Linear methods of dimension reduction for classification
For classification problems, traditional dimension reduction methods often take into account only the feature information, while ignoring the class label. This poses an opportunity for improvement. In this thesis, we explore new methods that aim to find linear orthogonal projections that maximize op...
- Autores:
-
Ramírez Garrido, Diego Alejandro
- Tipo de recurso:
- Trabajo de grado de pregrado
- Fecha de publicación:
- 2024
- Institución:
- Universidad de los Andes
- Repositorio:
- Séneca: repositorio Uniandes
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.uniandes.edu.co:1992/75195
- Acceso en línea:
- https://hdl.handle.net/1992/75195
- Palabra clave:
- Dimension Reduction
Dimensionality Reduction
Wasserstein Distance
Sinkhorn Divergence
Subgradient Descent
Binary Classification
Optimal Transport
Matemáticas
- Rights
- openAccess
- License
- Attribution 4.0 International
| id |
UNIANDES2_fefd20b46ce04db5120c32600eeae623 |
|---|---|
| oai_identifier_str |
oai:repositorio.uniandes.edu.co:1992/75195 |
| network_acronym_str |
UNIANDES2 |
| network_name_str |
Séneca: repositorio Uniandes |
| repository_id_str |
|
| dc.title.eng.fl_str_mv |
Linear methods of dimension reduction for classification |
| title |
Linear methods of dimension reduction for classification |
| spellingShingle |
Linear methods of dimension reduction for classification Dimension Reduction Dimensionality Reduction Wasserstein Distance Sinkhorn Divergence Subgradient Descent Binary Classification Optimal Transport Matemáticas |
| title_short |
Linear methods of dimension reduction for classification |
| title_full |
Linear methods of dimension reduction for classification |
| title_fullStr |
Linear methods of dimension reduction for classification |
| title_full_unstemmed |
Linear methods of dimension reduction for classification |
| title_sort |
Linear methods of dimension reduction for classification |
| dc.creator.fl_str_mv |
Ramírez Garrido, Diego Alejandro |
| dc.contributor.advisor.none.fl_str_mv |
Quiroz Salazar, Adolfo José |
| dc.contributor.author.none.fl_str_mv |
Ramírez Garrido, Diego Alejandro |
| dc.contributor.jury.none.fl_str_mv |
Junca Peláez, Mauricio José |
| dc.subject.keyword.eng.fl_str_mv |
Dimension Reduction Dimensionality Reduction Wasserstein Distance Sinkhorn Divergence Subgradient Descent Binary Classification Optimal Transport |
| topic |
Dimension Reduction Dimensionality Reduction Wasserstein Distance Sinkhorn Divergence Subgradient Descent Binary Classification Optimal Transport Matemáticas |
| dc.subject.themes.none.fl_str_mv |
Matemáticas |
| description |
For classification problems, traditional dimension reduction methods often take into account only the feature information, while ignoring the class label. This poses an opportunity for improvement. In this thesis, we explore new methods that aim to find linear orthogonal projections that maximize optimal transportation distances (Wasserstein and Sinkhorn) feature subsamples corresponding to the categories. These methods employ ubgradient ascent and stochastic subgradient ascent algorithms. We detail the calculation of the subgradient of these distances with respect to the projection and implement these methods in Python. To validate our approach, we test these methods on various datasets. Our results demonstrate that the proposed methods effectively enhance classification performance by incorporating class information into the dimension reduction process. |
| publishDate |
2024 |
| dc.date.accessioned.none.fl_str_mv |
2024-10-30T21:55:29Z |
| dc.date.available.none.fl_str_mv |
2024-10-30T21:55:29Z |
| dc.date.issued.none.fl_str_mv |
2024-10-30 |
| dc.type.none.fl_str_mv |
Trabajo de grado - Pregrado |
| dc.type.driver.none.fl_str_mv |
info:eu-repo/semantics/bachelorThesis |
| dc.type.version.none.fl_str_mv |
