploiting the structure of the Hessian. Numerical results show that our proposed algorithm converges faster while achieving similar accuracy compared with existing methods.","Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2027-12-01","The student, Chun Ying Hou, accepted the attached license on 2025-12-04 at 13:44.","The student, Chun Ying Hou, submitted this Thesis for approval on 2025-12-10 at 13:35.","This Thesis was approved for publication on 2025-12-10 at 18:59.","DSpace SAF Submission Ingestion Package generated from Vireo submission #23063 on 2026-02-19 at 20:10:02"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Scalable second-order Riemannian optimization for K-means clustering"]}]}],"canonical_facts":{"dc:contributor":["Zhang, Richard Y"],"dc:creator":["Hou, Chun Ying"],"dc:date":["2025-12","2025-12-10"],"dc:description":["Clustering is a fundamental problem in unsupervised learning. The classical K-means formulation for clustering is a worst-case NP-hard discrete optimization problem. Despite being NP-hard, the SDP relaxation of the discrete formulation is guaranteed to recover the true cluster whenever it is statistically solvable. In this thesis, we propose to solve the relaxed K-means problem as an unconstrained optimization problem on a smooth manifold. The proposed manifold can be parametrized by a product manifold with simple structures, allowing the application of second-order Riemannian algorithms. We show how to efficiently implement the cubic-regularized Riemannian Newton method by exploiting the structure of the Hessian. Numerical results show that our proposed algorithm converges faster while achieving similar accuracy compared with existing methods.","Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2027-12-01","The student, Chun Ying Hou, accepted the attached license on 2025-12-04 at 13:44.","The student, Chun Ying Hou, submitted this Thesis for approval on 2025-12-10 at 13:35.","This Thesis was approved for publication on 2025-12-10 at 18:59.","DSpace SAF Submission Ingestion Package generated from Vireo submission #23063 on 2026-02-19 at 20:10:02"],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/132799"],"dc:language":["en"],"dc:rights":["Copyright 2025 Chun Ying Hou"],"dc:subject":["K-means clustering","manifold optimization"],"dc:title":["Scalable second-order Riemannian optimization for K-means clustering"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Electrical & Computer Engr"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["M.S."],"thesis:institution_name":["University of Illinois Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:07Z"}