  We consider multi-task regression models where observations are assumed to be
a linear combination of several latent node and weight functions, all drawn
from Gaussian process (GP) priors that allow nonzero covariance between grouped
latent functions. We show that when these grouped functions are conditionally
independent given a group-dependent pivot, it is possible to parameterize the
prior through sparse Cholesky factors directly, hence avoiding their
computation during inference. Furthermore, we establish that kernels that are
multiplicatively separable over input points give rise to such sparse
parameterizations naturally without any additional assumptions. Finally, we
extend the use of these sparse structures to approximate posteriors within
variational inference, further improving scalability on the number of
functions. We test our approach on multi-task datasets concerning distributed
solar forecasting and show that it outperforms several multi-task GP baselines
and that our sparse specifications achieve the same or better accuracy than
non-sparse counterparts.
