using any of the kernels above, practical considerations of tuning the parameters and ensuring reasonable computation time must be considered. Regularization perspective Approached from the regularization perspective, parameter tuning is similar to the scalar-valued case and can generally be accomplished with cross validation. Solving the required linear system is typically expensive in memory and time. If the kernel is separable, a coordinate transform can convert to a block-diagonal matrix, greatly reducing the computational burden by solving D independent subproblems (plus the eigendecomposition of ). In particular, for a least squares loss function (Tikhonov regularization), there exists a closed form solution for : Bayesian perspective There are many works related to parameter estimation for Gaussian processes. Some methods such as maximization of the marginal likelihood (also known as evidence approximation, type II maximum likelihood, empirical Bayes), and least squares give point estimates of the parameter vector . There are also works employing a full Bayesian inference by assigning priors to and computing the posterior distribution through a sampling procedure. For non-Gaussian likelihoods, there is no closed form solution for the posterior distribution or for the marginal likelihood. However, the marginal likelihood can be approximated under a Laplace, variational Bayes or expectation propagation (EP) approximation frameworks for multiple output classification and used to find estimates for the hyperparameters. The main computational problem in the Bayesian viewpoint is the same as the one appearing in regularization theory of inverting the matrix This step is necessary for computing the marginal likelihood and the predictive distribution. For most proposed approximation methods to reduce computation, the computational efficiency gained is independent of the particular method employed (e.g. LMC, process convolution) used to compute the multi-output covariance matrix. A summary of different methods for reducing computational complexity in multi-output Gaussian processes is presented in. References Machine learning algorithms Kernel methods for machine learning