Ítem
Acceso Abierto
The nerve theorem in topological data analysis: applications in binary classification problems
Título de la revista
Autores
Luengas Fonseca, David Leonardo
Fecha
2025-08-11
Directores
Martínez Esparaza, Cristian Mauricio
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Editor
Universidad del Rosario
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Resumen
In 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.
Abstract
Palabras clave
Análisis topológico de datos , Homología persistente , Teorema del nervio , Clasificación binaria , Topología algebraíca
Keywords
Topological Data Analysis , Persistent Homology , Nerve Theorem , Binary Classification , Algebraic Topology
