I was reading this paper ("How well can we estimate a sparse vector?" by Candès and Davenport, link: http://arxiv.org/pdf/1104.5246v5.pdf). They consider the problem of estimating a $k$-sparse vector $x\in\mathbb{R}^{n}$ from measurements $y\in\mathbb{R}^{m}$, under the following model:
$y=Ax+z$,
with $A\in\mathbb{R}^{m\times n}$ and $z\sim\mathcal{N}(0,\sigma^{2}I)$ and establish lower bounds on the mean squared error (MSE) $\frac{1}{n}\|x-\hat{x}\|_{2}^{2}$ where $\hat{x}$ is the estimate of $x$.
On page 2 (section 1.1), they consider the case when the support $x$ is known and say that the MSE can be shown to be
$\mathbb{E}(\frac{1}{n}\|x-\hat{x}\|_{2}^{2})=(\frac{k\sigma^{2}}{m})(\frac{k}{n})$.
How can one prove this? If $\hat{x}$ is known to be the output of a particular optimization problem (or if we directly have an expression for it), we can work out the expression for the MSE. How do we proceed otherwise?