High dimensional errors-in-variables models with dependent measurements

We consider a parsimonious model for fitting observation data $X = X_0 + W$ with two-way dependencies; that is, we use the signal matrix $X_0$ to explain column-wise dependency in $X$, and the measurement error matrix $W$ to explain its row-wise dependency. In the matrix normal setting, we have the following representation where $X$ follows the matrix variate normal distribution with the Kronecker Sum covariance structure: ${\rm vec}\{X\} \sim \mathcal{N}(0, \Sigma)$ where $\Sigma = A \oplus B$, which is generalized to the subgaussian settings as follows. Suppose that we observe $y \in {\bf R}^f$ and $X \in {\bf R}^{f \times m}$ in the following 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, $\epsilon \in {\bf R}^m$ 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. Under sparsity and restrictive eigenvalue type of conditions, we show that one is able to recover a sparse vector $\beta^* \in {\bf R}^m$ from the following 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.