Bootstrap como estrategia para al estabilización de las soluciones sparse en modelos tensoriales. Aplicado al modelo CenetTucker

Abstract

[EN] This doctoral thesis addresses a key methodological challenge in high-dimensional data analysis: the instability of sparse solutions in penalized tensor models. Specifically, it proposes a theoretical and computational framework that integrates Bootstrap resampling techniques with the Elastic Net-penalized Tucker decomposition —known as the CenetTucker model— to enhance the stability and reproducibility of latent factor selection. The research is structured around three main pillars: (i) a comprehensive review of the theoretical foundations and limitations of sparse solutions in tensor-structured data, (ii) the formalization and implementation of a Bootstrap-based stabilization procedure tailored to the CenetTucker model, and (iii) the empirical evaluation of model stability through simulated experiments and real datasets. As an applied contribution, the thesis introduces an R package named GSparseBoot, which automates the model fitting, resampling, and computation of stability metrics —including variable inclusion frequency, Jaccard index, support variability, and stable selection index. While the package is not yet published on CRAN, its development is complete, and its public release is currently in process. Results demonstrate that incorporating Bootstrap significantly reduces the structural variability of penalized solutions without compromising interpretability or predictive performance. This improvement is particularly evident in scenarios involving high collinearity or weak latent structures, where traditional approaches tend to be unstable. Additionally, a set of tailored stability metrics is proposed to rigorously assess consistency across resampling replicates in multi-way contexts. This work offers an original methodological contribution at the intersection of computational statistics, tensor factorization, and regularization. It provides a solid mathematical foundation, a reproducible computational implementation, and practical tools to support scientific studies in genomics, neuroscience, sensory data analysis, and other domains where statistical reproducibility is paramount. Overall, this thesis advances the development of more robust and reliable statistical models in the era of complex, highdimensional data.