Tensor networks for scalable probabilistic modeling

Probabilistic or Bayesian modeling plays a fundamental role in engineering and science, providing a framework for integrating noisy measurements with predictive models through probability distributions.
While probabilistic methods have many benefits, such as recursive estimation and uncertainty quantification, they often come with substantial memory and compute requirements.
Computational challenges are particularly pronounced in large-scale settings, where data sets contain a high number of measurements, and for high-dimensional problems, which require exponentially many parameters to describe probability distributions.
These scenarios can suffer from the curse of dimensionality, which requires exponentially growing computing resources, making conventional approaches computationally intractable.

This dissertation addresses computational challenges by leveraging tensor networks (TNs) to develop computationally efficient probabilistic algorithms.
TNs, also known as tensor decompositions, extend matrix decomposition to higher dimensions by representing large multidimensional arrays, i.e., tensors, in a compact, decomposed format, defined by TN components and TN ranks.
Under the assumption of low-rank structure, TNs enable efficient storage and computation, making large-scale and high-dimensional problems more tractable, even on resource-constrained hardware such as conventional laptops.
The focus of this work is on scalable solutions for Bayesian estimation problems involving Gaussian distributions and exact inference, including recursive filtering and Gaussian process (GP) regression.