  Suppose that we observe $y \in \mathbb{R}^f$ and $X \in \mathbb{R}^{f \times
m}$ in the following errors-in-variables model: \begin{eqnarray*} y & = & X_0
\beta^* + \epsilon \\ X & = & X_0 + W \end{eqnarray*} where $X_0$ is a $f
\times m$ design matrix with independent subgaussian row vectors, $\epsilon \in
\mathbb{R}^f$ is a noise vector and $W$ is a mean zero $f \times m$ random
noise matrix with independent subgaussian column vectors, independent of $X_0$
and $\epsilon$. This model is significantly different from those analyzed in
the literature in the sense that we allow the measurement error for each
covariate to be a dependent vector across its $f$ observations. Such error
structures appear in the science literature when modeling the trial-to-trial
fluctuations in response strength shared across a set of neurons.
  Under sparsity and restrictive eigenvalue type of conditions, we show that
one is able to recover a sparse vector $\beta^* \in \mathbb{R}^m$ from the
model given a single observation matrix $X$ and the response vector $y$. We
establish consistency in estimating $\beta^*$ and obtain the rates of
convergence in the $\ell_q$ norm, where $q = 1, 2$ for the Lasso-type
estimator, and for $q \in [1, 2]$ for a Dantzig-type conic programming
estimator. We show error bounds which approach that of the regular Lasso and
the Dantzig selector in case the errors in $W$ are tending to 0.