info:eu-repo/semantics/acceptedVersion |
| dc.type.coar.none.fl_str_mv |
http://purl.org/coar/resource_type/c_7a1f |
| dc.type.content.none.fl_str_mv |
Text |
| dc.type.redcol.none.fl_str_mv |
http://purl.org/redcol/resource_type/TP |
| format |
http://purl.org/coar/resource_type/c_7a1f |
| status_str |
acceptedVersion |
| dc.identifier.uri.none.fl_str_mv |
https://hdl.handle.net/1992/75195 |
| dc.identifier.instname.none.fl_str_mv |
instname:Universidad de los Andes |
| dc.identifier.reponame.none.fl_str_mv |
reponame:Repositorio Institucional Séneca |
| dc.identifier.repourl.none.fl_str_mv |
repourl:https://repositorio.uniandes.edu.co/ |
| url |
https://hdl.handle.net/1992/75195 |
| identifier_str_mv |
instname:Universidad de los Andes reponame:Repositorio Institucional Séneca repourl:https://repositorio.uniandes.edu.co/ |
| dc.language.iso.none.fl_str_mv |
eng |
| language |
eng |
| dc.relation.references.none.fl_str_mv |
Boyd, Stephen, and Lieven Vandenberghe. *Convex Optimization.* Cambridge University Press, 2004. Clarke, Frank H. “Generalized Gradients and Applications.” *Transactions of the American Mathematical Society*, vol. 205, 1975, pp. 247–262. https://doi.org/10.1090/s0002-9947-1975-0367131-6. Accessed 14 Jan. 2021. Clarke, Frank H. *Optimization and Nonsmooth Analysis.* Wiley-Interscience, 1983. https://doi.org/10.1137/1.9781611971309 Cuturi, Marco. “Sinkhorn Distances: Lightspeed Computation of Optimal Transport.” *Advances in Neural Information Processing Systems*, vol. 26, 2013, pp. 2292–2300.https://doi.org/10.48550/arXiv.1306.0895 Devroye, Luc, et al. *A Probabilistic Theory of Pattern Recognition.* Springer Science \& Business Media, 2013. Munkres, James. “Algorithms for the Assignment and Transportation Problems.” *Journal of the Society for Industrial and Applied Mathematics*, vol. 5, no. 1, Mar. 1957, pp. 32–38. https://doi.org/10.1137/0105003. Accessed 26 July 2020. Peyré, Gabriel, and Marco Cuturi. *Computational Optimal Transport.* Foundations and Trends in Machine Learning, 2019. Sinkhorn, Richard, and Paul Knopp. “Concerning Nonnegative Matrices and Doubly Stochastic Matrices.” *Pacific Journal of Mathematics*, vol. 21, no. 2, 1967, pp. 343–348. https://doi.org/10.2140/pjm.1967.21.343 Accessed 30 July 2022. Vanderbei, Robert J. *Linear Programming: Foundations and Extensions.* Springer, 2021. Villani, Cédric. *Optimal Transport: Old and New.* Springer, 2009. Papailiopoulos, D. *ECE 901: Large-scale Machine Learning and Optimization.* Lecture 9. Scribed by Guangtong Bai \& Yuan-Ting Hsieh, Spring 2018. Janosi, Andras, Steinbrunn, William, Pfisterer, Matthias, and Detrano, Robert. *Heart Disease.* UCI Machine Learning Repository, 1988. https://doi.org/10.24432/C52P4X Kahn, Michael. *Diabetes.* UCI Machine Learning Repository. https://doi.org/10.24432/C5T59G Hotelling, H. “Analysis of a Complex of Statistical Variables into Principal Components.” *Journal of Educational Psychology*, vol. 24, no. 6, 1933, pp. 417–441. https://doi.org/10.1037/h0071325 Stein, Elias M. *Singular Integrals and Differentiability Properties of Functions.* Princeton University Press, 1970. Bottou, Léon. "Large-Scale Machine Learning with Stochastic Gradient Descent." In *Proceedings of COMPSTAT2010*, edited by Yves Lechevallier and Gilbert Saporta, Physica-Verlag HD, 2010, pp. 177-186. https://doi.org/10.1007/978-3-7908-2604-3_16 Polyak, Boris T. "Some Methods of Speeding up the Convergence of Iteration Methods." *USSR Computational Mathematics and Mathematical Physics*, vol. 4, no. 5, 1964, pp. 1-17. https://doi.org/10.1016/0041-5553(64)90137-5 Hinton, Geoffrey. "Lecture 6e rmsprop: Divide the Gradient by a Running Average of Its Recent Magnitude." *Coursera Lecture Notes*, 2012. Tieleman, Tijmen, and Geoffrey Hinton. "Lecture 6.5 - RMSProp: Divide the Gradient by a Running Average of Its Recent