  We propose two methods for exact Gaussian process (GP) inference and learning
on massive image, video, spatial-temporal, or multi-output datasets with
missing values (or "gaps") in the observed responses. The first method ignores
the gaps using sparse selection matrices and a highly effective low-rank
preconditioner is introduced to accelerate computations. The second method
introduces a novel approach to GP training whereby response values are inferred
on the gaps before explicitly training the model. We find this second approach
to be greatly advantageous for the class of problems considered. Both of these
novel approaches make extensive use of Kronecker matrix algebra to design
massively scalable algorithms which have low memory requirements. We
demonstrate exact GP inference for a spatial-temporal climate modelling problem
with 3.7 million training points as well as a video reconstruction problem with
1 billion points.
