  It is common in functional data analysis to look at a set of related
functions: a set of learning curves, a set of brain signals, a set of spatial
maps, etc. One way to express relatedness is through an additive model, whereby
each individual function $g_{i}\left(x\right)$ is assumed to be a variation
around some shared mean $f(x)$. Gaussian processes provide an elegant way of
constructing such additive models, but suffer from computational difficulties
arising from the matrix operations that need to be performed. Recently Heersink
& Furrer have shown that functional additive model give rise to covariance
matrices that have a specific form they called quasi-Kronecker (QK), whose
inverses are relatively tractable. We show that under additional assumptions
the two-level additive model leads to a class of matrices we call restricted
quasi-Kronecker, which enjoy many interesting properties. In particular, we
formulate matrix factorisations whose complexity scales only linearly in the
number of functions in latent field, an enormous improvement over the cubic
scaling of na\"ive approaches. We describe how to leverage the properties of
rQK matrices for inference in Latent Gaussian Models.
