Análisis topológico de datos
Year: 2020
Type: dissertation
Abstract: Topological data analysis is a branch of computational topology which uses algebra to \nobtain topological features from a data set. It has many applications in computer vision, shape \ndescription, time series analysis, biomedicine, drug design... \nThe first step to learn topological information from data is to build a filtration of nested \nsimplicial complexes estimating the global structure of the data set at different scales of precision. The second step is to study the evolution of the simplicial homology along this filtration, \nusing a tool from algebraic topology called persistent homology. When the filtration is indexed \nby a single parameter, we can describe discrete and complete invariants for persistent homology \nwith coefficients over a field, such as barcodes, persistent diagrams or persistent landscapes. \nUnfortunately, we don’t have a simple classification for persistent homology when the filtration \nis indexed by two or more parameters. The problem of classifying multiparameter persistent \nhomology is being widely studied nowadays, and we present some proposals for it. \nOne of the big problems of large sets is the presence of noisy, wrong or incomplete data. \nAlthough persistent homology is stable under small perturbation, we still need to give a statistical frame in order to separate insignificant features from topological signal and build some \nhypothesis tests applied to topological features. \nAnother usual problem of large sets is the high dimensionality. Even if we have captured \nthe main topological caracteristics of the data, we aren’t usually able to fully understand its \nstructure. In order to reduce dimensionality and obtain a better visualization we present the \nMapper algorithm, which groups the data using a filter function and summarizes the set as a \ngraph or a low dimensional simplicial complex.
Source: idUS (Universidad de Sevilla)
Author: Perera Lago, Javier
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7064264030