Topological Data Analysis is a quickly expanding field but one particular subfield, multidimensional persistence, has hit a dead end. Although TDA is a very active field, it has been proven that the one-dimensional persistence used in persistent homology cannot be generalized to higher dimensions. With this in mind, progress can still be made in the accuracy of approximating it. The central challenge lies in the multiple persistence parameters. Using more than one parameter at a time creates a multi-filtration of the data which cannot be totally ordered in the way that a single filtration can. The goal of this thesis is to contribute to the development of persistence heat maps by replacing the persistent betti number function (PBN) defined by Xia and Wei in 2015 with a new persistence summary function, the accumulated persistence function (APF) defined by Biscio and Moller in 2016. The PBN function fails to capture persistence in most cases and thus their heat maps lack important information. The APF, on the other hand, does capture persistence that can be seen in their heat maps. A heat map is a way to visually describe three dimensions with two spatial dimensions and color. In two-dimensional persistence heat maps, the two chosen parameters lie on the x- and y- axes. These persistence parameters define a complex on the data, and its topology is represented by the color. We use the method of heat maps introduced by Xia and Wei. We acquired an R script from Matthew Pietrosanu to generate our own heat maps with the second parameter being curvature threshold. We also use the accumulated persistence function introduced by Biscio and Moller, who provided an R script to compute the APF on a data set. We then wrote new code, building from the existing codes, to create a modified heat map. In all the examples in this thesis, we show both the old PBN and the new APF heat maps to illustrate their differences and similarities. We study the two-dimensional heat maps with respect to curvature applied to two types of parameterized knots, Lissajous knots and torus knots. We also show how both heat maps can be used to compare and contrast data sets. This research is important because the persistence heat map acts as a guide for finding topologically significant features as the data changes with respect to two parameters. Improving the accuracy of the heat map ultimately improves the efficiency of data analysis. Two-dimensional persistence has practical applications in analyses of data coming from proteins and DNA. The unfolding of proteins offers a second parameter of configuration over time, while tangled DNA may have a second parameter of curvature. The concluding argument of this thesis is that using the accumulated persistence function in conjunction with the persistent betti number function provides a more accurate representation of two-dimensional persistence than the PBN heat map alone.

TDA mapper is an algorithm used to visualize and analyze big data. TDA mapper is applied to a dataset, X, equipped with a filter function f from X to R. The output of the algorithm is an abstract graph (or simplicial complex). The abstract graph captures topological and geometric information of the underlying space of X. One of the interests in TDA mapper is to study whether or not a mapper graph is stable. That is, if a dataset X is perturbed by a small value, and denote the perturbed dataset by X∂, we would like to compare the TDA mapper graph of X to the TDA mapper graph of X∂. Given a topological space X, if the cover of the image of f satisfies certain conditions, Tamal Dey, Facundo Memoli, and Yusu Wang proved that the TDA mapper is stable. That is, the mapper graph of X differs from the mapper graph of X∂ by a small value measured via homology. The goal of this thesis is three-fold. The first is to introduce a modified TDA mapper algorithm. The fundamental difference between TDA mapper and the modified version is the modified version avoids the use of filter function. In comparing the mapper graph outputs, the proposed modified mapper is shown to capture more geometric and topological features. We discuss the advantages and disadvantages of the modified mapper. Tamal Dey, Facundo Memoli, and Yusu Wang showed that a filtration of covers induce a filtration of simplicial complexes, which in turn induces a filtration of homology groups. While Tamal Dey, Facundo Memoli, and Yusu Wang focused on TDA mapper's application to topological space, the second goal of this thesis is to show DBSCAN clustering gives a filtration of covers when TDA mapper is applied to a point cloud. Hence, DBSCAN gives a filtration of mapper graphs (simplicial complexes) and homology groups. More importantly, DBSCAN gives a filtration of covers, mapper graphs, and homology groups in three parameter directions: bin size, epsilon, and Minpts. Hence, there is a multi-dimensional filtration of covers, mapper graphs, and homology groups. We also note that single-linkage clustering is a special case of DBSCAN clustering, so the results proved to be true when DBSCAN is used are also true when single-linkage is used. However, complete-linkage does not give a filtration of covers in the direction of bin, hence no filtration of simpicial complexes and homology groups exist when complete-linkage is applied to cluster a dataset. In general, the results hold for any clustering algorithm that gives a filtration of covers. The third (and last) goal of this thesis is to prove that two multi-dimensional persistence modules (one: with respect to the original dataset, X; two: with respect to the ∂-perturbation of X) are 2∂-interleaved. In other words, the mapper graphs of X and X∂ differ by a small value as measured by homology.

