  In this article we investigate consistency of selection in regression models
via the popular Lasso method. Here we depart from the traditional linear
regression assumption and consider approximations of the regression function
$f$ with elements of a given dictionary of $M$ functions. The target for
consistency is the index set of those functions from this dictionary that
realize the most parsimonious approximation to $f$ among all linear
combinations belonging to an $L_2$ ball centered at $f$ and of radius
$r_{n,M}^2$. In this framework we show that a consistent estimate of this index
set can be derived via $\ell_1$ penalized least squares, with a data dependent
penalty and with tuning sequence $r_{n,M}>\sqrt{\log(Mn)/n}$, where $n$ is the
sample size. Our results hold for any $1\leq M\leq n^{\gamma}$, for any
$\gamma>0$.
