# Recovering a matrix instead of a vector

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?

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Isn't a matrix just a special case of a vector? That is, can't you "flatten" an $n\times n$ matrix into a vector in $\mathbb{R}^{n^2}$ and then solve the problem by the usual means? – Noah Stein Nov 23 '11 at 3:00
I'm not sure I agree with the assertion in the first sentence. It seems to depend strongly on the nature of corruption, and on the precision of measurements. – S. Carnahan Nov 23 '11 at 5:31
– rodms Jun 5 '14 at 21:19

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".