{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/139219"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/139219","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"From Proximal Point Method to Accelerated Methods on Riemannian Manifolds","abstract":"Recently, there has been significant effort to generalize successful ideas in Euclidean optimization to Riemannian optimization. However, one landmark result of Euclidean optimization has eluded the Riemannian setting: namely, a Riemannian analog of Nesterov's accelerated gradient method (AGM). In this thesis, we establish the first globally accelerated gradient method for Riemannian manifolds. Toward establishing our result, the first part of the thesis revisits Nesterov's AGM and develops a conceptually simple understanding of it based on the proximal point method (PPM). The main observation is that AGM is in fact an approximation of PPM, which results in simple derivations and analyses of different versions of AGM. The second part of the thesis then extends our simple approach to the Riemannian case. In our extension, we handle a technical hurdle inherent to the Riemannian case by introducing an appropriate notion of ``metric distortion.'' We control this distortion via a novel geometric inequality, which enables us to formulate and analyze global Riemannian acceleration.","abstract_html":"Recently, there has been significant effort to generalize successful ideas in Euclidean optimization to Riemannian optimization. However, one landmark result of Euclidean optimization has eluded the Riemannian setting: namely, a Riemannian analog of Nesterov's accelerated gradient method (AGM). In this thesis, we establish the first globally accelerated gradient method for Riemannian manifolds. Toward establishing our result, the first part of the thesis revisits Nesterov's AGM and develops a conceptually simple understanding of it based on the proximal point method (PPM). The main observation is that AGM is in fact an approximation of PPM, which results in simple derivations and analyses of different versions of AGM. The second part of the thesis then extends our simple approach to the Riemannian case. In our extension, we handle a technical hurdle inherent to the Riemannian case by introducing an appropriate notion of ``metric distortion.'' We control this distortion via a novel geometric inequality, which enables us to formulate and analyze global Riemannian acceleration.","abstract_has_math":false,"creators":["Ahn, Kwangjun"],"institution":"Massachusetts Institute of Technology","degree_name":"Master","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science","school":null,"contributors":[],"advisors":["Sra, Suvrit"],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-06","date_published":"2021-06","updated_at":"2026-07-22T22:21:16Z","subjects":[],"languages":[],"rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"rights_urls":["http://rightsstatements.org/page/InC-EDU/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1721.1/139219","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Sra, Suvrit"]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science"]},{"key":"dc:creator","label":"Author","values":["Ahn, Kwangjun"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2022-01-14T14:57:28Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-01-14T14:57:28Z"]},{"key":"dc:date.issued","label":"Date","values":["2021-06"]},{"key":"dc:publisher","label":"Institution","values":["Massachusetts Institute of Technology"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master","Master of Science in Electrical Engineering and Computer Science"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright - Educational Use Permitted","Copyright MIT"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/page/InC-EDU/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/1721.1/139219"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Recently, there has been significant effort to generalize successful ideas in Euclidean optimization to Riemannian optimization. However, one landmark result of Euclidean optimization has eluded the Riemannian setting: namely, a Riemannian analog of Nesterov's accelerated gradient method (AGM). In this thesis, we establish the first globally accelerated gradient method for Riemannian manifolds. Toward establishing our result, the first part of the thesis revisits Nesterov's AGM and develops a conceptually simple understanding of it based on the proximal point method (PPM). The main observation is that AGM is in fact an approximation of PPM, which results in simple derivations and analyses of different versions of AGM. The second part of the thesis then extends our simple approach to the Riemannian case. In our extension, we handle a technical hurdle inherent to the Riemannian case by introducing an appropriate notion of ``metric distortion.'' We control this distortion via a novel geometric inequality, which enables us to formulate and analyze global Riemannian acceleration."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["S.M."]},{"key":"dc:title","label":"Title","values":["From Proximal Point Method to Accelerated Methods on Riemannian Manifolds"]}]}],"canonical_facts":{"dc:contributor.advisor":["Sra, Suvrit"],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science"],"dc:creator":["Ahn, Kwangjun"],"dc:date.accessioned":["2022-01-14T14:57:28Z"],"dc:date.available":["2022-01-14T14:57:28Z"],"dc:date.issued":["2021-06"],"dc:description.abstract":["Recently, there has been significant effort to generalize successful ideas in Euclidean optimization to Riemannian optimization. However, one landmark result of Euclidean optimization has eluded the Riemannian setting: namely, a Riemannian analog of Nesterov's accelerated gradient method (AGM). In this thesis, we establish the first globally accelerated gradient method for Riemannian manifolds. Toward establishing our result, the first part of the thesis revisits Nesterov's AGM and develops a conceptually simple understanding of it based on the proximal point method (PPM). The main observation is that AGM is in fact an approximation of PPM, which results in simple derivations and analyses of different versions of AGM. The second part of the thesis then extends our simple approach to the Riemannian case. In our extension, we handle a technical hurdle inherent to the Riemannian case by introducing an appropriate notion of ``metric distortion.'' We control this distortion via a novel geometric inequality, which enables us to formulate and analyze global Riemannian acceleration."],"dc:description.degree":["S.M."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/139219"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"dc:rights.uri":["http://rightsstatements.org/page/InC-EDU/1.0/"],"dc:title":["From Proximal Point Method to Accelerated Methods on Riemannian Manifolds"],"dc:type":["Thesis"],"thesis:degree_name":["Master","Master of Science in Electrical Engineering and Computer Science"]},"updated_at":"2026-07-22T22:21:16Z"}