Is There a Sparse Representation for Noise?
Let's think about it in a different way - Generate Noise from a Dictionary.

Let's create a Dictionary $ A \in \mathbb{R}^{m \times n} $ where each of its rows is normalized (Has Euclidean Norm of $ 1 $) and generated by a Gaussian Random Vector.

Now, let's create $ N $ random vector $ {\left\{ {r}_{i} \right\}}_{i = 1}^{N} $ by:

$$ {r}_{i} = A {g}_{i} $$

Where $ {g}_{i} $ are random vectors $ {g}_{i} \in \mathbb{R}^{n} $ with only $ k \ll n $ non zero elements (Let's say they are pre defined and deterministic otherwise we'll have to deal with multiplication of random variables)..

Now, the output vectors are valid Gaussian Random Variables and given enough of them and some properties of $ A $ you'd be able to build a Dictionary which they are sparse with relation to it.  

Yet it means the dictionary is random by itself and next time when you get new set of random data it won't be able to represent it correctly (It doesn't generalize).

While the above is not rigorous (It means to develop some intuition, not analytic analysis) it tells you why when we deal with finite amount of data it can be done but for noise as in general it won't work.