Data-driven kernel representations for sampling with an unknown block dependence structure under correlation constraints

The multidimensional Gaussian kernel-density estimation (G-KDE) is a powerful tool to identify the distribution of random vectors when the maximal information is a set of independent realizations. For these methods, a key issue is the choice of the kernel and the optimization of the bandwidth matrix. To optimize these kernel representations, two adaptations of the classical G-KDE are presented. First, it is proposed to add constraints on the mean and the covariance matrix in the G-KDE formalism. Secondly, it is suggested to separate in different groups the components of the random vector of interest that could reasonably be considered as independent. This block by block decomposition is carried out by looking for the maximum of a cross-validation likelihood quantity that is associated with the block formation. This leads to a tensorized version of the classical G-KDE. Finally, it is shown on a series of examples how these two adaptations can improve the nonparametric representations of the densities of random vectors, especially when the number of available realizations is relatively low compared to their dimensions.