Least-squares support-vector-machines are a frequently used supervised learning method for nonlinear regression and classification. The method can be implemented by solving either its primal problem or dual problem. In the dual problem a linear system needs to be solved, yet for large-scale problems this can be impractical as current methods suffer from the \textit{curse of dimensionality}. This phenomena causes the computational and memory requirements to exceed the capabilities of standard computers for large datasets. In this thesis, a tensor network Bayesian learning method was developed to avoid these burdensome complexities. The developed method performs competitively with the current state-of-the-art, and unlike other low-rank approximation methods, allows for incorporation of user-knowledge, noise, early stopping, and yields confidence bounds on the obtained model.