  Understanding sub-cellular protein localisation is an essential component to
analyse context specific protein function. Recent advances in quantitative
mass-spectrometry (MS) have led to high resolution mapping of thousands of
proteins to sub-cellular locations within the cell. Novel modelling
considerations to capture the complex nature of these data are thus necessary.
We approach analysis of spatial proteomics data in a non-parametric Bayesian
framework, using mixtures of Gaussian process regression models. The Gaussian
process regression model accounts for correlation structure within a
sub-cellular niche, with each mixture component capturing the distinct
correlation structure observed within each niche. Proteins with a priori
labelled locations motivate using semi-supervised learning to inform the
Gaussian process hyperparameters. We moreover provide an efficient
Hamiltonian-within-Gibbs sampler for our model. As in other recent work, we
reduce the computational burden associated with inversion of covariance
matrices by exploiting the structure in the covariance matrix. A tensor
decomposition of our covariance matrices allows extended Trench and Durbin
algorithms to be applied it order to reduce the computational complexity of
inversion and hence accelerate computation. A stand-alone R-package
implementing these methods using high-performance C++ libraries is available
at: https://github.com/ococrook/toeplitz
