  Gaussian processes are frequently deployed as part of larger machine learning
and decision-making systems, for instance in geospatial modeling, Bayesian
optimization, or in latent Gaussian models. Within a system, the Gaussian
process model needs to perform in a stable and reliable manner to ensure it
interacts correctly with other parts of the system. In this work, we study the
numerical stability of scalable sparse approximations based on inducing points.
To do so, we first review numerical stability, and illustrate typical
situations in which Gaussian process models can be unstable. Building on
stability theory originally developed in the interpolation literature, we
derive sufficient and in certain cases necessary conditions on the inducing
points for the computations performed to be numerically stable. For
low-dimensional tasks such as geospatial modeling, we propose an automated
method for computing inducing points satisfying these conditions. This is done
via a modification of the cover tree data structure, which is of independent
interest. We additionally propose an alternative sparse approximation for
regression with a Gaussian likelihood which trades off a small amount of
performance to further improve stability. We provide illustrative examples
showing the relationship between stability of calculations and predictive
performance of inducing point methods on spatial tasks.
