Ítem
Acceso Abierto

The nerve theorem in topological data analysis: applications in binary classification problems

Mostrar el registro sencillo de la publicación
dc.contributor.advisorMartínez Esparza, Cristian Mauricio
dc.creatorLuengas Fonseca, David Leonardo
dc.creator.degreeProfesional en Matemáticas Aplicadas y Ciencias de la Computación
dc.creator.degreeLevelPregrado
dc.date.accessioned2025-08-22T16:38:09Z
dc.date.available2025-08-22T16:38:09Z
dc.date.created2025-08-11
dc.descriptionIn this thesis, we give a brief overview of Topological Data Analysis (TDA) from its math- ematical foundations to state-of-the-art applications. First, we present the necessary con- cepts from Topology, Algebra and Algebraic Topology that make TDA possible. Next, we introduce and prove the Nerve Theorem for convex and compact covers which underlies most of the methods used in TDA. We will also show the relevance of the Nerve Theorem in TDA from the perspective of Category Theory. Then, we explain how persistent homology is defined and used in TDA. Lastly, we present an application published in [1] of the Nerve Theorem and persistent homology that proves a lower bound for the sample size of data points that faithfully recovers the homology of the decision boundary manifold in binary classification problems.
dc.format.extent43 pp
dc.format.mimetypeapplication/pdf
dc.identifier.doihttps://doi.org/10.48713/10336_46318
dc.identifier.urihttps://repository.urosario.edu.co/handle/10336/46318
dc.language.isoeng
dc.publisherUniversidad del Rosario
dc.publisher.departmentEscuela de Ciencias e Ingeniería
dc.publisher.programPrograma de Matemáticas Aplicadas y Ciencias de la Computación - MACC
dc.rightsAttribution 4.0 International*
dc.rights.accesRightsinfo:eu-repo/semantics/openAccess
dc.rights.accesoAbierto (Texto Completo)
dc.rights.urihttp://creativecommons.org/licenses/by/4.0/*
dc.source.bibliographicCitationK. N. Ramamurthy, K. Varshney, and K. Mody, “Topological data analysis of decision boundaries with application to model selection,” in Proceedings of the 36th Interna- tional Conference on Machine Learning, K. Chaudhuri and R. Salakhutdinov, Eds., ser. Proceedings of Machine Learning Research, vol. 97, PMLR, Sep. 2019, pp. 5351– 5360. [Online]. Available: https://proceedings.mlr.press/v97/ramamurthy19a. html.
dc.source.bibliographicCitationEdelsbrunner, Letscher, and Zomorodian, “Topological persistence and simplifica- tion,” Discrete Computational Geometry, vol. 28, no. 4, pp. 511–533, Nov. 2002, issn: 1432-0444. doi: 10 . 1007 / s00454 - 002 - 2885 - 2. [Online]. Available: http : //dx.doi.org/10.1007/s00454-002-2885-2.
dc.source.bibliographicCitationJ. R. Munkres, Topology, 2nd ed. Prentice Hall, Inc., Jan. 2000, isbn: 0131816292. [Online]. Available: http://www.worldcat.org/isbn/0131816292.
dc.source.bibliographicCitationV. Nanda, Computational algebraic topology lecture notes, Mar. 2021.
dc.source.bibliographicCitationU. Bauer, M. Kerber, F. Roll, and A. Rolle, “A unified view on the functorial nerve theorem and its variations,” Expositiones Mathematicae, vol. 41, no. 4, p. 125 503, Dec. 2023, issn: 0723-0869. doi: 10.1016/j.exmath.2023.04.005. [Online]. Available: http://dx.doi.org/10.1016/j.exmath.2023.04.005.
dc.source.bibliographicCitationG. Singh, F. Memoli, and G. Carlsson, “Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition,” in Eurographics Symposium on Point-Based Graphics, M. Botsch, R. Pajarola, B. Chen, and M. Zwicker, Eds., The Eurographics Association, 2007, isbn: 978-3-905673-51-7. doi: /10.2312/SPBG/ SPBG07/091-100.
dc.source.bibliographicCitationM. Velásquez, Personal Communication, Jun. 14, 2025.