Magnitude." *COURSERA: Neural Networks for Machine Learning*, University of Toronto, 2012. Kingma, Diederik P., and Jimmy Ba. "Adam: A Method for Stochastic Optimization." In *Proceedings of the 3rd International Conference on Learning Representations (ICLR)*, 2015.https://doi.org/10.48550/arXiv.1412.6980 Volgenant, Ton, and R. Jonker. "A Branch and Bound Algorithm for the Symmetric Traveling Salesman Problem Based on the 1-tree Relaxation." *European Journal of Operational Research*, vol. 6, no. 4, 1981, pp. 447-458. https://doi.org/10.1016/0377-2217(81)90100-0 Edmonds, Jack, and Richard M. Karp. "Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems." *Journal of the ACM*, vol. 19, no. 2, 1972, pp. 248-264. https://doi.org/10.1145/321694.321699 Shannon, Claude E. "A Mathematical Theory of Communication." *Bell System Technical Journal*, vol. 27, no. 3, 1948, pp. 379-423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x |
| dc.rights.en.fl_str_mv |
Attribution 4.0 International |
| dc.rights.uri.none.fl_str_mv |
http://creativecommons.org/licenses/by/4.0/ |
| dc.rights.accessrights.none.fl_str_mv |
info:eu-repo/semantics/openAccess |
| dc.rights.coar.none.fl_str_mv |
http://purl.org/coar/access_right/c_abf2 |
| rights_invalid_str_mv |
Attribution 4.0 International http://creativecommons.org/licenses/by/4.0/ http://purl.org/coar/access_right/c_abf2 |
| eu_rights_str_mv |
openAccess |
| dc.format.extent.none.fl_str_mv |
54 páginas |
| dc.format.mimetype.none.fl_str_mv |
application/pdf |
| dc.publisher.none.fl_str_mv |
Universidad de los Andes |
| dc.publisher.program.none.fl_str_mv |
Matemáticas |
| dc.publisher.faculty.none.fl_str_mv |
Facultad de Ciencias |
| dc.publisher.department.none.fl_str_mv |
Departamento de Matemáticas |
| publisher.none.fl_str_mv |
Universidad de los Andes |
| institution |
Universidad de los Andes |
| bitstream.url.fl_str_mv |
https://repositorio.uniandes.edu.co/bitstreams/6b84d5c6-56cd-4eb9-b3cf-8024e83e62fb/download https://repositorio.uniandes.edu.co/bitstreams/6711caf2-5d86-4d68-a2e8-01136cc34e33/download https://repositorio.uniandes.edu.co/bitstreams/136472ce-4188-428e-9e74-9711bcd2666c/download https://repositorio.uniandes.edu.co/bitstreams/0a11f59d-83c5-499e-b1c8-f473adf27252/download https://repositorio.uniandes.edu.co/bitstreams/4b635b18-fdb2-46fc-9f13-4ef6bde21fc7/download https://repositorio.uniandes.edu.co/bitstreams/44ab51ee-b5b5-4a80-b105-f38f73c89f92/download https://repositorio.uniandes.edu.co/bitstreams/99ffe3bb-a03e-448a-89ef-515c05eb8ffc/download https://repositorio.uniandes.edu.co/bitstreams/ed6f09db-2ac8-4012-811d-7b7a0a7131eb/download |
| bitstream.checksum.fl_str_mv |
0175ea4a2d4caec4bbcc37e300941108 ae9e573a68e7f92501b6913cc846c39f e16bbe0f3f60ef09d2f08436b10e3c82 f31c9fcfa9ab4280de1d80371f63e986 30a21ea615d12756d53f82d9b82a658b 13c2e06b42c89281584508f3b9219df6 e7df4079db8e97692aea301bd02fce9b f22df99a365ea68942bb39442996fc09 |
| bitstream.checksumAlgorithm.fl_str_mv |
MD5 MD5 MD5 MD5 MD5 MD5 MD5 MD5 |
| repository.name.fl_str_mv |
Repositorio institucional Séneca |
| repository.mail.fl_str_mv |
adminrepositorio@uniandes.edu.co |
| _version_ |
1818112052178190336 |
| spelling |
Quiroz Salazar, Adolfo Josévirtual::20118-1Ramírez Garrido, Diego AlejandroJunca Peláez, Mauricio Josévirtual::20119-12024-10-30T21:55:29Z2024-10-30T21:55:29Z2024-10-30https://hdl.handle.net/1992/75195instname:Universidad de los Andesreponame:Repositorio Institucional Sénecarepourl:https://repositorio.uniandes.edu.co/For classification problems, traditional dimension reduction methods often take into account only the feature information, while ignoring the class label. This poses an opportunity for improvement. In this thesis, we explore new methods that aim to find linear orthogonal projections that maximize optimal transportation distances (Wasserstein and Sinkhorn) feature