When to use sparse overcomplete representations?

I'm seeing a lot of methods these days using sparse overcomplete dictionaries to represent the data.

Could someone provide an intuitive explanation why this should work? It seems counter to the idea of dimensionality reduction; the belief that there is a simpler set of latent variables that can explain your data.

Besides just trying to see if it works, is there any way to suspect your data would benefit from a sparse overcomplete representation as opposed to a dimensionality reduction?