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Sparse recovery under matrix uncertainty

We consider the model {eqnarray*}y=Xθ^*+ξ, Z=X+Ξ,{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, $Ξ$ is an $n\times p$ random noise matrix, $ξ\in\mathbb{R}^n$ is a noise independent of $Ξ$, and $θ^*$ 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 $θ^*$. 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 $θ^*$), 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 $θ^*$ 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.