subsamples corresponding to the categories. These methods employ ubgradient ascent and stochastic subgradient ascent algorithms. We detail the calculation of the subgradient of these distances with respect to the projection and implement these methods in Python. To validate our approach, we test these methods on various datasets. Our results demonstrate that the proposed methods effectively enhance classification performance by incorporating class information into the dimension reduction process.Pregrado54 páginasapplication/pdfengUniversidad de los AndesMatemáticasFacultad de CienciasDepartamento de MatemáticasAttribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Linear methods of dimension reduction for classificationTrabajo de grado - Pregradoinfo:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_7a1fTexthttp://purl.org/redcol/resource_type/TPDimension ReductionDimensionality ReductionWasserstein DistanceSinkhorn DivergenceSubgradient DescentBinary ClassificationOptimal TransportMatemáticasBoyd, Stephen, and Lieven Vandenberghe. *Convex Optimization.* Cambridge University Press, 2004.Clarke, Frank H. “Generalized Gradients and Applications.” *Transactions of the American Mathematical Society*, vol. 205, 1975, pp. 247–262. https://doi.org/10.1090/s0002-9947-1975-0367131-6. Accessed 14 Jan. 2021.Clarke, Frank H. *Optimization and Nonsmooth Analysis.* Wiley-Interscience, 1983. https://doi.org/10.1137/1.9781611971309Cuturi, Marco. “Sinkhorn Distances: Lightspeed Computation of Optimal Transport.” *Advances in Neural Information Processing Systems*, vol. 26, 2013, pp. 2292–2300.https://doi.org/10.48550/arXiv.1306.0895Devroye, Luc, et al. *A Probabilistic Theory of Pattern Recognition.* Springer Science \& Business Media, 2013.Munkres, James. “Algorithms for the Assignment and Transportation Problems.” *Journal of the Society for Industrial and Applied Mathematics*, vol. 5, no. 1, Mar. 1957, pp. 32–38. https://doi.org/10.1137/0105003. Accessed 26 July 2020.Peyré, Gabriel, and Marco Cuturi. *Computational Optimal Transport.* Foundations and Trends in Machine Learning, 2019.Sinkhorn, Richard, and Paul Knopp. “Concerning Nonnegative Matrices and Doubly Stochastic Matrices.” *Pacific Journal of Mathematics*, vol. 21, no. 2, 1967, pp. 343–348. https://doi.org/10.2140/pjm.1967.21.343 Accessed 30 July 2022.Vanderbei, Robert J. *Linear Programming: Foundations and Extensions.* Springer, 2021.Villani, Cédric. *Optimal Transport: Old and New.* Springer, 2009.Papailiopoulos, D. *ECE 901: Large-scale Machine Learning and Optimization.* Lecture 9. Scribed by Guangtong Bai \& Yuan-Ting Hsieh, Spring 2018.Janosi, Andras, Steinbrunn, William, Pfisterer, Matthias, and Detrano, Robert. *Heart Disease.* UCI Machine Learning Repository, 1988. https://doi.org/10.24432/C52P4XKahn, Michael. *Diabetes.* UCI Machine Learning Repository. https://doi.org/10.24432/C5T59GHotelling, H. “Analysis of a Complex of Statistical Variables into Principal Components.” *Journal of Educational Psychology*, vol. 24, no. 6, 1933, pp. 417–441. https://doi.org/10.1037/h0071325Stein, Elias M. *Singular Integrals and Differentiability Properties of Functions.* Princeton University Press, 1970.Bottou, Léon. "Large-Scale Machine Learning with Stochastic Gradient Descent." In *Proceedings of COMPSTAT2010*, edited by Yves Lechevallier and Gilbert Saporta, Physica-Verlag HD, 2010, pp. 177-186. https://doi.org/10.1007/978-3-7908-2604-3_16Polyak, Boris T. "Some Methods of Speeding up the Convergence of Iteration Methods." *USSR Computational Mathematics and Mathematical Physics*, vol. 4, no. 5, 1964, pp. 1-17. https://doi.org/10.1016/0041-5553(64)90137-5Hinton, Geoffrey. "Lecture 6e rmsprop: Divide the Gradient by a Running Average of Its Recent Magnitude." *Coursera Lecture Notes*, 2012.Tieleman, Tijmen, and Geoffrey Hinton. "Lecture 6.5 - RMSProp: Divide the