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

TL;DR

This paper provides a theoretical analysis of the ISDE algorithm, showing it effectively estimates multivariate densities by reducing dimensionality and converging under Kullback-Leibler loss, thus addressing the curse of dimensionality.

Contribution

The paper offers a novel theoretical framework for ISDE, demonstrating its convergence properties and its ability to reduce dimensionality in multivariate density estimation.

Findings

Convergence rate of ISDE under Kullback-Leibler loss.

ISDE reduces the curse of dimensionality by partitioning features.

Constants reflect combinatorial complexity reduction.

Abstract

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