dc.source.bibliographicCitationJ. R. Munkres, Elements of Algebraic Topology. Addison Wesley Publishing Company, 1984, isbn: 0201045869. [Online]. Available: http : / / www . worldcat . org / isbn / 0201045869.
dc.source.bibliographicCitationJ. W. Vick, Homology Theory. Springer New York, 1994, isbn: 9781461208815. doi: 10.1007/978-1-4612-0881-5. [Online]. Available: http://dx.doi.org/10.1007/ 978-1-4612-0881-5.
dc.source.bibliographicCitationA. Hatcher, Algebraic topology. Cambridge: Cambridge Univ. Press, 2000. [Online]. Available: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf.
dc.source.bibliographicCitationH. Edelsbrunner, Notes on alpha complexes, Oct. 2006. [Online]. Available: https: //courses.cs.duke.edu/fall06/cps296.1/Lectures/sec-III-4.pdf.
dc.source.bibliographicCitationH. Edelsbrunner and E. P. Mücke, “Three-dimensional alpha shapes,” ACM Trans. Graph., vol. 13, no. 1, pp. 43–72, Jan. 1994, issn: 0730-0301. doi: 10.1145/174462. 156635. [Online]. Available: https://doi.org/10.1145/174462.156635.
dc.source.bibliographicCitationH. Edelsbrunner and E. Mücke, Three-dimensional alpha shapes, 1994. arXiv: math/ 9410208 [math.CO]. [Online]. Available: https://arxiv.org/abs/math/9410208.
dc.source.bibliographicCitationH. Edelsbrunner and J. Harer, “Persistent homology -— a survey,” in Surveys on discrete and computational geometry, Amer Mathematical Society, vol. 453, 2008, p. 257.
dc.source.bibliographicCitationP. Niyogi, S. Smale, and S. Weinberger, “Finding the homology of submanifolds with high confidence from random samples,” Discrete and Computational Geometry, vol. 39, no. 1–3, pp. 419–441, Mar. 2008, issn: 1432-0444. doi: 10.1007/s00454-008- 9053-2. [Online]. Available: http://dx.doi.org/10.1007/s00454-008-9053-2.
dc.source.bibliographicCitationV. Guillemin and A. Pollack, Differential Topology (AMS Chelsea Publishing). AMS Chelsea Pub., 2010, isbn: 9780821851937. [Online]. Available: https : / / books . google.com.co/books?id=FdRhAQAAQBAJ.
dc.source.bibliographicCitationM. Bianchini and F. Scarselli, “On the complexity of neural network classifiers: A comparison between shallow and deep architectures,” IEEE Transactions on Neural Networks and Learning Systems, vol. 25, no. 8, pp. 1553–1565, Aug. 2014, issn: 2162- 2388. doi: 10.1109/tnnls.2013.2293637. [Online]. Available: http://dx.doi.org/ 10.1109/TNNLS.2013.2293637.
dc.source.instnameinstname:Universidad del Rosario
dc.source.reponamereponame:Repositorio Institucional EdocUR
dc.subjectAnálisis topológico de datos
dc.subjectHomología persistente
dc.subjectTeorema del nervio
dc.subjectClasificación binaria
dc.subjectTopología algebraíca
dc.subject.keywordTopological Data Analysis
dc.subject.keywordPersistent Homology
dc.subject.keywordNerve Theorem
dc.subject.keywordBinary Classification
dc.subject.keywordAlgebraic Topology
dc.titleThe nerve theorem in topological data analysis: applications in binary classification problems
dc.title.TranslatedTitleEl teorema del nervio en el análisis topológico de datos: aplicaciones en problemas de clasificación binaria
dc.typebachelorThesis
dc.type.hasVersioninfo:eu-repo/semantics/acceptedVersion
dc.type.spaTrabajo de grado
local.department.reportEscuela de Ciencias e Ingeniería
local.regionesBogotá
Archivos
Bloque original
Mostrando1 - 1 de 1
Miniatura
Nombre:
The_nerve_theorem_in_topological_data_analysis_LeonardoLuengas.pdf
Tamaño:
568.87 KB
Formato:
Adobe Portable Document Format
Descripción:
Descargar
Colecciones
Programa de Matemáticas Aplicadas y Ciencias de la Computación - MACC