This paper studies sparse density estimation via ℓ1 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 [10]. 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.
Publié le : 2010-08-15
Classification:
Adaptive estimation,
aggregation,
lasso,
minimax risk,
mixture models,
consistent model selection,
nonparametric density estimation,
oracle inequalities,
penalized least squares,
sparsity,
statistical learning,
62G08,
62C20,
62G05,
62G20
@article{1278861256,
author = {Bunea, Florentina and Tsybakov, Alexandre B. and Wegkamp, Marten H. and Barbu, Adrian},
title = {SPADES and mixture models},
journal = {Ann. Statist.},
volume = {38},
number = {1},
year = {2010},
pages = { 2525-2558},
language = {en},
url = {http://dml.mathdoc.fr/item/1278861256}
}
Bunea, Florentina; Tsybakov, Alexandre B.; Wegkamp, Marten H.; Barbu, Adrian. SPADES and mixture models. Ann. Statist., Tome 38 (2010) no. 1, pp. 2525-2558. http://gdmltest.u-ga.fr/item/1278861256/