Tesis Doctorals - Departament - Matemàtiques i Informàtica

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Methods and Benchmarks for Trustworthy AI in Breast Imaging
(Universitat de Barcelona, 2025-12-17) Garrucho Moras, Lidia; Lekadir, Karim, 1977-; Igual Muñoz, Laura; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] Breast cancer remains one of the most pressing global health challenges, with early detection and effective treatment planning relying heavily on advanced imaging and accurate interpretation. This PhD thesis addresses critical limitations in artificial intelligence (AI) applications for breast cancer imaging, specifically the lack of generalisability across clinical centres and the need for improved fairness across patient subgroups. The work focuses on two key modalities: digital mammography for early lesion detection, and dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) for tumor segmentation and treatment response prediction. Four core contributions are presented: (1) a deep learning-based domain generalization framework for robust mass detection across unseen mammography domains; (2) a GAN-based augmentation method to improve detection in women with dense breast tissue, addressing a known fairness gap; (3) an image synthesis and domain adaptation strategy to harmonize MRI protocol variability and improve tumor segmentation; and (4) the development of a large-scale, multi-centre breast DCEMRI dataset with expert annotations and clinical outcome labels. These efforts culminated in the organization of the international MAMA-MIA Challenge (MICCAI 2025)—the first benchmark to evaluate breast MRI segmentation and treatment response prediction with integrated subgroup-aware fairness metrics. Together, these contributions demonstrate that targeted algorithmic innovation, inclusive data design, and open benchmarking can substantially enhance the robustness, equity, and translational potential of AI in breast cancer imaging. This thesis lays the foundation for next-generation decision support systems that are not only technically reliable, but also ethically aligned with the goals of personalized and equitable cancer care.
Abelian Varieties of GLn-type and Galois Representations
(Universitat de Barcelona, 2025-11-28) Florit Zacarías, Enric; Dieulefait, L. V. (Luis Victor); Fité Naya, Francesc; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] In this thesis, the artimetic properties of abelian manifolds are considered in relation to their algebras of endomorphisms. More precisely, we study the good reductions of an abelian manifold defined over a field of numbers, as well as those of associated Galois representations. We also give some results of modularity of abelian manifolds with respect to modular Siegel forms. Chapter 1 gives some definitions and preliminary results on simple algebras and abelian manifolds. The work itself begins with Chapter 2, where we study the immersions of simple algebras. We have placed special emphasis on characterizing the existence of an immersion between two simple algebras. We give a specialization of a Chia-Fu Yu criterion for algebras on global and local bodies, which plays an important role in Chapters 3, 5 and 7. Chapter 3 studies the properties of abelian manifolds defined on finite fields under the hypothesis that the algebra of endomorphisms is noncommutative. We focus on classifying the 4-abelian manifolds with quaternionic multiplication. We prove the following result: an abelian manifold over a field of numbers with noncommutative endomorphisms is a non-simple modulus all first out of a finite set. This statement generalizes the analogous result for the so-called "false elliptical curves", and makes one of the directions of the Murty and Patankar Conjecture more precise. On the other hand, we also give an example of a 4-abelian manifold with exactly two geometrically simple reductions. In Chapter 4, we begin the description of Galois representations associated with abelian manifolds with some non-integer endomorphism. We describe the irreducible components of the Tate modulus in terms of the algebra of endomorphisms, and prove that Galois representations take values in a form of the algebraic group GLn. We also explain the nature of Weil's pairing in these irreducible components, which depends on the Albert type of the algebra of endomorphisms. Chapter 5 is based on a joint work with F. Fité and X. Guitart. We define the notion of genuinely GLn-type abelian manifold, generalizing the GL2-type abelian manifolds without potential complex multiplication considered by Ribet. The system of representations associated with these manifolds has the property of being absolutely irreducible when making a change of basis to a finite extension. We give a theory of building blocks for these varieties. Under a certain technical hypothesis, we also define the inner twists and the character of an abelian variety, called Nebentipus. We characterize the manifolds with symplectic or orthogonal Galois representation, which we genuinely call GSpn or GOn type, respectively. In this way, we extend the class of abelian varieties with such representations given earlier by Banaszak, Gajda and Krasoń. Finally, we show that a genuinely GL4 type abelian manifold is Siegel's modular if and only if it is genuinely GSp4 type. In Chapter 6 we construct a GL4 family of building blocks. These are given by the Jacobian women of certain curves of genus 2 with a Richelot isogeny in their Galois conjugates. Under certain conditions, Weil's constraint gives us examples of 4-abelian manifolds genuinely of type GSp4. The family includes examples of images of Galois representations in GSp4 and non-trivial Nebentype. Chapter 7 is based on a joint work with A. Pacetti. It deals with Galois representations associated with abelian k-manifolds. One of the main points is that we do not assume that all endomorphisms of the manifold are defined on the base field. We give a procedure to construct a representation of the absolute Galois group of k associated with a k-abelian manifold. In addition, we give some results on the induced Weil pairing to these representations. As an application, we demonstrate that abelian surfaces over Q with potential quaternionic multiplication are Siegel modulars.
Contributions to the Theory of Large Cardinals Beyond Choice
(Universitat de Barcelona, 2025-10-30) Mohammd, Marwan Salam; Bagaria, Joan; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] This thesis investigates large cardinals that are inconsistent with the Axiom of Choice. First, we characterize Berkeley cardinals in terms of a restricted form of Vopěnka’s Principle, and determine the consistency strength of several related theories. Next, we present a method for producing elementary embeddings from homomorphisms, which is then used to show that the Strongly Rigid Relation Principle is a weak Choice principle. We also provide a characterization of rank-Berkeley cardinals in terms of a strong failure of this principle. We then explore the connection between elementary embeddings from the universe into itself and eventually dominating functions, culminating in an alternative proof of Kunen’s Inconsistency Theorem. Finally, using the method of forcing, we establish the consistency (relative to large cardinals) of the successor of the first singular cardinal being supercompact in the transitive model of Hereditarily Ordinal Definable sets.
