Nonparametric estimation of a multivariate density under\n Kullback-Leibler loss with ISDE

In this paper, we propose a theoretical analysis of the algorithm ISDE,\nintroduced in previous work. From a dataset, ISDE learns a density written as a\nproduct of marginal density estimators over a partition of the features. We\nshow that under some hypotheses, the Kullback-Leibler loss between the proper\ndensity and the output of ISDE is a bias term plus the sum of two terms which\ngoes to zero as the number of samples goes to infinity. The rate of convergence\nindicates that ISDE tackles the curse of dimensionality by reducing the\ndimension from the one of the ambient space to the one of the biggest blocks in\nthe partition. The constants reflect a combinatorial complexity reduction\nlinked to the design of ISDE.\n