  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.
