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Gaussian processes are frequently deployed as part of larger machine learningand decision-making systems, for instance in geospatial modeling, Bayesianoptimization, or in latent Gaussian models. Within a system, the Gaussianprocess model needs to perform in a stable and reliable manner to ensure itinteracts correctly with other parts of the system. In this work, we study thenumerical stability of scalable sparse approximations based on inducing points.To do so, we first review numerical stability, and illustrate typicalsituations in which Gaussian process models can be unstable. Building onstability theory originally developed in the interpolation literature, wederive sufficient and in certain cases necessary conditions on the inducingpoints for the computations performed to be numerically stable. Forlow-dimensional tasks such as geospatial modeling, we propose an automatedmethod for computing inducing points satisfying these conditions. This is donevia a modification of the cover tree data structure, which is of independentinterest. We additionally propose an alternative sparse approximation forregression with a Gaussian likelihood which trades off a small amount ofperformance to further improve stability. We provide illustrative examplesshowing the relationship between stability of calculations and predictiveperformance of inducing point methods on spatial tasks.

Numerically Stable Sparse Gaussian Processes via Minimum Separation using Cover Trees