{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/113832"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/113832","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Machine learning in space forms: Embeddings, classification, and similarity comparisons","abstract":"DSpace SAF Submission Ingestion Package generated from Vireo submission #17190 on 2022-04-06 at 17:09:37","abstract_html":"DSpace SAF Submission Ingestion Package generated from Vireo submission #17190 on 2022-04-06 at 17:09:37","abstract_has_math":false,"creators":["Tabaghi, Puoya"],"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":["Dokmanic, Ivan","Milenkovic, Olgica","Hajek, Bruce","Raginsky, Maxim"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-04-29T21:34:24Z","date_published":"2022-04-29T21:34:24Z","updated_at":"2026-07-22T22:24:53Z","subjects":["Engineering"],"languages":["en","eng"],"rights":["Copyright 2021 Puoya Tabaghi"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/113832","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dokmanic, Ivan","Milenkovic, Olgica","Hajek, Bruce","Raginsky, Maxim"]},{"key":"dc:creator","label":"Author","values":["Tabaghi, Puoya"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2022-04-29T21:34:24Z","2021-12","2021-11-30"]},{"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":["Engineering"]}]},{"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 2021 Puoya Tabaghi"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/113832"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["DSpace SAF Submission Ingestion Package generated from Vireo submission #17190 on 2022-04-06 at 17:09:37","Made available in DSpace on 2022-04-29T21:34:24Z (GMT). No. of bitstreams: 4 TABAGHI-DISSERTATION-2021.pdf: 13424507 bytes, checksum: 35fe719f7737c19cfeb3d75f8d0c365c (MD5) main.zip: 67855657 bytes, checksum: c372adb6699c6ec16c35940217264b39 (MD5) LICENSE.txt: 4210 bytes, checksum: 98f09e1f83b3504149a1f7ce830cc08a (MD5) PROQUEST_LICENSE.txt: 4556 bytes, checksum: 7db926cf5dca163d6cee753fbca37f03 (MD5) Previous issue date: 2021-11-30","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-04-06 without embargo terms","The student, Puoya Tabaghi, accepted the attached license on 2021-11-29 at 13:02.","The student, Puoya Tabaghi, submitted this Dissertation for approval on 2021-11-29 at 13:27.","This Dissertation was approved for publication on 2021-11-30 at 09:07.","We take a non-Euclidean view at three classical machine learning subjects: low-dimensional embedding, classification, and similarity comparisons. We first introduce kinetic Euclidean distance matrices to solve kinetic distance geometry problems. In distance geometry problems (DGPs), the task is to find a geometric representation, that is, an embedding, for a collection of entities consistent with pairwise distance (metric) or similarity (nonmetric) measurements. In kinetic DGPs, the twist is that the points are dynamic. And our goal is to localize them by exploiting the information about their trajectory class. We show that a semidefinite relaxation can reconstruct trajectories from incomplete, noisy, time-varying distance observations. We then introduce another distance-geometric object: hyperbolic distance matrices. Recent works have focused on hyperbolic embedding methods for low-distortion embedding of distance measurements associated with hierarchical data. We derive a semidefinite relaxation to estimate the missing distance measurements and denoise them. Further, we formalize the hyperbolic Procrustes analysis, which uses extraneous information in the form of anchor points, to uniquely identify the embedded points. Next, we address the design of learning algorithms in mixed-curvature spaces. Learning algorithms in low-dimensional mixed-curvature spaces have been limited to certain non-Euclidean neural networks. Here, we study the problem of learning a linear classifier (a perceptron) in product of Euclidean, spherical, and hyperbolic spaces, i.e., space forms. We introduce a notion of linear separation surfaces in Riemannian manifolds and use a metric that renders distances in different space forms compatible with each other and integrates them into one classifier. Lastly, we show how similarity comparisons carry information about the underlying space of geometric graphs. We introduce the ordinal spread of a distance list and relate it to the ordinal capacity of their underlying space, a notion that quantifies the space's ability to host extreme patterns in nonmetric measurements. Then, we use the distribution of random ordinal spread variables as a practical tool to identify the underlying space form."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Machine learning in space forms: Embeddings, classification, and similarity comparisons"]}]}],"canonical_facts":{"dc:contributor":["Dokmanic, Ivan","Milenkovic, Olgica","Hajek, Bruce","Raginsky, Maxim"],"dc:creator":["Tabaghi, Puoya"],"dc:date":["2022-04-29T21:34:24Z","2021-12","2021-11-30"],"dc:description":["DSpace SAF Submission Ingestion Package generated from Vireo submission #17190 on 2022-04-06 at 17:09:37","Made available in DSpace on 2022-04-29T21:34:24Z (GMT). No. of bitstreams: 4 TABAGHI-DISSERTATION-2021.pdf: 13424507 bytes, checksum: 35fe719f7737c19cfeb3d75f8d0c365c (MD5) main.zip: 67855657 bytes, checksum: c372adb6699c6ec16c35940217264b39 (MD5) LICENSE.txt: 4210 bytes, checksum: 98f09e1f83b3504149a1f7ce830cc08a (MD5) PROQUEST_LICENSE.txt: 4556 bytes, checksum: 7db926cf5dca163d6cee753fbca37f03 (MD5) Previous issue date: 2021-11-30","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-04-06 without embargo terms","The student, Puoya Tabaghi, accepted the attached license on 2021-11-29 at 13:02.","The student, Puoya Tabaghi, submitted this Dissertation for approval on 2021-11-29 at 13:27.","This Dissertation was approved for publication on 2021-11-30 at 09:07.","We take a non-Euclidean view at three classical machine learning subjects: low-dimensional embedding, classification, and similarity comparisons. We first introduce kinetic Euclidean distance matrices to solve kinetic distance geometry problems. In distance geometry problems (DGPs), the task is to find a geometric representation, that is, an embedding, for a collection of entities consistent with pairwise distance (metric) or similarity (nonmetric) measurements. In kinetic DGPs, the twist is that the points are dynamic. And our goal is to localize them by exploiting the information about their trajectory class. We show that a semidefinite relaxation can reconstruct trajectories from incomplete, noisy, time-varying distance observations. We then introduce another distance-geometric object: hyperbolic distance matrices. Recent works have focused on hyperbolic embedding methods for low-distortion embedding of distance measurements associated with hierarchical data. We derive a semidefinite relaxation to estimate the missing distance measurements and denoise them. Further, we formalize the hyperbolic Procrustes analysis, which uses extraneous information in the form of anchor points, to uniquely identify the embedded points. Next, we address the design of learning algorithms in mixed-curvature spaces. Learning algorithms in low-dimensional mixed-curvature spaces have been limited to certain non-Euclidean neural networks. Here, we study the problem of learning a linear classifier (a perceptron) in product of Euclidean, spherical, and hyperbolic spaces, i.e., space forms. We introduce a notion of linear separation surfaces in Riemannian manifolds and use a metric that renders distances in different space forms compatible with each other and integrates them into one classifier. Lastly, we show how similarity comparisons carry information about the underlying space of geometric graphs. We introduce the ordinal spread of a distance list and relate it to the ordinal capacity of their underlying space, a notion that quantifies the space's ability to host extreme patterns in nonmetric measurements. Then, we use the distribution of random ordinal spread variables as a practical tool to identify the underlying space form."],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/113832"],"dc:language":["en","eng"],"dc:rights":["Copyright 2021 Puoya Tabaghi"],"dc:subject":["Engineering"],"dc:title":["Machine learning in space forms: Embeddings, classification, and similarity comparisons"],"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":"2026-07-22T22:24:53Z"}