Empirical Mode Decomposition and Sparsity
The idea of being sparse in a certain basis can be extended and one can talk about being sparse in a frame, or even, being sparse in a dictionary. Now we may lose orthogonality, have redundant signals in our dictionary and no longer have unique signal expansions; which is OK if we only care about sparsity.

The EMD expansion is sparse in the sense that you can express a signal as a linear combination of just a few IMFs.