Constructive Methods in KAM Theory for Quasi-Periodic Time-Dependent Hamiltonian Systems and Applications
(Universitat de Barcelona, 2025-11-04) Porras Flores, Pedro; Calleja Castillo, Renato Carlos; Haro, Àlex; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] In this work, we prove a KAM theorem and present an algorithm formulated in an a-posteriori format, using the parameterization method to look invariant tori in non-autonomous Hamiltonian systems with n degrees of freedom that depend periodically or quasi-periodically (QP) on time, with f external frequencies. Such a system is described by a Hamiltonian function in the 2n-dimensional phase space, M, that depends also on f angles, ϕ ∈ TR. We take advantage of the fibbered structure of the extended phase space M × TR. As a result of our approach, the parameterization of tori requires the last f variables, to be precise ϕ, while the first 2n components are determined by an invariance equation. This reduction decreases the dimension of the problem where the unknown is a parameterization from 2(n + f) to 2n. We employ a quasi-Newton method, in order to prove the KAM theorem. This iterative method begins with an initial parameterization of an approximately invariant torus, meaning it approximately satisfies the invariance equation. The approximation is refined by applying corrections that reduce quadratically the invariance equation error. This process converges to a torus in a complex strip of size ρ∞, provided suitable Diophantine (γ, τ ) conditions and a non-degeneracy condition on the torsion are met. Given the nature of the proof, this provides a numerical method that can be effectively implemented on a computer, We exhibit the algorithm with two models. The first is a Tokamak model [CVC+05, VL21], which proposes a control method to create barriers to the diffusion of magnetic field lines through a small modification in the magnetic perturbation. The second model [dCN00], known as the vorticity defect model, describes the nonlinear evolution of localized vorticity perturbations in a constant vorticity flow. This model was originally derived in the context of plasma physics and fluid dynamics.
Astronomical observation Scheduling Problem: a comprehensive study and novel metaheuristic solutions
(Universitat de Barcelona, 2025-11-12) Nakhjiri, Nariman; Salamó Llorente, Maria; Sànchez i Marrè, Miquel, 1964-; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] As the complexity of large-scale astronomical surveys increases, the need for intelligent and adaptive scheduling systems has become critical to maximizing scientific return. The Astronomical Observation Scheduling (AOS) problem represents a complex and highly constrained case of combinatorial optimization, characterized by strict computational time limits and frequent changes. Its unique structure and challenges motivate focused research to develop flexible and scalable scheduling solutions. This research is organized into two main phases, which together comprise its four core contributions. The first phase focuses on heuristic research with the aim of developing efficient heuristic strategies that effectively address the specific constraints and structure of the AOS problem. The first core contribution is the introduction of the Conflict Resolution Unit (CRU) heuristic algorithm and its variants, designed to fulfill the objectives of this phase. The second and principal phase focuses on metaheuristic research, aiming to design algorithms that are both flexible and scalable, and capable of addressing the diverse level of complexities and changes in AOS. To ensure that these metaheuristics are well-adapted to the problem, they incorporate the heuristics developed in the first phase as core components. The second contribution is the Accumulative Planner (AP) algorithm, which integrates the CRU heuristic with a greedy strategy to form a fast, primarily local optimization algorithm for AOS, with competent results. These results were used as a baseline for further improvements. The third core contribution is the Hybrid Accumulative Planner (HAP) algorithm, developed to overcome the limitations of AP. HAP uses a modified version of CRU and a multi-start strategy to enable a broader and more robust search process. The fourth and final core contribution is the Forgetful Swarm Optimization (FSO) algorithm. It combines another CRU variant with a Destroy-and-Repair strategy and a Swarm Intelligence framework to deliver a capable global optimization method. FSO is designed to balance the search in exploration and exploitation, achieving high-quality results within a reasonable computational time, while preserving the precision of domain-specific heuristics. The core contributions are adapted to a real-world example of AOS problems and evaluated using its available datasets. These datasets present a variety of test scenarios with diverse characteristics, highlighting the challenges that the algorithms must address. Besides the core contributions, the evaluation includes other adapted algorithms for this real-world problem to provide a better perspective on relative performance. These include an Evolutionary Algorithm, an Iterated Local Search, and a Hill-Climbing Greedy. The results show the effectiveness of the proposed heuristic, CRU, and its variants in handling different tasks and constraints of AOS. Furthermore, all metaheuristics that leverage CRU as a core component produce high-quality solutions. The mostly local optimization algorithm of AP competes with global approaches in terms of solution quality, even surpassing them in some cases, while operating at a fraction of their computational cost. On the other hand, FSO consistently outperforms all other algorithms across the evaluated datasets, with a significantly lower computational cost than other global optimization algorithms, such as the evaluated Evolutionary Algorithm. The HAP algorithm performs between the two other proposed metaheuristics in terms of both solution quality and computational cost. Additionally, this thesis presents an algorithm design framework that formalizes the development process leading to these solutions. This work advances the state of the art in AOS research. The novel proposals, in particular the FSO algorithm, aim to set a benchmark for future studies.
