LEARNING FUNCTIONS DEFINED OVER SETS OF VECTORS WITH KERNEL METHODS

We consider the problem of learning time-consuming functions defined over unordered sets of vectors.Such functions arise frequently, in particular in the context of networks of devices whose number is not fixed and that interact with each other.A working example is the modeling of a wind farm.Unordered sets of vectors are a mix of integer and continuous input variables suitable for functions that are permutation-invariant.The time-consuming aspect of the functions is, classically, treated by approximating them with a Gaussian process.This study addresses the problem of defining valid and efficient covariance kernels over clouds of points in the context of Gaussian process surrogate modeling.We review methods for defining such kernels.These kernels are compared on a set of analytical functions inspired from different engineering problems, such as the design of experiments and the modeling of wind farms production.The extrapolation properties of the kernels are tested on geometrically transformed clouds.We show that modeling 2D clouds of points as supports of discrete uniform distributions should be preferred to a Gaussian representation of the clouds.A detailed investigation of the good performance of MMD-based kernels illustrates how they adapt their hyperparameters to the geometrical properties of the studied functions.