Orientador: Alice Kimie Miwa Libardi / Banca: Thiago de Melo / Banca: Washington Mio / Resumo: O objetivo desta dissertação é apresentar uma versão detalhada do artigo: "A Tutorial on Spectral Clustering" de U. von Luxburg sobre agrupamentos através de grafos Laplacianos, suas propriedades e mostrar alguns resultados da teoria de agrupamentos. Além disso, serão apresentados três algoritmos de agrupamentos e ilustraremos um deles com uma aplicação no cultivo da soja em diferentes condições de cultivo / Abstract: The main goal of this dissertation is to present a detailed version of the paper: " A Tutorial on Spectral Clustering" of U. von Luxburg on clusters, through Laplacian graphs, their properties and to show some results of the cluster theory. In addition, it will be presented three clustering algorithms and we will illustrate one of them with an application in the soybean cultivation, under different conditions / Mestre

This dissertation explores how students’ beliefs and attitudes interact with their identities as physics people, motivated by calls to increase participation in science, technology, engineering, and mathematics (STEM) careers. This work combines several theoretical frameworks, including Identity theory, Future Time Perspective theory, and other personality traits to investigate associations between these factors. An enriched understanding of how these attitudinal factors are associated with each other extends prior models of identity and link theoretical frameworks used in psychological and educational research. The research uses a series of quantitative and qualitative methodologies, including linear and logistic regression analysis, thematic interview analysis, and an innovative analytic technique adapted for use with student educational data for the first time: topological data analysis via the Mapper algorithm. Engineering students were surveyed in their introductory engineering courses. Several factors are found to be associated with physics identity, including student interest in particular engineering majors. The distributions of student scores on these affective constructs are simultaneously represented in a map of beliefs, from which the existence of a large “normative group” of students (according to their beliefs) is identified, defined by the data as a large concentration of similarly minded students. Significant differences exist in the demographic representation of this normative group compared to other students, which has implications for recruitment efforts that seek to increase diversity in STEM fields. Select students from both the normative group and outside the normative group were selected for subsequent interviews investigating their associations between physics and engineering, and how their physics identities evolve during their engineering careers. Further analyses suggest a more complex model of physics and engineering identity which is not necessarily uniform for all engineering students, including discipline-specific differences that should be further investigated. Further, the use of physics identity as a model to describe engineering student choices may be limited in applicability to early college. Interview analysis shows that physics recognition beliefs become contextualized in engineering as students begin to view physics as an increasingly distinct domain from engineering.

abstract: Topological methods for data analysis present opportunities for enforcing certain invariances of broad interest in computer vision: including view-point in activity analysis, articulation in shape analysis, and measurement invariance in non-linear dynamical modeling. The increasing success of these methods is attributed to the complementary information that topology provides, as well as availability of tools for computing topological summaries such as persistence diagrams. However, persistence diagrams are multi-sets of points and hence it is not straightforward to fuse them with features used for contemporary machine learning tools like deep-nets. In this paper theoretically well-grounded approaches to develop novel perturbation robust topological representations are presented, with the long-term view of making them amenable to fusion with contemporary learning architectures. The proposed representation lives on a Grassmann manifold and hence can be efficiently used in machine learning pipelines. The proposed representation.The efficacy of the proposed descriptor was explored on three applications: view-invariant activity analysis, 3D shape analysis, and non-linear dynamical modeling. Favorable results in both high-level recognition performance and improved performance in reduction of time-complexity when compared to other baseline methods are obtained. / Dissertation/Thesis / Masters Thesis Electrical Engineering 2017

