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In recent years, Deep Learning has gained popularity for its ability to solve complex classification tasks, increasingly delivering better results thanks to the development of more accurate models, the availability of huge volumes of data and the improved computational capabilities of modern computers. However, these improvements in performance also bring efficiency problems, related to the storage of datasets and models, and to the waste of energy and time involved in both the training and inference processes. In this context, data reduction can help reduce energy consumption when training a deep learning model. In this paper, we present up to eight different methods to reduce the size of a tabular training dataset, and we develop a Python package to apply them. We also introduce a representativeness metric based on topology to measure how similar are the reduced datasets and the full training dataset. Additionally, we develop a methodology to apply these data reduction methods to image datasets for object detection tasks. Finally, we experimentally compare how these data reduction methods affect the representativeness of the reduced dataset, the energy consumption and the predictive performance of the model.

Simplicial map neural networks (SMNNs) are topology-based neural networks with interesting properties such as universal approximation ability and robustness to adversarial examples under appropriate conditions. However, SMNNs present some bottlenecks for their possible application in high-dimensional datasets. First, SMNNs have precomputed fixed weight and no SMNN training process has been defined so far, so they lack generalization ability. Second, SMNNs require the construction of a convex polytope surrounding the input dataset. In this paper, we overcome these issues by proposing an SMNN training procedure based on a support subset of the given dataset and replacing the construction of the convex polytope by a method based on projections to a hypersphere. In addition, the explainability capacity of SMNNs and effective implementation are also newly introduced in this paper.

En este trabajo fin de grado se estudian tantos los métodos actuales como el contexto en el ámbito de empresas, investigaciones y organizaciones que se cierne sobre el término Inteligencia Artificial Sostenible (Green AI) dando un enfoque más profundo a la reducción de datos en la fase de entrenamiento, ya que, al tener menos datos se realizarán menos operaciones y el entrenamiento será a la vez que más rápido, más sostenible. El documento presenta un análisis de trabajos relacionados abarcando desde empresas hasta artículos de investigación, pero introduciendo anteriormente los conceptos necesarios para su entendimiento. He realizado el diseño e implementación de dos métodos, el código se puede encontrar en GitHub, uno de los métodos que he implementado trata sobre cómo dividir el conjunto de manera representativa y el otro sobre cómo escoger datos con mayor información. Finalmente, hice una comparativa entre ambos métodos.

Attempts to incorporate topological information in supervised learning tasks have resulted in the creation of several techniques for vectorizing persistent homology barcodes. In this paper, we study thirteen such methods. Besides describing an organizational framework for these methods, we comprehensively benchmark them against three well-known classification tasks. Surprisingly, we discover that the best-performing method is a simple vectorization, which consists only of a few elementary summary statistics. Finally, we provide a convenient web application which has been designed to facilitate exploration and experimentation with various vectorization methods.

Data quality is crucial for the successful training, generalization and performance of artificial intelligence models. Furthermore, it is known that the leading approaches in artificial intelligence are notoriously data-hungry. In this paper, we propose the use of small training datasets towards faster training. Specifically, we provide a novel topological method based on morphisms between persistence modules to measure the training data quality with respect to the complete dataset. This way, we can provide an explanation of why the chosen training dataset will lead to poor performance.

Local Interpretable Model-Agnostic Explanations (LIME) are a well-known approach to provide local interpretability to Machine Learning models. LIME uses an exponential smoothing kernel based on the kernel width value, which defines the width of the local neighbourhood. In this paper, we study the influence of the distances for these local explanations, and we explore the choice of kernel width to guarantee a fair performance comparison between the distances.

The automatic recognition of a person’s emotional state has become a very active research field that involves scientists specialized in different areas such as artificial intelligence, computer vi sion, or psychology, among others. Our main objective in this work is to develop a novel approach, using persistent entropy and neural networks as main tools, to recognise and classify emotions from talking-face videos. Specifically, we combine audio-signal and image-sequence information to com pute a topology signature (a 9-dimensional vector) for each video. We prove that small changes in the video produce small changes in the signature, ensuring the stability of the method. These topological signatures are used to feed a neural network to distinguish between the following emotions: calm, happy, sad, angry, fearful, disgust, and surprised. The results reached are promising and competitive, beating the performances achieved in other state-of-the-art works found in the literature.

