Gauss-Legendre Features for Gaussian Process Regression

Gaussian processes provide a powerful probabilistic kernel learning framework, which allows learning high quality nonparametric regression models via methods such as Gaussian process regression. Nevertheless, the learning phase of Gaussian process regression requires massive computations which are not realistic for large datasets. In this paper, we present a Gauss-Legendre quadrature based approach for scaling up Gaussian process regression via a low rank approximation of the kernel matrix. We utilize the structure of the low rank approximation to achieve effective hyperparameter learning, training and prediction. Our method is very much inspired by the well-known random Fourier features approach, which also builds low-rank approximations via numerical integration. However, our method is capable of generating high quality approximation to the kernel using an amount of features which is poly-logarithmic in the number of training points, while similar guarantees will require an amount that is at the very least linear in the number of training points when random Fourier features. Furthermore, the structure of the low-rank approximation that our method builds is subtly different from the one generated by random Fourier features, and this enables much more efficient hyperparameter learning. The utility of our method for learning with low-dimensional datasets is demonstrated using numerical experiments.