  We consider a $l_1$-penalization procedure in the non-parametric Gaussian
regression model. In many concrete examples, the dimension $d$ of the input
variable $X$ is very large (sometimes depending on the number of observations).
Estimation of a $\beta$-regular regression function $f$ cannot be faster than
the slow rate $n^{-2\beta/(2\beta+d)}$. Hopefully, in some situations, $f$
depends only on a few numbers of the coordinates of $X$. In this paper, we
construct two procedures. The first one selects, with high probability, these
coordinates. Then, using this subset selection method, we run a local
polynomial estimator (on the set of interesting coordinates) to estimate the
regression function at the rate $n^{-2\beta/(2\beta+d^*)}$, where $d^*$, the
"real" dimension of the problem (exact number of variables whom $f$ depends
on), has replaced the dimension $d$ of the design. To achieve this result, we
used a $l_1$ penalization method in this non-parametric setup.