One of the main drawbacks of the practical use of neural networks is the long time required in the training process. Such a training process consists of an iterative change of parameters trying to minimize a loss function. These changes are driven by a dataset, which can be seen as a set of labeled points in an n-dimensional space. In this paper, we explore the concept of a representative dataset which is a dataset smaller than the original one, satisfying a nearness condition independent of isometric transformations. Representativeness is measured using persistence diagrams (a computational topology tool) due to its computational efficiency. We theoretically prove that the accuracy of a perceptron evaluated on the original dataset coincides with the accuracy of the neural network evaluated on the representative dataset when the neural network architecture is a perceptron, the loss function is the mean squared error, and certain conditions on the representativeness of the dataset are imposed. These theoretical results accompanied by experimentation open a door to reducing the size of the dataset to gain time in the training process of any neural network

En este trabajo, repasamos detalladamente el planteamiento de un problema de aprendizaje supervisado en general, sobre datos en forma de grafo en particular, y la construcción de la arquitectura de Graph Neural Networks para la resolución del mismo. En concreto, nos centramos en SEAL, una metodología basada en Graph Neural Networks para predicción de aristas. Además, explicaremos el método GAug-M para data augmentation en problemas de clasificación de nodos, que consiste en añadir o eliminar aristas en base a las probabilidades obtenidas de un modelo predictor de aristas, para enriquecer la información del grafo y aumentar el rendimiento del modelo de clasificación. En este TFM se propone un nuevo uso de la metodología SEAL como modelo predictor de aristas junto con GAug-M para realizar data augmentation en un grafo. Evaluaremos el rendimiento de usar esta combinación, y lo compararemos con el rendimiento obtenido en [37] con otro modelo predictor de aristas, en los mismos datasets y con el mismo modelo de clasificación de nodos., In this work, we explain in detail the formulation of a supervised learning problem in general, over graph data in particular, and the arquitecture of Graph Neural Networks for solving them. In particular, we focus on SEAL, a Graph Neural Network framework for predicting the existence of a link between nodes. We also talk about GAug-M, a method for data augmentation in node classification problems, which consists in adding or removing links based in precalculated probabilities, in order to improve the graph information, and also improve the performance of the node classification model. In this work, we propose a new use of SEAL as the link predictor, together with GAug-M for performing data augmentation in a graph. We evaluate the performance of this combination, and we compare it with the performance obtained in [37] with a different link prediction model, in the same datasets and with the same model for node classification. Moreover, we explain how to use this arquitecture for performing data augmentation on a graph, whose nodes we want to classify, using the GAug-M method, explained and tested in [37] with other link prediction arquitectures different from SEAL. . We evaluate the performance when using SEAL as the link predictor, and we compare it with the efficiency obtained in [37] by using other predictor models, in the same datasets and with the same node classification model.

A continuación, se expone en este documento la memoria del trabajo de fin de grado basado en el análisis de datos de COVID-19 en España a través del algoritmo BallMapper. La idea general de este trabajo se basa en sacar conclusiones que no habían sido expuestas anteriormente sobre el COVID-19, gracias a la aplicación del algoritmo mencionado, y confirmar otras que ya se habían hallado. Para conseguir esto, antes, se han debido escoger los conjuntos de datos sobre los que realizar las pruebas, realizando una limpieza y preprocesado de los datos y aplicando finalmente el algoritmo BallMapper. Tras la aplicación, se han analizado los resultados obtenidos para concluir con las ideas generales del estudio. Gracias a este estudio se han conseguido hallar nuevas deducciones sobre la situación de pandemia vivida durante el año 2020 y que aún sigue muy presente en nuestras vidas., Next, the report of the final degree project based on the analysis of COVID-19 data in Spain through the BallMapper algorithm is exposed in this document. The general idea of this work is based on drawing conclusions that have not been previously exposed about COVID-19, thanks to the application of the aforementioned algorithm, and confirming others that have already been found. To achieve this, before, the data sets on which the tests were carried out had to be chosen, cleaning and preprocessing the data and finally applying the BallMapper algorithm. After the application, the results obtained have been analyzed to conclude with the general ideas of the study. Thanks to this study, new deductions have been obtained about the pandemic situation experienced during the year 2020 and that is still very present in our lives.

Broadly speaking, an adversarial example against a classification model occurs when a small perturbation on an input data point produces a change on the output label assigned by the model. Such adversarial examples represent a weakness for the safety of neural network applications, and many different solutions have been proposed for minimizing their effects. In this paper, we propose a new approach by means of a family of neural networks called simplicial-map neural networks constructed from an Algebraic Topology perspective. Our proposal is based on three main ideas. Firstly, given a classification problem, both the input dataset and its set of one-hot labels will be endowed with simplicial complex structures, and a simplicial map between such complexes will be defined. Secondly, a neural network characterizing the classification problem will be built from such a simplicial map. Finally, by considering barycentric subdivisions of the simplicial complexes, a decision boundary will be computed to make the neural network robust to adversarial attacks of a given size.