Generative Deep Learning for Cancer Image analysis
(Universitat de Barcelona, 2025-10-22) Osuala, Richard; Lekadir, Karim, 1977-; Díaz, Oliver; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] Accurate and timely detection of cancerous lesions in medical imaging is essential for effective treatment. However, the diagnosis remains challenging due to, among others, tumor heterogeneity, imaging constraints, and observer variability. Deep learning architectures, such as convolutional and transformer-based networks, have been showing promise in improving cancer image analysis by learning complex patterns within features from raw imaging data, allowing for earlier, more precise detection and consistent, data-driven decision-making. Despite its potential, clinical adoption of deep learning is restricted by the need for large, annotated training datasets, which are scarce due to privacy and labeling cost constraints, as well as due to its variability in performance when applied in settings of domain shift, imaging artifacts, variation in imaging protocols, and patient populations. This thesis identifies and addresses the challenges in deep learning for cancer imaging through five core publications that propose novel frameworks, methods, and solutions. First, a large-scale survey of generative models in cancer imaging is conducted leading to the identification of key challenges and problems in the field alongside ideation of potential solutions. Based on these findings, the SynTRUST meta-analysis framework is derived to assess the trustworthiness and maturity level of cancer image synthesis studies and solutions. Second, conditional generative adversarial networks (GANs) are applied to the challenges of scarcity of cancer images and tumor annotations, by simulating dynamic contrast-enhanced breast magnetic resonance imaging (DCEMRI) sequences. Without relying on physical contrast agents or DCE-MRI data from real patients, this approach enables unsupervised tumor detection, localization, and characterization, along with providing synthetic training data for increasing the robustness of downstream task models such as tumor segmentation models. Third, a multi-conditional latent diffusion models is developed to translate non-contrast enhanced MRI images into variable time-dependent synthetic DCE-MRI images localizing tumors and predicting their contrast enhancement kinetic patterns. Addressing the need for respective quantitative evaluation metrics, the Fréchet Radiomics Distance (FRD) is proposed to measure (synthetic) image quality based on biomarker variability. Fourth, a mass malignancy-conditioned generative adversarial network (MCGAN) is proposed to generated synthetic data as privacy-preservation mechanism for training deep learning models without individual patient data. Via comparison and combination with differential privacy, the synthetic mammography data is shown to improve the performance in multiple privacy-preserving cancer classification scenarios. Fifth, the medigan library is introduced as sharing platform for pretrained generative models that enable researchers to generate high-quality synthetic data across diverse imaging modalities without requiring direct access to sensitive patient data. Additionally, medigan’s generative models are comprehensive analyzed based on the Fréchet Inception Distance (FID) using both radiology domain-specific and standard domain-invariant feature extractors. In conclusion, this thesis highlights the potential of generative deep learning to address the key challenges in cancer imaging, presenting novel methods in contrast enhancement simulation, tumor localization, privacy-preserving cancer classification, and tools for sharing and assessing generative models, paving the way for these models towards integration into clinical practice to the end of advancing healthcare for both individual patients and society at large.
Deep Learning Solutions for Semantic Segmentation of Topographic Airborne LiDAR Point Clouds
(Universitat de Barcelona, 2026-02-13) Carós, Mariona; Vitrià i Marca, Jordi; Seguí Mesquida, Santi; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] Airbome LiDAR has become a key technology for large-scale mapping, providing dense three-dimensional point clouds that capture both natural landscapes and human-made infrastructure. Transforming these raw data into semantic information, however, remains a challenge due to irregular sampling patterns, strong class imbalance, and distribution shifts across regions and acquisition conditions. This thesis addresses these challenges by developing deep learning solutions tailored to the semantic segmentation of topographic airborne LiDAR point clouds, with an emphasis on operational applicability in national mapping workflows. We first introduce object-centric methods for detecting and segmenting vertical objects in cluttered scenes, proposing a constrained sampling that enhances the characterization of vertical structures embedded in vegetation. We then investigate training and inference procedures to handle variability in density and scale, demonstrating how inductive biases and uncertainty-based inference substantially improve robustness without requiring architectural modifications. To reduce reliance on costly manual annotation, we adapt self-supervised learning to airborne LiDAR data, showing that Barlow Twins pre-training improves downstream segmentation, particularly for underrepresented classes. Finally, we explore domain adaptation and incremental learning, integrating LoRA into PointNet++ to achieve parameter-efficient fine-tuning. We show how LoRA facilitates the addition of new semantic categories with minimal overhead. Beyond methodological advances, this research contributes new resources to the community, including the release of the TerLiDAR dataset. In addition, several of the proposed methods have also been transferred to productive workflows at the Institut Cartografíe i Geologic de Catalunya, where they support tasks such as refining digital surface models, detecting missing transmission towers, recovering filtered power lines, and classifying wind turbines. Together, these contributions demonstrate how deep learning can be scaled and adapted for reliable, real-world airborne LiDAR semantic segmentation, bridging the gap between research innovation and productive workflows.
Training More Efficient Neural Networks
(2025-12-04) Riera Molina, Carles Roger; Puertas i Prats, Eloi
[eng] This thesis centers on a critical reassessment of standard practices in the training and pruning of deep neural networks, with the ultimate goal of exploring more efficient, interpretable, and theoretically grounded alternatives. Traditionally, deep learning models rely on a range of architectural and optimization techniques-such as residual connections (ResNet) or batch normalization-that, while effective in facilitating training and improving convergence, can introduce significant limitations. These methods often promote the use of overparameterized networks, where many parameters are underused or entirely inactive, and where certain training data points are effectively ignored, producing zero outputs or gradients. In this context, the first objective of the thesis is to show that it is possible to train neural networks effectively and robustly without relying on these conventional strategies, as long as both parameters and data are more fully utilized. This goal is realized through the introduction of two contributions: Linked Neurons and Jumpstart. Linked Neurons represent a family of activation functions that combine various nonlinear behaviors while sharing parameters, ensuring that every weight receives meaningful gradients and avoiding the problem of dead units. This enables improved model performance without increasing network size. In parallel, Jumpstart is a regularization technique designed to penalize both dead and purely linear units, thereby promoting effective nonlinear activation across all neurons. This regularization ensures that every unit contributes actively to the learning process, improving gradient flow and enabling more efficient use of training data. This mechanism not only enhances the capacity utilization of the model but also allows for the training of deeper networks without the need for traditional architectural crutches such as ResNet or BatchNorm-fulfilling one of the thesis's core objectives. The second objective is to revisit the Lottery Ticket Hypothesis (LTH), which posits that within a randomly initialized neural network, there exist subnetworks-or "winning tickets"-that can be trained in isolation to achieve performance comparable to the full model. However, this hypothesis currently depends on iterative pruning and the rewinding of selected weights to their original initialization, introducing a strong dependency on initialization and a significant computational overhead. With the integration of Jumpstart, the thesis demonstrates that both rewinding and iterative pruning can be eliminated, as enhanced gradient flow reduces sensitivity to initialization and allows sparse networks to be trained directly. Finally, the third objective of the thesis is to replace heuristic pruning strategies-such as magnitude-based pruning-that lack theoretical justification and may harm model performance. In response, the thesis propases a novel pruning algorithm based on the analysis of how each unit permutes the dataset samples. This method identifies and removes redundant units while preserving the original decision function of the model, yielding a sparse yet functional representation of the network. Taken together, this thesis offers an alternative and complementary vision to current deep learning practice, addressing efficiency, robustness, and theoretical grounding in the training and compression of deep neural networks.
