This talk will review recent developments in the use of statistical methods for inversion of data that are acquired compressively. A particular focus will be placed on dictionary learning and its connection to mixture models. It will be explained how these methods can be used to achieve very efficient inversion, with state-of-the-art accuracy. The basic framework will be demonstrated as applied to real compressively acquired data. In particular, we will consider compressive measurements with respect to time, spectrum and focus. © 2014 OSA.