Simplicial-map neural networks are a recent neural network architecture induced by simplicial maps defined between simplicial complexes. It has been proved that simplicial-map neural networks are universal approximators and that they can be refined to be robust to adversarial attacks. In this paper, the refinement toward robustness is optimized by reducing the number of simplices (i.e., nodes) needed. We have shown experimentally that such a refined neural network is equivalent to the original network as a classification tool but requires much less storage.

In this paper, we define a new flavour of well-composedness, called strong Euler well composedness. In the general setting of regular cell complexes, a regular cell complex of dimension n is strongly Euler well-composed if the Euler characteristic of the link of each boundary cell is 1, which is the Euler characteristic of an (n−1)-dimensional ball. Working in the particular setting of cubical complexes canonically associated with nD pictures, we formally prove in this paper that strong Euler well-composedness implies digital well-composedness in any dimension n ≥ 2 and that the converse is not true when n ≥ 4

Machine learning is drawn from a dataset that needs to be explained. Following this aim, a model is described to make predictions. These datasets can be of very different nature, as well as the problems they pose: supervised or unsupervised classification, regression... In this work, we have studied, from a computational topology point of view, two machine learning’s pillars: the datasets considered as sets of =-dimensional points, and a specific model, artificial neural networks. we have applied topological data analysis techniques to classify literary trends from a literature dataset and to reduce the size of datasets controlling the loss of information, and, finally, we have developed a new neural network architecture based on simplicial complexes and the maps defined between them, and we have also proved certain properties such as that the new family of neural networks are universal approximators and robust to "adversarial examples"., En aprendizaje automático partimos de un conjunto de datos que queremos explicar. Con ese objetivo, se diseña un modelo que permite hacer predicciones. Estos conjuntos de datos pueden ser de muy distinta naturaleza, así como los problemas que plantean: clasificación supervisada o no supervisada, regresión... En este trabajo, hemos estudiado, desde el punto de vista de la topología computacional, dos pilares del aprendizaje automático: los conjuntos de datos vistos como un conjunto de puntos =-dimensionales; y un modelo en concreto, las redes neuronales artificales. Más concretamente, hemos aplicado técnicas propias del análisis topológico de datos para clasificar tendencias literarias de un conjunto de datos literario, hemos estudiado cómo reducir un conjunto de datos de manera que podamos controlar la información que perdemos y, finalmente, hemos desarrollado una nueva arquitectura de red neuronal basada en complejos simpliciales y en las aplicaciones que se definen entre ellos, demostrando que esta nueva familia de redes neuronales son aproximadores universales y robustas ante "adversarial examples".

In this paper, we de ne a new avour of well-composedness, called Euler well-composedness, in the general setting of regular cell complexes: A regular cell complex is Euler well-composed if the Euler characteristic of the link of each boundary vertex is 1. A cell decomposi- tion of a picture I is a pair of regular cell complexes K(I), K( I) such that K(I) (resp. K( I)) is a topological and geometrical model represent- ing I (resp. its complementary, I). Then, a cell decomposition of a pic- ture I is self-dual Euler well-composed if both K(I) and K( I) are Euler well-composed. We prove in this paper that, rst, self-dual Euler well- composedness is equivalent to digital well-composedness in dimension 2 and 3, and second, in dimension 4, self-dual Euler well-composedness implies digital well-composedness, though the converse is not true.

Filtration simplification consists of simplifying a given filtration while simultaneously controlling the perturbation in the associated persistence diagrams. In this paper, we propose a filtration simplification algorithm for orientable 2-dimensional (2D) manifolds with or without boundary ( meshes ) represented by2D combinatorial maps. Given a lower-star filtration of the mesh, faces are added into contiguous clusters according to a “height” function and a parameter . Faces in the same cluster are merged into a single face, resulting in a lower resolution mesh and a simpler filtration. We prove that the parameter bounds the perturbation in the original persistence diagrams, and we provide experiments demonstrating thecomputational advantages of the simplification process.

This paper proposes a method for learning the process of curve fitting through a general class of totally positive rational bases. The approximation is achieved by finding suitable weights and control points to fit the given set of data points using a neural network and a training algorithm, called AdaMax algorithm, which is a first-order gradient-based stochastic optimization. The neural network presented in this paper is novel and based on a recent generalization of rational curves which inherit geometric properties and algorithms of the traditional rational Bézier curves. The neural network has been applied to different kinds of datasets and it has been compared with the traditional least-squares method to test its performance. The obtained results show that our method can generate a satisfactory approximation.

