  Gaussian process regression (GPR) is a powerful machine learning method which
has recently enjoyed wider use, in particular in physical sciences. In its
original formulation, GPR uses a square matrix of covariances among training
data and can be viewed as linear regression problem with equal numbers of
training data and basis functions. When data are sparse, avoidance of
overfitting and optimization of hyperparameters of GPR are difficult, in
particular in high-dimensional spaces where the data sparsity issue cannot
practically be resolved by adding more data. Optimal choice of hyperparameters,
however, determines success or failure of the application of the GPR method. We
show that parameter optimization is facilitated by rectangularization of the
defining equation of GPR. On the example of a 15-dimensional molecular
potential energy surface we demonstrate that this approach allows effective
hyperparameter tuning even with very sparse data.
