Eduardo Paluzo-Hidalgo

Álvaro Torras-Casas, Eduardo Paluzo-Hidalgo, Rocio Gonzalez-Diaz

Data quality is crucial for the successful training, generalization and performance of machine learning models. We propose to measure the quality of a subset concerning the dataset it represents, using topological data analysis techniques. Specifically, we define the persistence matching diagram, a topological invariant derived from combining embeddings with persistent homology. We provide an algorithm to compute it using minimum spanning trees. Also, the invariant allows us to understand whether the subset ``represents well" the clusters from the larger dataset or not, and we also use it to estimate bounds for the Hausdorff distance between the subset and the complete dataset. In particular, this approach enables us to explain why the chosen subset is likely to result in poor performance of a supervised learning model.

Eduardo Paluzo-Hidalgo, Yuichi Ike

We introduce a theoretical framework that connects multi-chart autoencoders in manifold learning with the classical theory of vector bundles and characteristic classes. Rather than viewing autoencoders as producing a single global Euclidean embedding, we treat a collection of locally trained encoder-decoder pairs as a learned atlas on a manifold. We show that any reconstruction-consistent autoencoder atlas canonically defines transition maps satisfying the cocycle condition, and that linearising these transition maps yields a vector bundle coinciding with the tangent bundle when the latent dimension matches the intrinsic dimension of the manifold. This construction provides direct access to differential-topological invariants of the data. In particular, we show that the first Stiefel-Whitney class can be computed from the signs of the Jacobians of learned transition maps, yielding an algorithmic criterion for detecting orientability. We also show that non-trivial characteristic classes provide obstructions to single-chart representations, and that the minimum number of autoencoder charts is determined by the good cover structure of the manifold. Finally, we apply our methodology to low-dimensional orientable and non-orientable manifolds, as well as to a non-orientable high-dimensional image dataset.

Víctor Toscano-Durán, Javier Perera-Lago, Eduardo Paluzo-Hidalgo et al.

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.

Javier Perera-Lago, Víctor Toscano-Durán, Eduardo Paluzo-Hidalgo et al.

Machine learning algorithms are fundamental components of novel data-informed Artificial Intelligence architecture. In this domain, the imperative role of representative datasets is a cornerstone in shaping the trajectory of artificial intelligence (AI) development. Representative datasets are needed to train machine learning components properly. Proper training has multiple impacts: it reduces the final model's complexity, power, and uncertainties. In this paper, we investigate the reliability of the $\varepsilon$-representativeness method to assess the dataset similarity from a theoretical perspective for decision trees. We decided to focus on the family of decision trees because it includes a wide variety of models known to be explainable. Thus, in this paper, we provide a result guaranteeing that if two datasets are related by $\varepsilon$-representativeness, i.e., both of them have points closer than $\varepsilon$, then the predictions by the classic decision tree are similar. Experimentally, we have also tested that $\varepsilon$-representativeness presents a significant correlation with the ordering of the feature importance. Moreover, we extend the results experimentally in the context of unseen vehicle collision data for XGboost, a machine-learning component widely adopted for dealing with tabular data.

Eduardo Paluzo-Hidalgo

This paper introduces a topological framework for interpreting the internal representations of Multilayer Perceptrons (MLPs). We construct a simplicial tower, a sequence of simplicial complexes connected by simplicial maps, that captures how data topology evolves across network layers. Our approach enables bi-persistence analysis: layer persistence tracks topological features within each layer across scales, while MLP persistence reveals how these features transform through the network. We prove stability theorems for our topological descriptors and establish that linear separability in latent spaces is related to disconnected components in the nerve complexes. To make our framework practical, we develop a combinatorial algorithm for computing MLP persistence and introduce trajectory-based visualisations that track data flow through the network. Experiments on synthetic and real-world medical data demonstrate our method's ability to identify redundant layers, reveal critical topological transitions, and provide interpretable insights into how MLPs progressively organise data for classification.

Rocio Gonzalez-Diaz, Miguel A. Gutiérrez-Naranjo, Eduardo Paluzo-Hidalgo

In this paper, we present SIMAP, a novel layer integrated into deep learning models, aimed at enhancing the interpretability of the output. The SIMAP layer is an enhanced version of Simplicial-Map Neural Networks (SMNNs), an explainable neural network based on support sets and simplicial maps (functions used in topology to transform shapes while preserving their structural connectivity). The novelty of the methodology proposed in this paper is two-fold: Firstly, SIMAP layers work in combination with other deep learning architectures as an interpretable layer substituting classic dense final layers. Secondly, unlike SMNNs, the support set is based on a fixed maximal simplex, the barycentric subdivision being efficiently computed with a matrix-based multiplication algorithm.

Eduardo Paluzo-Hidalgo, Miguel A. Gutiérrez-Naranjo, Rocio Gonzalez-Diaz

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 an effective implementation are also newly introduced in this paper.

Eduardo Paluzo-Hidalgo, Guillermo Aguirre-Carrazana, Rocio Gonzalez-Diaz

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 vision 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 compute a topology signature(a 9-dimensional vector) for each video. We prove that small changes in the video produce small changes in the signature. These topological signatures are used to feed a neural network to distinguish between the following emotions: neutral, calm, happy, sad, angry, fearful, disgust, and surprised. The results reached are promising and competitive, beating the performance reached in other state-of-the-art works found in the literature.

Eduardo Paluzo-Hidalgo, Rocio Gonzalez-Diaz, Miguel A. Gutiérrez-Naranjo

In this paper, we use topological data analysis (TDA) tools such as persistent homology, persistent entropy and bottleneck distance, to provide a {\it TDA-based summary} of any given set of texts and a general method for computing a distance between any two literary styles, authors or periods. To this aim, deep-learning word-embedding techniques are combined with these tools in order to study the topological properties of texts embedded in a metric space. As a case of study, we use the written texts of three poets of the Spanish Golden Age: Francisco de Quevedo, Luis de Góngora and Lope de Vega. As far as we know, this is the first time that word embedding, bottleneck distance, persistent homology and persistent entropy are used together to characterize texts and to compare different literary styles.

Rocio Gonzalez-Diaz, Miguel A. Gutiérrez-Naranjo, Eduardo Paluzo-Hidalgo

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.

Rocio Gonzalez-Diaz, Miguel A. Gutiérrez-Naranjo, Eduardo Paluzo-Hidalgo

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 labelled points in an n-dimensional space. In this paper, we explore the concept of are 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 prove that the accuracy of the learning process of a neural network on a representative dataset is "similar" to 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 by experimentation open a door to reducing the size of the dataset to gain time in the training process of any neural network.