  We consider the model {eqnarray*}y=X\theta^*+\xi, Z=X+\Xi,{eqnarray*} where
the random vector $y\in\mathbb{R}^n$ and the random $n\times p$ matrix $Z$ are
observed, the $n\times p$ matrix $X$ is unknown, $\Xi$ is an $n\times p$ random
noise matrix, $\xi\in\mathbb{R}^n$ is a noise independent of $\Xi$, and
$\theta^*$ is a vector of unknown parameters to be estimated. The matrix
uncertainty is in the fact that $X$ is observed with additive error. For
dimensions $p$ that can be much larger than the sample size $n$, we consider
the estimation of sparse vectors $\theta^*$. Under matrix uncertainty, the
Lasso and Dantzig selector turn out to be extremely unstable in recovering the
sparsity pattern (i.e., of the set of nonzero components of $\theta^*$), even
if the noise level is very small. We suggest new estimators called matrix
uncertainty selectors (or, shortly, the MU-selectors) which are close to
$\theta^*$ in different norms and in the prediction risk if the restricted
eigenvalue assumption on $X$ is satisfied. We also show that under somewhat
stronger assumptions, these estimators recover correctly the sparsity pattern.
