Convolutional Gaussian Processes
We present a practical way of introducing convolutional structure into Gaussian processes, making them more suited to high-dimensional inputs like images. The main contribution of our work is the construction of an inter-domain inducing point approximation that is well-tailored to the convolutional kernel. This allows us to gain the generalisation benefit of a convolutional kernel, together with fast but accurate posterior inference. We investigate several variations of the convolutional kernel, and apply it to MNIST and CIFAR-10, where we obtain significant improvements over existing Gaussian process models. We also show how the marginal likelihood can be used to find an optimal weighting between convolutional and RBF kernels to further improve performance. This illustration of the usefulness of the marginal likelihood may help automate discovering architectures in larger models.
The paper presents a method to incorporate convolutional structure in GPs, in order to make them more competitive for image classification problems. This is achieved by constructing a convolutional kernel through a latent function g() shared by different patches in the image. 
Approximate inference is based on variational methods and very promising results are given in MNIST and CIFAR datasets. 
The paper is clearly written and  it presents a quite extensive set of experimental results. Regarding the main construction of the convolutional kernel (see eq. 13 and 14) I am not sure if 
the concept of inter-domain inducing variables is really necessary or useful here. The latent function of the GP model is essentially g() which is used inside the likelihood through this patch-sharing, i.e. through the definition of f(x). Under this viewpoint the inducing variables are just points of the latent function, i.e. points of g(), and you don't really need to refer to inter-domains in order to describe the variational sparse approximation. Also having 
presented the method based on g() (and discussing f(x) only as a complementary viewpoint) will probably make it easier for the reader to understand the weighted and colour extensions.   
Can you clarify further the arguments in lines 215-220? In particular, it is not clear why you avoid the inversion of a 2 M × 2 M matrix given that q(u1,u2) is coupled. Section 3 in Supplementary material, that tries to clarify this point, is very unclear and it contains also mistakes (e.g. in equation 13). 
Overall I believe that this is a good paper since it improves a lot on GP classification by incorporating convolutional structure. Still of course GPs cannot compete with Deep NNs, but this paper introduces some new ideas and it makes good steps forward to making GP classifiers more competitive for image classification.