Topological Data Analysis Across Domains: From Point Clouds to Graphs
(Universitat de Barcelona, 2025-12-18) Ferrà Marcús, Aina; Casacuberta, Carles; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] This thesis builds on the foundational principle of topological data analysis (TDA), i.e., tracking the evolution of homological features in filtered spaces, and demonstrates the broad applicability of TDA through five studies conducted in distinct scientific contexts. In each study, the practical part was carried out in collaboration with experts from the cor-responding fields: neuroscience, artificial intelligence, complex systems, and cardiovascular medicine. Across these studies, TDA techniques are integrated with machine learning mod-els, yielding improvements in classification accuracy and interpretability of the resulting pipelines. Besides the experimental analyses, our work contributes novel methodologies and theo-retical insights. Specifically: (1) We develop a TDA-based classifier based on the variation of persistence descriptors when new points are added to a point cloud, and show that the classifier's accuracy can be used to estímate an inherent dimension of a data set, which, in our case, carne from a behavioral neuroscience study. (2) We design a method of importance attribution by selecting the most informative landscape levels for a neural network classifier of time series. (3) We prove that topological radiomics extracted from cardiovascular magnetic resonance images serve as a complement to standard collections of radiomic variables, achieving comparable accuracies with shorter feature vectors and less training time. (4) We implement extended persistence of cycles in graphs and introduce a new algorithm to replace an edge-weigthed graph with a vertex-weigthed one with the same persistence diagram, and we built a database of almost 800 000 graphs on which a neural network model was trained for latent dimensionality estimation of real-world net-works. (5) We enhance Mapper graphs with quantitave indices, which are used to achieve statistical significance using a dataset from a study of hemodynamic response to cardiac resynchronization therapy.
Large and iterated finite group actions on aspherical manifolds
(Universitat de Barcelona, 2025-07-11) Daura Serrano, Jordi; Mundet i Riera, Ignasi; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] Finite transformation group theory investigates the finite symmetries of topological objects, such as manifolds or CW-complexes. In this thesis, we focus on actions on closed topological manifolds and adopt the following approach: instead of directly studying the action properties of a finite group G to a manifold M, we focus on the properties of action restricted to certain subgroups H of bounded index. Several problems align with this philosophy, such as determining whether the group of homeomorphisms of a manifold is Jordan, calculating the discrete degree of symmetry of a manifold, determining whether a manifold is quasi-asymmetric, and studying the number and size of isotropy subgroups for finite group actions on manifolds. In the first part of the thesis, we provide solutions to these problems for two general classes of manifolds, namely: (1) Closed, connected and aspherical manifolds, whose fundamental group has a group of external Minkowski automorphisms (a group G is Minkowski if there exists a constant C such that every finite subgroup H of G has order at most C). (2) Closed, connected and orientable manifolds that admit a non-zero degree application to a nilmanifold. We show that the group of external automorphisms of a lattice of a connected Lie group is Minkowski, which allows us to apply our results to locally homogeneous aspherical closed manifolds. In addition, we provide the earliest known examples of manifolds M and M' with isomorphic cohomology rings such that Homeo(M) is Jordan but Homeo(M') is not. We establish two stiffness results for the discrete degree of symmetry: if M is a closed, connected, aspherical manifold and the external automorphism group of the fundamental group of M is Minkowski, or if M admits a non-zero degree application to a nilmanifold and its fundamental group is virtually solvable, then M is homeomorphic to a torus if its discrete degree of symmetry is equal to the dimension of M. In the second part, we refine the concept of group actions to explore in greater depth the topological and cohomology rigidity of closed and connected manifolds. This framework allows us to analyze in more detail the structure of closed aspherical manifolds and those that admit a non-zero degree application to a nilmanifold. We define new invariants, such as the iterated length of a manifold, which is closely related to its self-coatings, and introduce a refined version of the discrete degree of symmetry, called the discrete degree of iterate symmetry. We show that if M is a closed, oriented manifold that admits a non-zero degree application to a nilmanifold of nilpotency class 2, and both manifolds have the same discrete degree of iterated symmetry, then the rational cohomology of M is isomorphic to that of the nilmanifold. Furthermore, if the fundamental group of M is virtually solvable, then M is homeomorphic to the nilmanifold. We also prove that if M is a locally homogeneous closed aspherical manifold with a discrete degree of iterated symmetry equal to its dimension, then M is homeomorphic to a nilmanifold of nilpotency class 2.
