SPADES and mixture models

TL;DR

This paper introduces SPADES, a sparse density estimation method using $\

Contribution

It demonstrates that SPADES can effectively recover mixture components and adaptively estimate densities with high probability, offering a computationally efficient tuning parameter selection.

Findings

SPADES recovers mixture components with high probability.

SPADES provides minimax adaptive density estimates.

The proposed tuning method reduces computational cost.

Abstract

This paper studies sparse density estimation via $\ell_1$ penalization (SPADES). We focus on estimation in high-dimensional mixture models and nonparametric adaptive density estimation. We show, respectively, that SPADES can recover, with high probability, the unknown components of a mixture of probability densities and that it yields minimax adaptive density estimates. These results are based on a general sparsity oracle inequality that the SPADES estimates satisfy. We offer a data driven method for the choice of the tuning parameter used in the construction of SPADES. The method uses the generalized bisection method first introduced in \citebb09. The suggested procedure bypasses the need for a grid search and offers substantial computational savings. We complement our theoretical results with a simulation study that employs this method for approximations of one and two-dimensional densities with mixtures. The numerical results strongly support our theoretical findings.