Learning unions of orthonormal bases with thresholded singular value decomposition

We propose a new method to learn overcomplete dictionaries for sparse coding structured as unions of orthonormal bases. The interest of such a structure is manifold. Indeed, it seems that many signals or images can be modeled as the superimposition of several layers with sparse decompositions in as many bases. Moreover, in such dictionaries, the efficient block coordinate relaxation (BCR) algorithm can be used to compute sparse decompositions. We show that it is possible to design an iterative learning algorithm that produces a dictionary with the required structure. Each step is based on the coefficients estimation, using a variant of BCR, followed by the update of one chosen basis, using singular value decomposition. We assess experimentally how well the learning algorithm recovers dictionaries that may or may not have the required structure, and to what extent the noise level is a disturbing factor.