  The Linear Model of Co-regionalization (LMC) is a very general model of
multitask gaussian process for regression or classification. While its
expressivity and conceptual simplicity are appealing, naive implementations
have cubic complexity in the number of datapoints and number of tasks, making
approximations mandatory for most applications. However, recent work has shown
that under some conditions the latent processes of the model can be decoupled,
leading to a complexity that is only linear in the number of said processes. We
here extend these results, showing from the most general assumptions that the
only condition necessary to an efficient exact computation of the LMC is a mild
hypothesis on the noise model. We introduce a full parametrization of the
resulting \emph{projected LMC} model, and an expression of the marginal
likelihood enabling efficient optimization. We perform a parametric study on
synthetic data to show the excellent performance of our approach, compared to
an unrestricted exact LMC and approximations of the latter. Overall, the
projected LMC appears as a credible and simpler alternative to state-of-the art
models, which greatly facilitates some computations such as leave-one-out
cross-validation and fantasization.