Emotion recognition consists of a series of processes to detect human emotions from facial human expressions. Humans interact with each other primarily through speech, but also through body gestures to emphasize a certain part of the conversation and exhibit emotions. The automatic recognition of a person’s emotional state has become a very active research eld that involves scientists specialized in di erent areas such as arti cial intelligence, computer vision or psychology, among others. Our main objective in thiswork is to develop a novel approach, based on topological data analysis, for the recognition and classi cation of emotions that combines features extracted from videos and audios, in their respective spaces of representation, of people expressing emotions., El área de reconocimiento de emociones consiste en una serie de procesos para detectar emociones humanas a partir de expresiones faciales. Los humanos interactuamos entre nosotros utilizando el habla, pero también a través de los gestos corporales para así enfatizar una cierta parte de la conversación y exhibir emociones. El reconocimiento automático del estado emocional de una persona se ha convertido en un campo activo de investigación que involucra científicos especializados en diferentes áreas tales como la inteligencia artificial, la visión por ordenador o la psicología, entre otras. Nuestro principal objetivo en este trabajo es desarrollar un novedoso enfoque, basado en el análisis topológico de datos, para el reconocimiento y clasificación de emociones que combine características extraídas de videos y audios, en sus respectivos espacios de representación, de personas expresando emociones.

It is well-known that artificial neural networks are universal approximators. The classical existence result proves that, given a continuous function on a compact set embedded in an n-dimensional space, there exists a one-hidden-layer feed-forward network that approximates the function. In this paper, a constructive approach to this problem is given for the case of a continuous function on triangulated spaces. Once a triangulation of the space is given, a two-hidden-layer feed-forward network with a concrete set of weights is computed. The level of the approximation depends on the refinement of the triangulation.

The canon of the baroque Spanish literature has been thoroughly studied with philological techniques. The major representatives of the poetry of this epoch are Francisco de Quevedo and Luis de Góngora y Argote. They are commonly classified by the literary experts in two different streams: Quevedo belongs to the Conceptismo and Góngora to the Culteranismo. Besides, traditionally, even if Quevedo is considered the most representative of the Conceptismo, Lope de Vega is also considered to be, at least, closely related to this literary trend. In this paper, we use Topological Data Analysis techniques to provide a first approach to a metric distance between the literary style of these poets. As a consequence, we reach results that are under the literary experts’ criteria, locating the literary style of Lope de Vega, closer to the one of Quevedo than to the one of Góngora.

It is well known that Artificial Neural Networks are universal approximators. The classical result proves that, given a continuous function on a compact set on an n-dimensional space, then there exists a one-hidden-layer feedforward network which approximates the function. Such result proves the existence, but it does not provide a method for finding it. In this paper, a constructive approach to the proof of this property is given for the case of two-hidden-layer feedforward networks. This approach is based on an approximation of continuous functions by simplicial maps. Once a triangulation of the space is given, a concrete architecture and set of weights can be obtained. The quality of the approximation depends on the refinement of the covering of the space by simplicial complexes.

Emotion recognition and classification is a very active area of research. In this paper, we present a first approach to emotion classification using persistent entropy and support vector machines. A topology-based model is applied to obtain a single real number from each raw signal. These data are used as input of a support vector machine to classify signals into 8 different emotions (calm, happy, sad, angry, fearful, disgust and surprised).

One of the main drawbacks of the practical use of neural networks is the long time needed in the training process. Such training process consists in an iterative change of parameters trying to minimize a loss function. These changes are driven by a dataset, which can be seen as a set of labeled points in an n-dimensional space. In this paper, we explore the concept of representative dataset which is smaller than the original dataset and satisfies a nearness condition independent of isometric transformations. The representativeness is measured using persistence diagrams due to its computational efficiency. We also prove that the accuracy of the learning process of a neural network on a representative dataset is comparable with the accuracy on the original dataset when the neural network architecture is a perceptron and the loss function is the mean squared error. These theoretical results accompanied with experimentation open a door to the size reduction of the dataset in order to gain time in the training process of any neural network.

Neural networks present big popularity and success in many fields. The large training time process problem is a very important task nowadays. In this paper, a new approach to get over this issue based on reducing dataset size is proposed. Two algorithms covering two different shape notions are shown and experimental results are given.