High dimensional errors-in-variables models with dependent measurements

Understand

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 $y$.