Tensor-Based Kernel Methods
F. Wesel (TU Delft - Team Kim Batselier)
J.W. van Wingerden – Promotor (TU Delft - Team Jan-Willem van Wingerden)
Borbala Hunyadi – Promotor (TU Delft - Signal Processing Systems)
K. Batselier – Copromotor (TU Delft - Team Kim Batselier)
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Abstract
In today's data-driven landscape, the capacity to efficiently process and analyze vast datasets is crucial across various domains, including healthcare, climate modeling, and finance. Despite the growing need for scalable and interpretable machine learning models, traditional approaches, particularly kernel machines, face significant challenges due to the curse of dimensionality. As data complexity increases, the computational and memory demands of kernel machines often become prohibitive, limiting their applicability in high-dimensional applications. This thesis addresses these challenges by investigating the integration of tensor networks (TNs) with kernel machines, aiming to enhance scalability, efficiency without sacrificing predictive power and interpretability.
We propose that TNs, with their ability to represent high-dimensional data through low-rank structures, can effectively alleviate the limitations of kernel machines. Our research is structured around three central inquiries: first, we examine how TNs can accelerate kernel machine scalability while accurately approximating kernel functions; second, we elucidate the theoretical links between TN-constrained kernel machines and Gaussian processes, providing insights into convergence and generalization; finally, we introduce a novel optimization framework characterizing a specific TN, the multi-linear singular value decomposition (MLSVD), in terms of primal and dual problems.