Bridging Natural Language and Hierarchical Multivariate Data Visualisation to Support Data Analysis
(Universitat de Barcelona, 2025-04-23) Kavaz, Ecem; Rodríguez Santiago, Inmaculada; Puig Puig, Anna; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] Tracking and analysing the vast amounts of data generated from social networks and digital platforms presents important challenges, not only due to the overwhelming volume but also the complex relationships embedded within the data. This thesis addresses these challenges through data visualisation techniques, focusing on hierarchical and multivariate data, where visual clutter and effective use of space are key concerns. Furthermore, the rise of Visual Natural Language Interfaces (V-NLIs), also referred to in this thesis as VisChatbots, offers new opportunities to facilitate the interaction with data visualisations via natural language. This thesis contributes to the fields of Hierarchical Multivariate Data Visualisation and Visualisation-oriented Natural Language Interfaces. Specifically, we introduce a novel categorization algorithm to classify hierarchical data, from which we propose the most suitable visual designs for their visualisation. Additionally, we propose a new incremental design methodology for Vis-Chatbots, called VisChat. This structured approach guides the development of chatbots integrated into visualisation platforms, establishing smooth communication among stakeholders—end users, designers, and developers—and introducing new design artefacts, such as the VisAgent persona, visualisation conversation patterns, and conversational transcripts that help guide and validate the design of the VisChatbot. Following the VisChat methodology, we have integrated a VisChatbot into a platform for visualising hierarchical and multivariate data. To validate our proposal, we present a case study on the analysis of hate speech in online news articles, where the suitability of the proposed visualisations was evaluated, as well as the capability of the visualisation chatbot to enable users to easily explore and understand, through Natural Language interactions, both the structural relationships and the feature-based relationships within the data. In conclusion, this thesis not only advances data visualisation techniques for multivariate hierarchical data but also establishes a framework for integrating natural language interfaces intov isual analysis platforms, thereby promoting a more efficient and effective analysis of data.
Boosting the Artificial Intelligence solutions training phase by means of process simulation methods
(Universitat de Barcelona, 2025-05-13) Abió Rojo, Albert; Pujol Vila, Oriol; Bonada Bo, Francesc; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] Despite the emergence of Industry 4.0 and the rise of a data-driven manufacturing paradigm, the acquisition of valuable data in a cost-efficient and sustainable manner for manufacturing processes remains a challenge for many companies. Conducting non-productive tests on the production line in an industrial plant result in a waste of raw materials, energy, human resources, and time. Furthermore, executing high fidelity manufacturing simulations entails a significant temporal and computational burden. Consequently, these drawbacks hinder the creation of knowledge in manufacturing processes and the development of technologies that aim to enhance and influence in the process performance, such as optimization or AI-based tools. This is especially critical for tools that benefit from the availability of large volumes of data and real-time responses, like Digital Twins and Reinforcement Learning agents. Therefore, it is necessary to provide methods that facilitate data generation in industrial environments. This dissertation is devoted to present a set of general methods to companies and manufacturers to boost the data generation phase in the industrial context. Concretely, we focus on a fast and efficient way to model manufacturing processes through the development of Machine Learning-based Surrogate Models. We propose different general theoretical frameworks implementing or combining machine learning techniques for surrogate modeling applicable in distinct manufacturing process. The thesis demonstrates that the proposed methods enable significant cost and time reductions in different practical manufacturing applications while maintaining high accuracy in modeling and predicting process variables. We investigate the importance of the data chosen to construct the Surrogate Models and the transfer of the knowledge in the Surrogate Models from simulation to real plants by means of Trans fer Learning. Overall, this supposes an improvement of the presented surrogate modeling methods and it facilitates the deployment of Surrogate Models in real-world industrial plants. The developed models during the thesis are a valuable asset in other studies, acting as a virtual environment to train Reinforcement Learning agents in hot stamping or supporting a Digital Twin of the high pressure die casting process. The thesis helps to advance towards the innovation of data-driven manufacturing by providing practical and efficient solutions in the direction of a better understanding of the manufacturing processes, leading to an enhancement in their performance and sustainability.
Flow map parameterization methods for invariant tori in Hamiltonian systems
(Universitat de Barcelona, 2025-06-05) Fernández-Mora, Álvaro; Haro, Àlex; Mondelo González, José María; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] Given a dynamic system, it is important to identify the invariant objects that organize long-term behavior, as well as their dynamic connections. Both in theory and in applications. The objective of this thesis is to advance in the development of Kolmogorov-Arnold-Moser (KAM) type techniques within the framework of the parameterization method and its application to problems of celestial mechanics. We have developed KAM iterative schemes for the calculation of partially hyperbolic invariant torus and their invariant bundles in quasiperiodic Hamiltonian systems. We look for invariant bulls and bundles under adequate time-1 maps, which allow us to reduce the dimension of the bull to be calculated by one. The computational cost of manipulating functions grows exponentially with the number of variables in the parameterization. Therefore, reduction by flow maps is computationally advantageous, although it requires numerical integration. However, this integration can be easily parallelized. If the parameterization is approximated with N Fourier coefficients, the iterative step requires O(N) of storage and O(N log N) operations, in contrast to standard Newtonian methods, which need O(N^2) of storage and O(N^3) operations. This gain in efficiency comes from the geometric properties of phase space (i.e., symplectic geometry), systems (symplectic accuracy), torus (isotropy, Lagrangianity), as well as dynamical properties (reducibility). In particular, the reducibility of the linearized dynamics around the torus to a triangular matrix by blocks is known as automatic reducibility and is an important property both in theory and in applications. The algorithms have been implemented and applied to the Three-Body Elliptic Restricted Problem (ERTBP) to compute an extensive set of non-resonant three-dimensional invariant torus along with their invariant bundles. From these results, we have obtained an a posteriori theorem for partially hyperbolic invariant bulls and their rank 1 invariant bundles in quasiperiodic Hamiltonian systems. The approach followed allows the theorem to be applied to autonomous, periodic and quasiperiodic Hamiltonian systems, and constitutes the demonstration of the convergence of methods based on flow maps. In addition, we simultaneously obtain both stable and unstable bundles, providing a clear geometric view of the tangent space to the torus. The proof is based on geometric properties of a symplectic nature, which hold approximately when the parameterizations approximately satisfy their equations of invariance. We have obtained geometric lemmas that control error in the KAM iterative process. The new error in the invariance equations is controlled with explicit constants, which requires a careful treatment of the loss of analyticity at each iterative step. The demonstration concludes by obtaining convergence conditions for the KAM iterative process. The a posteriori theorem obtained allows computer-aided proofs to be carried out. Partially hyperbolic bulls have associated stable and unstable varieties, whiskers, where dynamics converge exponentially fast in the future and in the past, respectively. The stable and unstable bundles with the linear approximations of these varieties. We have also developed KAM schemes to compute high-order Fourier-Taylor expansions of whiskers in autonomous and quasiperiodic Hamiltonian systems. Unlike order-to-order methods, which first calculate the torus and its bundles before calculating the whiskers on an order-by-order basis, the approach followed simultaneously computes both the torus and the whiskers using the same KAM iterative method. This unified framework improves the efficiency of whisker calculation by doubling the number of correct terms in expansion in each iteration. The algorithms have been applied to the calculation of high-order expansions of partially hyperbolic non-resonant invariant torus in the circular and three-body elliptical constrained problems.
