Spectral Mixture Kernels for Multi-Output Gaussian Processes
Early approaches to multiple-output Gaussian processes (MOGPs) relied on linear combinations of independent, latent, single-output Gaussian processes (GPs). This resulted in cross-covariance functions with limited parametric interpretation, thus conflicting with the ability of single-output GPs to understand lengthscales, frequencies and magnitudes to name a few. On the contrary, current approaches to MOGP are able to better interpret the relationship between different channels by directly modelling the cross-covariances as a spectral mixture kernel with a phase shift. We extend this rationale and propose a parametric family of complex-valued cross-spectral densities and then build on Cramér's Theorem (the multivariate version of Bochner's Theorem) to provide a principled approach to design multivariate covariance functions. The so-constructed kernels are able to model delays among channels in addition to phase differences and are thus more expressive than previous methods, while also providing full parametric interpretation of the relationship across channels. The proposed method is first validated on synthetic data and then compared to existing MOGP methods on two real-world examples.
The paper formulates a new type of covariance function for multi-output Gaussian processes. Instead of directly specifying the cross-covariance functions, the authors propose to specify the cross-power spectral densities and using the inverse Fourier transform to find the corresponding cross-covariances. The paper follows a similar idea to the one proposed by [6] for single output Gaussian processes, where the power spectral density was represented through a mixture of Gaussians and then back-transformed to obtaining a powerful kernel function with extrapolation abilities. The authors applied the newly established covariance for a synthetic data example, and for two real data examples, the weather dataset and the Swiss Jura dataset for metal concentrations. 
Extending the idea from [6] sounds like a clever way to build new types of covariance functions for multiple-outputs. One of the motivations in [6] was to build a kernel function with the ability to perform extrapolation. It would have been interesting to provide this covariance function for multiple-outputs with the same ability. The experimental section only provides results with very common datasets, and the results obtained are similar to the ones obtained with other covariance functions for multiple-outputs. The impact of the paper could be increased by including experiments in datasets for which none of the other models are suitable. 
Did you test for the statistical significance of the results in Tables 1 and 2. It seems that, within the standard deviation, almost all models provide a similar performance.