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Homological Illusions of Persistence and StabilityMorozov, Dmitriy 04 August 2008 (has links)
<p>In this thesis we explore and extend the theory of persistent homology, which
captures topological features of a function by pairing its critical values. The
result is represented by a collection of points in the extended plane called
persistence diagram.</p><p>We start with the question of ridding the function of topological noise as
suggested by its persistence diagram. We give an algorithm for hierarchically
finding such epsilon-simplifications on 2-manifolds as well as answer the question
of when it is impossible to simplify a function in higher dimensions.</p><p>We continue by examining time-varying functions. The original algorithm computes
the persistence pairing from an ordering of the simplices in a triangulation and
takes worst-case time cubic in the number of simplices. We describe how to
maintain the pairing in linear time per transposition of consecutive simplices.
A side effect of the update algorithm is an elementary proof of the stability of
persistence diagrams. We introduce a parametrized family of persistence diagrams
called persistence vineyards and illustrate the concept with a vineyard
describing a folding of a small peptide. We also base a simple algorithm to
compute the rank invariant of a collection of functions on the update procedure.</p><p>Guided by the desire to reconstruct stratified spaces from noisy samples, we use
the vineyard of the distance function restricted to a 1-parameter family of
neighborhoods of a point to assess the local homology of a sampled stratified space at
that point.
We prove the correctness of
this assessment under the assumption of a sufficiently dense sample. We also
give an algorithm that constructs the vineyard and makes the local assessment in
time at most cubic in the size of the Delaunay triangulation of the point
sample.</p><p>Finally, to refine the measurement of local homology the thesis extends the
notion of persistent homology
to sequences of kernels, images, and cokernels of maps induced
by inclusions in a filtration of pairs of spaces.
Specifically,
we note that persistence in this context is well defined,
we prove that the persistence diagrams are stable,
and we explain how to compute them.
Additionally, we use image persistence to cope with functions on noisy domains.</p> / Dissertation
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Estrutura e estabilidade de módulos de persistência / Structure and stability of persistence modulesSilva, Fernando Gasparotto da [UNESP] 14 August 2017 (has links)
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Previous issue date: 2017-08-14 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / O intuito deste trabalho é de integrar os aspectos aplicado e teórico da Homologia Persistente, uma ferramenta popular da Topological Data Analysis (TDA). Para isso, são apresentados e demonstrados os resultados fundamentais da teoria embasada na topologia algébrica que permitem o desenvolvimento de algoritmos e paradigmas computacionais para obter diagramas de persistência. Dessa forma, iniciaremos explorando como decodificar as informações contidas em um módulo de persistência, entendendo os conceitos de multiconjuntos, módulos de persistência e cálculos Quiver. Em seguida, o caminho contrário será explorado, onde os dados são codificados em diagramas de persistência a fim de extrair suas características topológicas, aprofundando os conceitos de funções de Morse, Homologia Persistente, diagramas de persistência, dualidade e simetria, bem como estabilidade. Por último, encerramos demonstrando duas possíveis aplicações da teoria no âmbito computacional no campo da Biologia. / The goal of this work is to integrate applied and theoretical aspects of Persistence Homology, a popular tool in Topological Data Analysis (TDA). For this, we present and prove fundamental theoretical results based on algebraic topology, which allow us to develop algorithms and computational paradigms to obtain persistence diagrams. In this way, we start exploring how to decode the information contained in a persistence module, understanding the concepts of multiset, persistence modules and Quiver alculations. Then, the opposite path will be explored, where the data are encoded in persistence diagrams in order to extract their topological characteristics, going deep into the concepts of Morse functions, persistent homology, persistence diagrams, duality and symmetry, as well as stability. Finally, we conclude with two possible applications, one from computational theory, and the second one in the field of biology. / CNPq: 135622/2015-8
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Topological data analysis: applications in machine learning / Análise topológica de dados: aplicações em aprendizado de máquinaCalcina, Sabrina Graciela Suárez 05 December 2018 (has links)
Recently computational topology had an important development in data analysis giving birth to the field of Topological Data Analysis. Persistent homology appears as a fundamental tool based on the topology of data that can be represented as points in metric space. In this work, we apply techniques of Topological Data Analysis, more precisely, we use persistent homology to calculate topological features more persistent in data. In this sense, the persistence diagrams are processed as feature vectors for applying Machine Learning algorithms. In order to classification, we used the following classifiers: Partial Least Squares-Discriminant Analysis, Support Vector Machine, and Naive Bayes. For regression, we used Support Vector Regression and KNeighbors. Finally, we will give a certain statistical approach to analyze the accuracy of each classifier and regressor. / Recentemente a topologia computacional teve um importante desenvolvimento na análise de dados dando origem ao campo da Análise Topológica de Dados. A homologia persistente aparece como uma ferramenta fundamental baseada na topologia de dados que possam ser representados como pontos num espaço métrico. Neste trabalho, aplicamos técnicas da Análise Topológica de Dados, mais precisamente, usamos homologia persistente para calcular características topológicas mais persistentes em dados. Nesse sentido, os diagramas de persistencia são processados como vetores de características para posteriormente aplicar algoritmos de Aprendizado de Máquina. Para classificação, foram utilizados os seguintes classificadores: Análise de Discriminantes de Minimos Quadrados Parciais, Máquina de Vetores de Suporte, e Naive Bayes. Para a regressão, usamos a Regressão de Vetores de Suporte e KNeighbors. Finalmente, daremos uma certa abordagem estatística para analisar a precisão de cada classificador e regressor.
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