Periodic boundary points for transcendental Fatou components
(Universitat de Barcelona, 2025-09-18) Jové Campabadal, Anna; Fagella Rabionet, Núria; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] This thesis is framed in the field of Complex Dynamics, which studies discrete dynamical systems generated by the iteration of holomorphic functions. More precisely, given a transcendent, integer or meromorphic function, we consider the discrete dynamical system generated by it. Then the complex plane is divided into two totally invariant sets: the Fatou set, where the dynamics are stable; and Julia's ensemble, its complement, where the dynamics are chaotic. The Fatou set is open and generally has infinite related components, called Fatou components, and they are periodic, pre-periodic, or wandering. One of the basic results in Complex Dynamics (demonstrated by Fatou and Julia for rational functions) is that the Julia set is the closure of the repulsing periodic points of the function. This result was generalized by Baker by integer functions, and by Baker, Kotus, and Lü by transcendent meromorphic functions. We note that, given an invariant Fatou component, then its boundary is an invariant closed subset of the Julia set. So, the next question arises naturally: given a meromorphic function, and it is a periodic Fatou component, are the periodic points dense at their boundary? Note that, although the periodic points are dense in the Julia set, a priori they could accumulate at the boundary from its complement, without being at the boundary For example, if the Fatou component is a rotation domain with a locally connected boundary, then there is no periodic point. However, F. Przytycki and A. Zdunik showed that, by rational functions, rotation domains (i.e. Siegel's disks and Herman's rings) are the only exceptions for which the periodic points are not dense at the boundary. In particular, they gave a positive answer to the previous question for attraction or parabolic basins of rational functions. The work of F. Przytycki and A. Zdunik already shows us that the answer to such an elementary question is far from simple. Indeed, an exhaustive study of the boundaries of such Fatou components (which may not be locally connected) is necessary, combining tools of dynamics, measurement theory and conformal analysis. In the particular case of simply connected attraction basins, the proof is based on the properties of the function at the boundary from the point of view of the theory of measurement and Lyapunov's exponents, as well as precise estimates of the distortion of the Riemann application and the finite Blaschke products in the unit circle, and Pesin's theory conforms. For components of Fatou not limited to transcendent functions, the situation is even more delicate, due to the presence of the essential singularity, and most of the above techniques cannot be applied. Moreover, since the boundary of the Fatou component is not compact, it is not compact, nor is the existence of periodic points on the boundary evident. In view of the above questions, and the existing previous work to understand the boundaries of transcendent Fatou components, the following conjecture naturally arises, which is a large open problem in transcendent dynamics. Let it be a meromorphic function, and let it be a simply connected periodic Fatou component, such that it is not univalent. Then, there is a periodic point on the border of such a component of Fatou. In addition, if it is an attractor or parabolic basin, or a doubly parabolic Baker domain, then the periodic points are dense at the boundary. This thesis should be understood as significant progress in proving the above conjecture. Indeed, we demonstrate the existence and density of periodic points at the boundary of Fatou components under very weak hypotheses in the postsingular set, together with additional results in relation to boundary dynamics, escape points and accessibility. During the thesis, new techniques have been demonstrated, such as estimates in the distortion of internal functions and Pesin's theory for transcendent functions.
Deep Learning Approaches for Human Activity Understanding
(Universitat de Barcelona, 2025-06-12) Zhang, Zejian; Escalera Guerrero, Sergio; Palmero Cantariño, Cristina; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] Understanding human activities is crucial for developing practical applications that benefit society. Temporal action localization (TAL) in untrimmed videos is one of the most challenging tasks in this field. While significant progress has been made over the years, the methods developed are still far from being suitable for real-world use, and TAL remains an ongoing challenge. This thesis aims to address this challenge task through three contributions. First, we propose a dual hierarchical model capable of extracting and fusing both local, fine-grained boundary details and broader, high-level semantic contexts for TAL. In this method, the second hierarchical design enables the model to uncover actions of varying durations, leveraging the features learned from the first hierarchy. Our findings show that fusing temporal contexts at different scales is essential for precise TAL. In this approach, the model utilizes the self-attention mechanism in Transformer encoders. However, due to the quadratic complexity of self-attention, methods relying on it may struggle to handle real-world-length videos. Next, we present a comprehensive experimental comparison to determine which temporal feature encoder should be selected under different conditions. We analyzed 12 models, equipped with pure Transformer encoders, pure Mamba Blocks, and combinations of both into a unified encoder for TAL. The experimental results suggest that the choice of encoder depends heavily on the specific dataset. Nevertheless, the pure Mamba Block emerges as the preferred option for unknown datasets due to its performance and lower complexity. Finally, we introduce UDIVA-HHOI, a novel large-scale audio-visual dyadic human-human-object interaction dataset. This dataset provides rich, extremely short-duration and concurrent actions, featuring both low-level physical actions and high-level goal-oriented actions and the objects involved in these actions—elements not typically represented in commonly used TAL benchmarks. UDIVA-HHOI opens up new possibilities for addressing the detection of complex interactive actions in real-world scenarios. Our preliminary study confirms its potential, and our analysis also offers recommendations for selecting an appropriate feature encoder for future research on this new benchmark, with the Mamba Block being the preferred choice.
