Sparse signal representations play an important role in many applications in signal processing.  Recently, attention has turned to learning sparse signal representations that are directly adapted to data. In particular, there has been extensive research on learning synthesis dictionaries and some recent work on learning analysis dictionaries. These offer improved performance over analytical dictionaries in various applications.

We consider instead the transform model, which has been classically used with analytical sparsifying transforms, such as DCT, wavelets, curvelets, and finite differences. We describe a new formulation and algorithms for data-driven learning of sparsifying transforms, and illustrate their advantages.