In this paper, the authors present several approaches to tackle the difficult problem of designing proxy models for probability density functions. 

The first approach involves standard kernel regression, but the authors also introduce a new kernel estimator based on the Hellinger distance. 

In the second approach, a functional decomposition point of view is proposed. 

The idea is to build a low-dimensional approximation basis for the space of pdfs, which is adapted to the experimental design. 
Once the basis is computed, pdf prediction is achieved by building proxy models for the projection coefficients in the basis. 

The authors suggest three techniques for constructing an approximation basis: constrained PCA, Modified Magic Points and Alternate Quadratic Minimization.

This research work is highly motivated by industrial applications, and I really enjoy the large overview of numerical examples on both toy and industrial cases. 

Such comparisons on a variety of problems are essential for practitioners. 

In particular, one of the conclusions is that the kernel density approach performs well in a low dimensional setting but suffers from the curse of dimensionality, just as in standard kernel regression. Interestingly, the novel kernel estimate based on the Hellinger distance seems to outperform the standard one.


Concerning the decomposition methods, the ultimate step (i.e. building the final surrogate model) is unfortunately not performed in the paper. Indeed, as stated by the authors, this would imply designing proxy models for the decomposition coefficients subject to constraints in order to produce pdfs. 

This is a complex task: even if recent research demonstrated that some constraints can be incorporated in Gaussian Process metamodels, it is still unclear if these results can be extended to the type of constraints which are encountered here. 


In the future I expect that we will see very interesting research on this topic, surely combining decomposition steps with dedicated constrained surrogate models and maybe borrowing ideas from new approaches on RKHS embeddings of probability density functions. 

This paper and all its numerical illustrations will then serve as a benchmark for the new papers to come.




Sébastien Da Veiga (SNECMA, SAFRAN)