  The LASSO is a recent technique for variable selection in the regression
model \bean y & = & X\beta + z, \eean where $X\in \R^{n\times p}$ and $z$ is a
centered gaussian i.i.d. noise vector $\mathcal N(0,\sigma^2I)$. The LASSO has
been proved to achieve remarkable properties such as exact support recovery of
sparse vectors when the columns are sufficently incoherent and low prediction
error under even less stringent conditions. However, many matrices do not
satisfy small coherence in practical applications and the LASSO estimator may
thus suffer from what is known as the slow rate regime.
  The goal of the present paper is to study the LASSO from a slightly different
perspective by proposing a mixture model for the design matrix which is able to
capture in a natural way the potentially clustered nature of the columns in
many practical situations. In this model, the columns of the design matrix are
drawn from a Gaussian mixture model. Instead of requiring incoherence for the
design matrix $X$, we only require incoherence of the much smaller matrix of
the mixture's centers.
  Our main result states that $X\beta$ can be estimated with the same precision
as for incoherent designs except for a correction term depending on the maximal
variance in the mixture model.
