Sparse Gaussian processes using pseudo-inputs

We present a new Gaussian process (GP) regression model whose covariance is parameterized by the the locations of M pseudo-input points, which we learn by a gradient based optimization. We take M << N, where N is the number of real data points, and hence obtain a sparse regression method which has O(M^2N) training cost and O(M^2) prediction cost per test case. We also find hyperparameters of the covariance function in the same joint optimization. The method can be viewed as a Bayesian regression model with particular input dependent noise. The method turns out to be closely related to several other sparse GP approaches, and we discuss the relation in detail. We finally demonstrate its performance on some large data sets, and make a direct comparison to other sparse GP methods. We show that our method can match full GP performance with small M, i.e. very sparse solutions, and it significantly outperforms other approaches in this regime.