Adaptive Dantzig density estimation

TL;DR

This paper introduces an adaptive Dantzig density estimation method that uses $\, ext{l}_1$-minimization with a novel constraint, providing theoretical guarantees and demonstrating improved performance over traditional methods.

Contribution

It proposes a new adaptive Dantzig constraint for density estimation, extending the applicability of $\, ext{l}_1$-based methods with theoretical oracle inequalities and practical calibration strategies.

Findings

The method achieves oracle inequalities under coherence assumptions.

The approach outperforms classical procedures in numerical experiments.

Theoretical results are validated for the associated Lasso estimate.

Abstract

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 $\ell_1$-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.