  Recently there has been an increasing interest in methods that deal with
multiple outputs. This has been motivated partly by frameworks like multitask
learning, multisensor networks or structured output data. From a Gaussian
processes perspective, the problem reduces to specifying an appropriate
covariance function that, whilst being positive semi-definite, captures the
dependencies between all the data points and across all the outputs. One
approach to account for non-trivial correlations between outputs employs
convolution processes. Under a latent function interpretation of the
convolution transform we establish dependencies between output variables. The
main drawbacks of this approach are the associated computational and storage
demands. In this paper we address these issues. We present different sparse
approximations for dependent output Gaussian processes constructed through the
convolution formalism. We exploit the conditional independencies present
naturally in the model. This leads to a form of the covariance similar in
spirit to the so called PITC and FITC approximations for a single output. We
show experimental results with synthetic and real data, in particular, we show
results in pollution prediction, school exams score prediction and gene
expression data.
