  We consider a Gaussian sequence space model $X_{\lambda}=f_{\lambda} +
\xi_{\lambda},$ where $\xi $ has a diagonal covariance matrix
$\Sigma=\diag(\sigma_\lambda ^2)$. We consider the situation where the
parameter vector $(f_{\lambda})$ is sparse. Our goal is to estimate the unknown
parameter by a model selection approach. The heterogenous case is much more
involved than the direct model. Indeed, there is no more symmetry inside the
stochastic process that one needs to control since each empirical coefficient
has its own variance. The problem and the penalty do not only depend on the
number of coefficients that one selects, but also on their position. This
appears also in the minimax bounds where the worst coefficients will go to the
larger variances. However, with a careful and explicit choice of the penalty we
are able to select the correct coefficients and get a sharp non-asymptotic
control of the risk of our procedure. Some simulation results are provided.
