Adaptive Dantzig density estimation

This paper deals with the problem of density estimation. We aim at building an estimate of an unknown density as a linear combination of functions of a dictionary. Inspired by Cand`es and Tao’s approach, we propose an ${\mathrm{\ell }}_1$-minimization under an adaptive Dantzig constraint coming from sharp concentration inequalities. This allows to consider a wide class of dictionaries. Under local or global coherence assumptions, oracle inequalities are derived. These theoretical results are also proved to be valid for the natural Lasso estimate associated with our Dantzig procedure. Then, the issue of calibrating these procedures is studied from both theoretical and practical points of view. Finally, a numerical study shows the significant improvement obtained by our procedures when compared with other classical procedures.