ANALYSIS OPERATOR LEARNING FOR OVERCOMPLETE COSPARSE REPRESENTATIONS

We consider the problem of learning a low-dimensional signal model from a collection of training samples. The mainstream ap-proach would be to learn an overcomplete dictionary to provide good approximations of the training samples using sparse synthe-sis coefficients. This famous sparse model has a less well known counterpart, in analysis form, called the cosparse analysis model. In this new model, signals are characterized by their parsimony in a transformed domain using an overcomplete analysis operator. We propose to learn an analysis operator from a training corpus using a constrained optimization program based on L1 optimization. We derive a practical learning algorithm, based on projected subgradi-ents, and demonstrate its ability to robustly recover a ground truth analysis operator, provided the training set is of sufficient size. A local optimality condition is derived, providing preliminary theoret-ical support for the well-posedness of the learning problem under appropriate conditions. 1