It is known that given corrupt measurements $y = Af+e \in \mathbb{R}^m$ with $f \in \mathbb{R}^n$ and $\f\_0 < m < n$, one can recover $f$ exactly by solving a convex optimization problem. What if $f$ is instead a square matrix? Can we recover a matrix from corrupt measurements instead of just a vector?

First, I emphasize S. Carnahan's comment: Exact recovery from noisy measurements is not that simple. "Exact recovery" usually means "recovery of the exact support of $f$". Moreover, sparsity assumptions for $f$ and special assumptions for $A$ and the size of $e$ are needed. To address your question: This again depends on a lot of things. Of course you can view this as $n$ multiple instances of the original problem and basically use the previous theory. Other structural assumption lead to other results (e.g. having a "joint sparsity pattern in the columns of $f$"). Buzzwords here are "joint sparsity" or "multiple measurement vectors". 

