No description available.

Dans un contexte où des quantités toujours plus colossales de données sont disponibles,extraire des informations significatives et non triviales devient toujours plus difficile. Afin d’améliorer la classification, régression, ou encore l’analyse exploratoire de données, l’approche fournie par l’analyse topologique de données (TDA) est de rechercher la présence de formes dans le jeu de données.Dans cette thèse nous étudions les propriétés des modules de persistance multidimensionnelle dans le but d’obtenir une meilleure compréhension des sommandes et décompositions de ces derniers. Nous introduisons un foncteur qui plonge la catégorie des représentations de carquois dont le graphe est un arbre enraciné dans la catégorie des modules de persistance indexé sur ℝ². Nous enrichissons la structure de module de persistance provenant de l’application du foncteur cohomologie à une filtration en une structure d’algèbre de persistance.Enfin, nous généralisons l’approche de Crawley Beovey à la multipersistance et identifions une classe de modules de persistance indexé sur ℝ² qui possède des descripteurs simples et analogues au théorème de décomposition existant en persistance1-dimensionnelle. / In a context where huge amounts of data are available, extracting meaningful and non trivial information is getting harder. In order to improve the tasks of classification, regression, or exploratory analysis, the approach provided by topological data analysisis to look for the presence of shapes in data set.In this thesis, we investigate the properties of multidimensional persistence modules in order to obtain a better understanding of the summands and decompositions of such modules. We introduce a functor that embeds the representations category of any quiver whose graph is a rooted tree into the category of ℝ²-indexed persistence modules. We also enrich the structure of persistence module arising from the cohomology of a filtration to a structure of persistence algebra.Finally, we generalize the approach of Crawley Beovey to multipersistence and identify a class of persistencemodules indexed on ℝ² which have simple descriptor and an analog of the decomposition theorem available in one dimensional persistence.

Orientador: Thiago de Melo / Resumo: Neste trabalho estudamos certos funtores sobre grafos, chamados de representáveis ou motívicos. Esses funtores não mudam os vértices de um grafo, mas apenas suas setas (as arestas direcionadas). Quaisquer tais funtores podem ser estendidos para networks (uma generalização de espaços métricos). Funtores de clustering sobre grafos dão origem a funtores de hierarchical clustering sobre networks. Mais ainda, podemos modificar a definição de funtor representável para criar filtrações de complexos simpliciais, que tem como caso particular os complexos de Vietoris-Rips e Cech. Isso faz com que possamos aplicar o funtor de homologia ˇ simplicial e obter um diagrama de persistência, como usual em Análise Topológica de Dados. Obtivemos resultados de estabilidade com respeito à distância bottleneck e à distância network, quando uma certa condição é imposta nos motivos de um funtor representável. Algumas operações sobre grafos (e.g., produtos e suspensão) também podem ser estendidas para networks, e três fórmulas de Künneth foram obtidas. Finalmente, alguns algoritmos e códigos para casos especiais são fornecidos com exemplos. / Doutor

