  We develop a novel framework to accelerate Gaussian process regression (GPR).
In particular, we consider localization kernels at each data point to
down-weigh the contributions from other data points that are far away, and we
derive the GPR model stemming from the application of such localization
operation. Through a set of experiments, we demonstrate the competitive
performance of the proposed approach compared to full GPR, other localized
models, and deep Gaussian processes. Crucially, these performances are obtained
with considerable speedups compared to standard global GPR due to the
sparsification effect of the Gram matrix induced by the localization operation.
