Compression Sensing for Blind Source Separation
I will sketch an idea how to use Sparse Represenattion (Dictionary Learning) for BSS.

Let's say we have $ \mathcal{S} = \mathcal{S}_{1} \cup \mathcal{S}_{2} $ where $ \mathcal{S}_{1} = \left\{ {x}_{i} \right\}_{i = 1}^{m} $ and $ \mathcal{S}_{2} = \left\{ {y}_{i} \right\}_{i = 1}^{n} $.

So if we assume different sources for $ \mathcal{S}_{1} $ and $ \mathcal{S}_{2} $ then what we can do is learn a dictionary (For example, using K-SVD) for $ {z}_{i} \in \mathcal{S} $.

After we learned this Dictionary we can look at the representations of each signal.
If the our assumption was correct, the atoms used for signals from $ \mathcal{S}_{1} $ will be different than atoms used for signals from $ \mathcal{S}_{2} $.