Sparse density estimation with l1 penalties

Authors:
Florentina Bunea;Alexandre B. Tsybakov;Marten H. Wegkamp
Affiliations:
Florida State University, Tallahassee, FL;Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI, France;Florida State University, Tallahassee, FL
Venue:
COLT'07 Proceedings of the 20th annual conference on Learning theory
Year:
2007

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Abstract

This paper studies oracle properties of l1-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.