Submitted by Marco Antonio de Freitas Contessoto (marco_contessoto@hotmail.com) on 2018-06-19T06:27:03Z No. of bitstreams: 1 Marcoeditado.pdf: 1251669 bytes, checksum: 5fe5c25a4002aeefa7831bd4137fb1f8 (MD5) / Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2018-06-19T14:26:54Z (GMT) No. of bitstreams: 1 contessoto_maf_me_sjrp.pdf: 1242012 bytes, checksum: e5b5acc9695b0f3103a68a1f1f32edac (MD5) / Made available in DSpace on 2018-06-19T14:26:54Z (GMT). No. of bitstreams: 1 contessoto_maf_me_sjrp.pdf: 1242012 bytes, checksum: e5b5acc9695b0f3103a68a1f1f32edac (MD5) Previous issue date: 2018-03-09 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / O objetivo dessa dissertação é apresentar parte do artigo [2] de Gunnar Carlsson, onde se discute a adaptação de métodos da teoria usual de homologia para problemas de reconhecimento topológico de padrões em conjuntos de dados. Esta adaptação conduz aos conceitos de homologia de persistência e de barcodes. Atualmente, várias aplicações são obtidas com o uso deste método. Apresentaremos alguns casos onde a homologia de persistência é usada, ilustrando diferentes modos em que podem ser aplicados. Descreveremos, também baseado no artigo de Carlsson, um novo método para estudar a persistência de características topológicas através de uma família de conjuntos de dados, chamado persistência zig-zag . Este método generaliza a teoria de homologia de persistência e chama atenção de situações que não são cobertas pela outra teoria. Além disso, são apresentadas algumas aplicações dessa ferramenta para a obtenção de informações de alguns conjuntos de dados / The main goal of this work is to present a part of the Gunnar Carlsson paper [2], where the adaptation of the theory of usual homology to topological pattern recognition problems in point cloud data sets is discussed. This adaptation leads to the concepts of persistence homology and barcodes. Several applications have been obtained using this method. We will present some cases where persistence homology is used, illustrating different ways in which the method can be applied. We will describe,alsobasedintheCarlsson’spaper,anewmethodtostudythepersistence oftopologicalfeaturesthroughpointclouddatasets,calledzig-zagpersistence. This method generalizes the homology persistent theory and we will pay attention to situations that are not covered by the other theory. In addition, some applications of this tool are presented to obtain information from some data sets. / 2016/25659-3

Submitted by Larissa Moura null (moura.larie@gmail.com) on 2018-02-26T11:39:11Z No. of bitstreams: 1 moura_larissa_sjrp.pdf: 1591130 bytes, checksum: 7997e476e0c0da8c86b51d6ce91c8898 (MD5) / Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2018-02-26T19:05:03Z (GMT) No. of bitstreams: 1 moura_l_me_sjrp.pdf: 1591130 bytes, checksum: 7997e476e0c0da8c86b51d6ce91c8898 (MD5) / Made available in DSpace on 2018-02-26T19:05:04Z (GMT). No. of bitstreams: 1 moura_l_me_sjrp.pdf: 1591130 bytes, checksum: 7997e476e0c0da8c86b51d6ce91c8898 (MD5) Previous issue date: 2018-02-16 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo desta dissertação é apresentar uma versão detalhada do artigo: “A Tutorial on Spectral Clustering” de U. von Luxburg sobre agrupamentos através de grafos Laplacianos, suas propriedades e mostrar alguns resultados da teoria de agrupamentos. Além disso, serão apresentados três algoritmos de agrupamentos e ilustraremos um deles com uma aplicação no cultivo da soja em diferentes condições de cultivo. / The main goal of this dissertation is to present a detailed version of the paper: “ A Tutorial on Spectral Clustering” of U. von Luxburg on clusters, through Laplacian graphs, their properties and to show some results of the cluster theory. In addition, it will be presented three clustering algorithms and we will illustrate one of them with an application in the soybean cultivation, under different conditions.