Approximate option pricing for jump-diffusion stochastic volatility models
(Universitat de Barcelona, 2025-01-21) Makumbe, Zororo Stanelake; Vives i Santa Eulàlia, Josep, 1963-; El-Khatib, Youssef; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] The following is a summary of the above-mentioned thesis. The thesis covers alternative stochastic models, risk management, and option price decomposition. 1.1 Alternative Stochastic Models Several alternative asset pricing models are explored, each designed to capture market dynamics better. The models we consider include: 1. The Hybrid Stochastic Local Volatility (SLV) model entitled the Heston-CEV model with jumps (HCEVJ). This model combines the strengths of the Constant Elasticity of Variance (CEV) and Heston models, making volatility dependent on time, asset price, and an underlying stochastic process. 2. The 2-factor stochastic volatility model with jumps (2FSVJ) with multiple factors to drive the volatility process, offering richer dynamics. 3. Lastly, we consider the infinite activity Heston-Lévy model to account for large changes in asset prices due to jumps, thus capturing large market movements. The HCEVJ model is a general model with the CEV, Heston, Heston-CEV, Bates, and Heston-CEV with jumps as special cases. We calibrate each of these models to the EURO STOXX 50 European option quotes of 30 September 2014 using a brute-force grid search algorithm to facilitate comparisons. Monte Carlo methods are employed to model the asset price movements and verify model properties. The analysis shows that the HCEVJ model deepens the volatility smile by introducing an additional parameter that controls the intensity of the volatility smile. Moreover, the HCEVJ model exhibits the leverage effect, volatility clustering and price jumps, offering a more comprehensive representation of asset price dynamics. The properties of the 2FSVJ and Heston-Lévy models are not tested, however. 1.2 Malliavin Calculus, Hedging, and Option Greeks A key element of options pricing is hedging and option sensitivities. In this thesis, we use Malliavin Calculus to obtain faster estimates of the first-order Greeks of the option prices for the HCEVJ model. Additionally, we use the Clark-Ocone formula to find an explicit formula to hedge a portfolio. 1.3 Pricing Methods The third and main objective is to explore option pricing methods. The thesis reviews existing approaches, such as Monte Carlo simulations, Fourier integral methods, and decomposition techniques, and compares them. Decomposition methods outperform Monte Carlo simulations and Fourier integral methods, particularly under simple jump structures. In the 2FSVJ case, a first-order and a second-order decomposition formula are derived under a general jump structure. We choose log-normal and double-exponential jumps and carry out numerical experiments. The double-exponential jump case shows poorer accuracy and speed performance. Furthermore, the decomposition pricing model was more accurate under short-maturity and out-of-the-money conditions. In the Heston-Lévy case, we build on existing research on decomposition formulas by considering infinite activity jumps in asset returns. The study addresses the pricing of options in models where the return process includes both stochastic volatility and jumps of infinite activity but finite variation. The key contribution of this work is two exact decomposition formulas for option pricing under this model. The first decomposition formula expands the pricing model using a series of expectations involving the underlying asset and volatility terms and correction terms for the jumps in the asset price. However, this approach remains computationally challenging, as obtaining a tractable version of the formula is an open problem. The second Decomposition Formula uses methods from Lévy process estimation to simplify the infinite activity problem by approximating it as a finite activity jump process. Error bounds are provided for this approximation, and numerical estimates are compared to benchmark prices. The method, while accurate, relies on Monte Carlo simulations making it computationally slow.
Characterization and Mitigation of Algorithmic Bias in Recommender Systems
(Universitat de Barcelona, 2024-12-17) Gómez Yepes, Elizabeth; Salamó Llorente, Maria; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] Recommender Systems are critical in helping users navigate large amounts of information by providing personalized suggestions. However, these systems can exhibit biases, especially when data imbalances exist, leading to unfair recommendations that favor more popular or majority items over those from minority groups. This thesis explores the identification, characterization, and mitigation of algorithmic bias within Recommender Systems. This research focuses on addressing biases that arise from data imbalances and how these biases can lead to unfair treatment of certain groups, particularly in terms of visibility and exposure in recommendations. The primary goal of the thesis is to mitigate algorithmic bias in Recommender Systems to produce fairer and more equitable recommendation lists, through techniques of post-processing bias mitigation (e.g., re-ranking recommendation results to ensure fairness). This includes identifying and categorizing biases in datasets, designing strategies to mitigate these biases, and developing techniques to optimize recommendation algorithms to reduce bias. The main contributions of this thesis are five, divided into two thematic parts. The first thematic part focuses on Provider Fairness and the second thematic part on Fairness from Multiple Perspectives. Regarding the first thematic part, two contributions have been made. In the first, a Binary Approach was adopted, by categorizing geographic bias or imbalance associated with the country of production of the items and identifying two groups of providers (majority versus rest), and based on the distribution observed in the original training set, the recommendations are adjusted to align with these groups, with the aim of mitigating disparity bias. In the second contribution, we explain the process of categorization and bias mitigation using a Multi-Class Approach. We explore how recommendation algorithms can exacerbate biases by promoting items from certain regions, which could disadvantage underrepresented geographic groups. Concerning the second thematic part, three contributions have been made. The first contribution introduces CONFIGRE, a novel methodology designed to ensure fairness in Recommender Systems by balancing visibility between coarse- and fine-grained demographic groups. In second contribution we present MOReGln, a new approach for managing multiple objectives in Recommender Systems. This method specifically addresses the challenge of achieving both global balance and individual fairness in recommendations. Finally, in an additional contribution, we develop a new dataset (AMBAR, in the music domain) that includes sensitive attributes at various levels of granularity. Furthermore, we extend two real-world datasets (MovieLenslM and Book-Crossing) with geographic information to study the link between geographic imbalance and disparate impact. This thesis advances on the identification, characterization, mitigation and evaluation of biases in collaborative Recommender Systems. It addresses existing gaps in the analysis of geographical biases in different group settings: from binary groups, multi-class groups to different levels of granularity of groups. The outlined contributions establish a basis for further advancements and effective mitigation of biases without significantly compromising accuracy. Our findings, developed software, and resources presented in this dissertation are available to the community to facilitate further research and knowledge transfer.