A ciliate is a phylum of protozoa that has two types of nuclei, macronuclei and micronuclei. There may be more than one of each type of nucleus in the organism [1]. The macronucleus is the structure where protein synthesis and cell metabolism occur [1]. The micronucleus stores genetic information and is mobilized during a sexual reproduction process called conjugation [1]. The somatic macronucleus (MAC) is developed from the germ-line micronucleus (MIC) through genome rearrangement during a sexual reproduction process called conjugation [6, 8]. Segments of the MIC that form the MAC during conjugation are called macronuclear destined sequences (MDSs) [8]. During sequencing each MDS is given coordinates where the MDS sequences begin and end in the MIC. The orientation of a MDS in the MIC can be taken to be positive or negative. If the direction of the MDS in the MIC agrees with the direction in the MAC then the orientation is positive otherwise it is a negative orientation. In this thesis we analyze various aspects of the gene assembly during the rearrangment process of the ciliate Oxytricha trifallax that were recently sequenced [15]. Some of the properties analyzed include overlapping MDSs, orientation, MDSs starting and ending position in the MIC and the gaps of overlapping MDS pairs. A gap of an overlapping MDS pair is the order difference of two MDSs for a particular MAC contig that overlap in the MIC contig. We use 120 MAC contigs from [15] that have overlaps among their own MDSs. These 120 MAC contigs make up the data set we call D4. We explore the patterns of overlapping MDSs in the MIC in D4. To quantify such patterns, we associate a vector V (An) to each MAC contig An, where V (An) = (v1(An), v2(An), v3(An)) is a vector in R3. The first entry is the number of overlapping MDS pairs divided by the number of MDSs. The second entry is the sum of gaps of overlapping MDS pairs divided by the sum of all possible gaps. The final entry is the total number of overlapping base pairs divided by the total length of the MAC contig. We computed the distance matrixM = (dij) where dij is the Euclidean distance between V (Ai) and V (Aj). The MAC contig vectors and M were computed using Python. To analyze D4 we applied Topological Data Analysis (TDA). TDA uses topological constructs to assess shapes in data [3, 12]. From the data entries of the distance matrix M = (dij) we applied a Vietoris-Rips filtration to generate the barcodes of the persistent homology in dimensions 0, 1 and 2. The persistence barcode of 0-dimensional homology illustrates clusters of the data while the 1-dimensional homology represents non-trivial loops in the simplicial complex [3, 13]. The application of TDA on the ciliate Oxytricha trifallax identified ten MAC contig clusters at epsilon= 0.1 in D4 and several loops that were persistent for two or three epsilon values. Other TDA methods can be applied to the Vietoris-Rips filtration to further identify which MAC contigs appear in each cluster.

No description available.

No description available.

No description available.

No description available.

No description available.

Artificial neural networks at the present time gain notable popularity and show astounding results in many machine learning tasks. This, however, also results in a drawback that the understanding of the processes happening inside of learning algorithms decreases. In many cases, the process of choosing a neural network architecture for a problem comes down to selection of network layers by intuition and to manual tuning of network parameters. Therefore, it is important to build a strong theoretical base in this area, both to try to reduce the amount of manual work in the future and to get a better understanding of capabilities of neural networks. In this master thesis, the ideas of applying different topological and geometric methods for the analysis of neural networks were investigated. Despite the difficulties which arise from the novelty of the approach, such as limited amount of related studies, some promising methods of network analysis were established and tested on baseline machine learning datasets. One of the most notable results of the study reveals how neural networks preserve topological features of the data when it is projected into space with low dimensionality. For example, the persistence for MNIST dataset with added rotations of images gets preserved after the projection into 3D space with the use of simple autoencoders; on the other hand, autoencoders with a relatively high weight regularization parameter might be losing this ability. / Artificiella neuronnät har för närvarande uppnått märkbar popularitet och visar häpnadsväckande resultat i många maskininlärningsuppgifter. Dock leder detta också till nackdelen att förståelsen av de processer som sker inom inlärningsalgoritmerna minskar. I många fall måste man använda intuition och ställa in parametrar manuellt under processen att välja en nätverksarkitektur. Därför är det viktigt att bygga en stark teoretisk bas inom detta område, både för att försöka minska manuellt arbete i framtiden och för att få en bättre förståelse för kapaciteten hos neuronnät. I detta examensarbete undersöktes idéerna om att tillämpa olika topologiska och geometriska metoder för analys av neuronnät. Många av svårigheterna härrör från det nya tillvägagångssättet, såsom en begränsad mängd av relaterade studier, men några lovande nätverksanalysmetoder upprättades och testades på standarddatauppsättningar för maskininlärning. Ett av de mest anmärkningsvärda resultaten av examensarbetet visar hur neurala nätverk bevarar de topologiska egenskaperna hos data när den projiceras till vektorrum med låg dimensionalitet. Till exempel bevaras den topologiska persistensen för MNIST-datasetet med tillagda rotationer av bilder efter projektion i ett tredimensionellt vektorrum med användning av en basal autoencoder; å andra sidan kan autoencoders med en relativt hög viktregleringsparameter förlora denna egenskap.