Dans le cadre de l'apprentissage automatique, l'utilisation de représentations alternatives, ou descripteurs, pour les données est un problème fondamental permettant d'améliorer sensiblement les résultats des algorithmes. Parmi eux, les descripteurs topologiques calculent et encodent l'information de nature topologique contenue dans les données géométriques. Ils ont pour avantage de bénéficier de nombreuses bonnes propriétés issues de la topologie, et désirables en pratique, comme par exemple leur invariance aux déformations continues des données. En revanche, la structure et les opérations nécessaires à de nombreuses méthodes d'apprentissage, comme les moyennes ou les produits scalaires, sont souvent absents de l'espace de ces descripteurs. Dans cette thèse, nous étudions en détail les propriétés métriques et statistiques des descripteurs topologiques les plus fréquents, à savoir les diagrammes de persistance et Mapper. En particulier, nous montrons que le Mapper, qui est empiriquement un descripteur instable, peut être stabilisé avec une métrique appropriée, que l'on utilise ensuite pour calculer des régions de confiance et pour régler automatiquement ses paramètres. En ce qui concerne les diagrammes de persistance, nous montrons que des produits scalaires peuvent être utilisés via des méthodes à noyaux, en définissant deux noyaux, ou plongements, dans des espaces de Hilbert en dimension finie et infinie. / In the context of supervised Machine Learning, finding alternate representations, or descriptors, for data is of primary interest since it can greatly enhance the performance of algorithms. Among them, topological descriptors focus on and encode the topological information contained in geometric data. One advantage of using these descriptors is that they enjoy many good and desireable properties, due to their topological nature. For instance, they are invariant to continuous deformations of data. However, the main drawback of these descriptors is that they often lack the structure and operations required by most Machine Learning algorithms, such as a means or scalar products. In this thesis, we study the metric and statistical properties of the most common topological descriptors, the persistence diagrams and the Mappers. In particular, we show that the Mapper, which is empirically instable, can be stabilized with an appropriate metric, that we use later on to conpute confidence regions and automatic tuning of its parameters. Concerning persistence diagrams, we show that scalar products can be defined with kernel methods by defining two kernels, or embeddings, into finite and infinite dimensional Hilbert spaces.

abstract: Over the past decade, machine learning research has made great strides and significant impact in several fields. Its success is greatly attributed to the development of effective machine learning algorithms like deep neural networks (a.k.a. deep learning), availability of large-scale databases and access to specialized hardware like Graphic Processing Units. When designing and training machine learning systems, researchers often assume access to large quantities of data that capture different possible variations. Variations in the data is needed to incorporate desired invariance and robustness properties in the machine learning system, especially in the case of deep learning algorithms. However, it is very difficult to gather such data in a real-world setting. For example, in certain medical/healthcare applications, it is very challenging to have access to data from all possible scenarios or with the necessary amount of variations as required to train the system. Additionally, the over-parameterized and unconstrained nature of deep neural networks can cause them to be poorly trained and in many cases over-confident which, in turn, can hamper their reliability and generalizability. This dissertation is a compendium of my research efforts to address the above challenges. I propose building invariant feature representations by wedding concepts from topological data analysis and Riemannian geometry, that automatically incorporate the desired invariance properties for different computer vision applications. I discuss how deep learning can be used to address some of the common challenges faced when working with topological data analysis methods. I describe alternative learning strategies based on unsupervised learning and transfer learning to address issues like dataset shifts and limited training data. Finally, I discuss my preliminary work on applying simple orthogonal constraints on deep learning feature representations to help develop more reliable and better calibrated models. / Dissertation/Thesis / Doctoral Dissertation Electrical Engineering 2020

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