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This paper presents a method for approximate Gaussian process (GP) regressionwith tensor networks (TNs). A parametric approximation of a GP uses a linearcombination of basis functions, where the accuracy of the approximation dependson the total number of basis functions $M$. We develop an approach that allowsus to use an exponential amount of basis functions without the correspondingexponential computational complexity. The key idea to enable this is usinglow-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 theweights of our model by solving a Bayesian inference problem. Finally, weproject the resulting weights back to the original space to make GPpredictions. The benefit of our approach comes from the projection to a smallersubspace: It modifies the shape of the basis functions in a way that it seesfit based on the given data, and it allows for efficient computations in thesmaller subspace. In an experiment with an 18-dimensional benchmark data set,we show the applicability of our method to an inverse dynamics problem.

Projecting basis functions with tensor networks for Gaussian process regression