End-to-End AI Solutions for Capsule Endoscopy: Enhancing Efficiency and Accuracy in Gastrointestinal Diagnostics
(Universitat de Barcelona, 2025-01-22) Gilabert Roca, Pere; Seguí Mesquida, Santi; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] Artificial Intelligence (AI) models are fundamentally transforming the way clinicians carry out their daily tasks. By streamlining various processes, AI offers a more robust and consistent method for reviewing medical procedures. This thesis is dedicated to the development of AI applications for Capsule Endoscopy (CE), a small device that patients swallow, which is equipped with both a light and a camera to traverse the digestive system, capturing detailed images of internal organs. Once these images are captured, physicians are tasked with meticulously reviewing an extensive number of frames to identify potential pathologies, a process that is both time-consuming and tedious. In this thesis, we aim to enhance the entire review pipeline from end to end, providing support to physicians at multiple stages of the process. These stages include data collection, data labeling, assessing the usability of the videos (particularly in determining whether intestinal residues may hinder the process), identifying the entry and exit points of the small and large intestines, and most crucially, detecting polyps as early indicators of Colorectal Cancer (CRC). By employing advanced techniques such as Active Learning (AL) for data labeling and Vision Transformer (ViT) for polyp detection, we significantly improve upon existing systems in the literature, achieving state-of-the-art results. Additionally, the integration of AI into CE holds the promise of not only improving diagnostic accuracy but also reducing the workload for clinicians, allowing them to focus on more complex cases. This technological advancement has the potential to revolutionize gastrointestinal diagnostics, leading to earlier detection of diseases and, ultimately, better patient outcomes. Furthermore, this thesis led to the initiation of two clinical studies. The first was a controlled study that evaluated the performance of the polyp detection application. The second is a larger study involving over 600 patients, testing an enhanced version of the application, which is currently under development.
Relational methods in algebraic logic
(Universitat de Barcelona, 2025-01-21) Fornasiere, Damiano; Gispert Brasó, Joan; Moraschini, Tommaso; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] This thesis is concerned with three instances of relational methods in algebraic logic. First, determining which partially ordered sets are isomorphic to the spectrum of a Heyting algebra. This is an open question related to the classical problem of representing partially ordered sets as spectra of bounded distributive lattices or, equivalently, commutative rings with unit. We prove that a root system (the order dual of a forest) is isomorphic to the spectrum of a Heyting algebra if and only if it satisfies a simple order theoretic condition, known as "having enough gaps", and each of its nonempty chains has an infimum. This strengthens Lewis' characterisation of the root systems which are spectra of commutative rings with unit. While a similar characterisation for arbitrary forests currently seems out of reach, we show that a well-ordered forest is isomorphic to the spectrum of a Heyting algebra if and only if it has enough gaps and each of its nonempty chains has a supremum. Second, Sahlqvist theorem provides sufficient syntactic conditions for a normal modal logic to be complete with respect to an elementary class of Kripke frames. We extend Sahlqvist theory to the fragments of the intuitionistic propositional calculus that include the conjunction connective. This allows us to introduce a Sahlqvist theory of intuitionistic character amenable to arbitrary protoalgebraic deductive systems. As an application, we obtain a Sahlqvist theorem for the fragments of the intuitionistic propositional calculus that include the implication connective and for the extensions of the intuitionistic linear logic. Third, Blok's celebrated dichotomy theorem proves that each normal modal logic shares its Kripke frames with exactly one or continuurn-many logics. It is an outstanding open problem to characterise the number of logics having the same posets of an axiomatic extension of the intuitionistic propositional calculus. We solve this question in the case of implicative logics, the axiomatic extensions of the implicative fragment of the propositional intuitionistic logic. In this case, a trichotomy holds: every irnplicative logics shares its posets exactly with 1, N0, or 2(No) many logics.
Geometric realizations using regular subdivisions: Construction of many polytopes, sweep polytopes, s-permutahedra
(Universitat de Barcelona, 2024-06-07) Philippe, Eva; Padrol Sureda, Arnau; Santos, Francisco, 1968-; Universitat de Barcelona. Departament de Matemàtiques i Informàtica
[eng] This thesis concerns three problems of geometric realizations of combinatorial structures via polytopes and polyhedral subdivisions. A polytope is the convex hull of a finite set of points in a Euclidean space Rd. It is endowed with a combinatorial structure coming from its faces. A subdivision is a collection of polytopes whose faces intersect properly and such that their union is convex. It is regular if it can be obtained by taking the lower faces of a lifting of its vertices in one dimension higher. We first present a new geometric construction of many combinatorially different polytopes of fixed dimension and number of vertices. This construction relies on showing that certain polytopes admit many regular triangulations. It allows us to improve the best known lower bound on the number of combinatorial types of polytopes. We then study the projections of permutahedra, that we call sweep polytopes because they model the possible orderings of a fixed point configuration by hyperplanes that sweep the space in a constant direction. We also introduce and study a combinatorial abstraction of these structures: the sweep oriented matroids, that generalize Goodman and Pollack’s theory of allowable sequences to dimensions higher than 2. Finally, we provide geometric realizations of the s-weak order, a combinatorial structure that generalizes the weak order on permutations, parameterized by a vector s ∈ (Z>0)n. In particular, we answer Ceballos and Pons’s conjecture that the s-weak order can be realized as the edge-graph of a polytopal complex that is moreover a subdivision of a permutahedron.

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