2009/05/06 by Karine Bertin, Bertin, Karine, Erwan Le Pennec +3 · 1 citation
Mathematics · #62G05 #62G07 #62G20 #FOS: Mathematics #Statistical Methods and Inference #Statistics Theory (math.ST) #math.ST #msc:62G05 #msc:62G07 #msc:62G20 #stat.TH
paper · pdf · doi:10.48550/arxiv.0905.0884
arxiv created 2009/05/06 · openalex publication_date 2009/05/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Candès and Tao's approach, we propose an ℓ1-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.