Linearly constrained Gaussian processes
We consider a modification of the covariance function in Gaussian processes to correctly account for known linear operator constraints. By modeling the target function as a transformation of an underlying function, the constraints are explicitly incorporated in the model such that they are guaranteed to be fulfilled by any sample drawn or prediction made. We also propose a constructive procedure for designing the transformation operator and illustrate the result on both simulated and real-data examples.
Summary of the Paper:
This paper describes a mechanism to incorporate linear operator constraints in the framework of Gaussian process regression. For this, the mean function and the covariance function of the Gaussian processes are changed. The aim of this transformation is to guarantee that samples from the GP posterior distribution satisfy the constraints indicated. These constraints are typically in the form of partial derivatives, although any linear operator can be considered in practice, e.g., integration too. Traditional methods incorporated these constraints by introducing additional observations. This has the limitation that is more expensive and restricted to the observations made. The framework proposed is evaluated in a synthetic problem and in a real problem, showing benefits with respect to the data augmentation strategy.
Detailed comments:
Quality:
I think the quality of the paper is good in general. It is a very well written paper. Furthermore, all the key points are carefully described. It also has a strong related work section. The weakest point is however, the section on experiments in which only a synthetic dataset and a real dataset is considered.
Clarity:
The clarity of the paper is high.
Originality:
As far as I know the paper seems original. There are some related methods in the literature. However, they simply augment the observed data with virtual instances that have the goal of guaranteeing the constraints imposed.
Significance:
I think the problem addressed by the authors is relevant and important for the machine learning community. However, I have the feeling that the authors have not success in noting this. The examples used by the authors are a bit simple. For example they only consider a single real example and only the linear operator of derivatives. I have the feeling that this paper may have potential applications in probabilistic numeric methods, in which often a GP is used. 
Summing up, I think that this is a good paper. However, the weak experimental section questions its significance. Furthermore, I find difficult to find practical applications within the machine learning community. The authors should have made a bigger effort on showing this. I would consider it hence a borderline paper.