  Interest in multioutput kernel methods is increasing, whether under the guise
of multitask learning, multisensor networks or structured output data. From the
Gaussian process perspective a multioutput Mercer kernel is a covariance
function over correlated output functions. One way of constructing such kernels
is based on convolution processes (CP). A key problem for this approach is
efficient inference. Alvarez and Lawrence (2009) recently presented a sparse
approximation for CPs that enabled efficient inference. In this paper, we
extend this work in two directions: we introduce the concept of variational
inducing functions to handle potential non-smooth functions involved in the
kernel CP construction and we consider an alternative approach to approximate
inference based on variational methods, extending the work by Titsias (2009) to
the multiple output case. We demonstrate our approaches on prediction of school
marks, compiler performance and financial time series.