Gradient by a Running Average of Its Recent Magnitude." *COURSERA: Neural Networks for Machine Learning*, University of Toronto, 2012.Kingma, Diederik P., and Jimmy Ba. "Adam: A Method for Stochastic Optimization." In *Proceedings of the 3rd International Conference on Learning Representations (ICLR)*, 2015.https://doi.org/10.48550/arXiv.1412.6980Volgenant, Ton, and R. Jonker. "A Branch and Bound Algorithm for the Symmetric Traveling Salesman Problem Based on the 1-tree Relaxation." *European Journal of Operational Research*, vol. 6, no. 4, 1981, pp. 447-458. https://doi.org/10.1016/0377-2217(81)90100-0Edmonds, Jack, and Richard M. Karp. "Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems." *Journal of the ACM*, vol. 19, no. 2, 1972, pp. 248-264. https://doi.org/10.1145/321694.321699Shannon, Claude E. "A Mathematical Theory of Communication." *Bell System Technical Journal*, vol. 27, no. 3, 1948, pp. 379-423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x202013233Publicationhttps://scholar.google.es/citations?user=qwMDh-4AAAAJvirtual::20118-1https://scholar.google.es/citations?user=CoIlxH0AAAAJvirtual::20119-10000-0003-4033-3400virtual::20118-10000-0002-5541-0758virtual::20119-1https://scienti.minciencias.gov.co/cvlac/visualizador/generarCurriculoCv.do?cod_rh=0001497101virtual::20118-1https://scienti.minciencias.gov.co/cvlac/visualizador/generarCurriculoCv.do?cod_rh=0000155861virtual::20119-11be19e5b-39c2-4d92-b44f-b9b4a48991cavirtual::20118-11be19e5b-39c2-4d92-b44f-b9b4a48991cavirtual::20118-11e5c3dc6-4d9c-406b-9f99-5c91523b7e49virtual::20119-11e5c3dc6-4d9c-406b-9f99-5c91523b7e49virtual::20119-1CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8908https://repositorio.uniandes.edu.co/bitstreams/6b84d5c6-56cd-4eb9-b3cf-8024e83e62fb/download0175ea4a2d4caec4bbcc37e300941108MD51LICENSElicense.txtlicense.txttext/plain; charset=utf-82535https://repositorio.uniandes.edu.co/bitstreams/6711caf2-5d86-4d68-a2e8-01136cc34e33/downloadae9e573a68e7f92501b6913cc846c39fMD52ORIGINALautorizacion tesis Diego Ramirez.pdfautorizacion tesis Diego Ramirez.pdfHIDEapplication/pdf365846https://repositorio.uniandes.edu.co/bitstreams/136472ce-4188-428e-9e74-9711bcd2666c/downloade16bbe0f3f60ef09d2f08436b10e3c82MD53Linear Methods of Dimension Reduction for Classification.pdfLinear Methods of Dimension Reduction for Classification.pdfapplication/pdf778417https://repositorio.uniandes.edu.co/bitstreams/0a11f59d-83c5-499e-b1c8-f473adf27252/downloadf31c9fcfa9ab4280de1d80371f63e986MD54TEXTautorizacion tesis Diego Ramirez.pdf.txtautorizacion tesis Diego Ramirez.pdf.txtExtracted texttext/plain2024https://repositorio.uniandes.edu.co/bitstreams/4b635b18-fdb2-46fc-9f13-4ef6bde21fc7/download30a21ea615d12756d53f82d9b82a658bMD55Linear Methods of Dimension Reduction for Classification.pdf.txtLinear Methods of Dimension Reduction for Classification.pdf.txtExtracted texttext/plain55616https://repositorio.uniandes.edu.co/bitstreams/44ab51ee-b5b5-4a80-b105-f38f73c89f92/download13c2e06b42c89281584508f3b9219df6MD57THUMBNAILautorizacion tesis Diego Ramirez.pdf.jpgautorizacion tesis Diego Ramirez.pdf.jpgGenerated Thumbnailimage/jpeg10933https://repositorio.uniandes.edu.co/bitstreams/99ffe3bb-a03e-448a-89ef-515c05eb8ffc/downloade7df4079db8e97692aea301bd02fce9bMD56Linear Methods of Dimension Reduction for Classification.pdf.jpgLinear Methods of Dimension Reduction for Classification.pdf.jpgGenerated Thumbnailimage/jpeg5875https://repositorio.uniandes.edu.co/bitstreams/ed6f09db-2ac8-4012-811d-7b7a0a7131eb/downloadf22df99a365ea68942bb39442996fc09MD581992/75195oai:repositorio.uniandes.edu.co:1992/751952024-10-31 03:08:28.004http://creativecommons.org/licenses/by/4.0/Attribution 4.0 Internationalopen.accesshttps://repositorio.uniandes.edu.coRepositorio institucional Sénecaadminrepositorio@uniandes.edu.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 |