  This paper presents a method for approximate Gaussian process (GP) regression
with tensor networks (TNs). A parametric approximation of a GP uses a linear
combination of basis functions, where the accuracy of the approximation depends
on the total number of basis functions $M$. We develop an approach that allows
us to use an exponential amount of basis functions without the corresponding
exponential computational complexity. The key idea to enable this is using
low-rank TNs. We first find a suitable low-dimensional subspace from the data,
described by a low-rank TN. In this low-dimensional subspace, we then infer the
weights of our model by solving a Bayesian inference problem. Finally, we
project the resulting weights back to the original space to make GP
predictions. The benefit of our approach comes from the projection to a smaller
subspace: It modifies the shape of the basis functions in a way that it sees
fit based on the given data, and it allows for efficient computations in the
smaller subspace. In an experiment with an 18-dimensional benchmark data set,
we show the applicability of our method to an inverse dynamics problem.
