On signal reconstruction in white noise using dictionaries

Assume that we observe a Gaussian vector $Y=X\beta+\sigma\xi$, where $X$ is a known $p\times n$ matrix with $p\ge n$, $\beta\in\mathbb R^n$ is an unknown vector, and $\xi\in\mathbb R^n$ is a standard Gaussian white noise. The problem is to reconstruct $X\beta$ from observations $Y$, provided that $\beta$ is a sparse vector.