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 approach would be to learn an overcomplete dictionary to provide good approximations of the training samples using sparse synthesis 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 subgradients, 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 theoretical support for the well-posedness of the learning problem under appropriate conditions.