Sparse density estimation with $\ell_1$ penalties.
Florentina Bunea, Alexandre Tsybakov and Marten Wegkamp
In: COLT 2007, 13-15 Jun 2007, San Diego, USA.
This paper studies oracle properties of $\ell_1$-penalized
estimators of a probability density. We show that the penalized
least squares estimator satisfies sparsity oracle inequalities,
i.e., bounds in terms of the number of non-zero components of the
oracle vector. The results are valid even when the dimension of the
model is (much) larger than the sample size. They are applied to
estimation in sparse high-dimensional mixture models, to
nonparametric adaptive density estimation and to the problem of
aggregation of density estimators.