Efficient learning of dictionaries with low-rank atoms

Sparsity-based techniques have been popular in many applications in signal and image processing. In particular, the data-driven adaptation of sparse signal models such as the synthesis model has shown promise in applications. However, dictionary learning problems are typically nonconvex and NP-hard, and the usual alternating minimization approaches for learning are often expensive and lack convergence guarantees. In this work, we investigate efficient methods for learning structured synthesis dictionaries. In particular, we model the atoms (columns) of the dictionary, after reshaping, as low-rank. We propose a block coordinate descent algorithm for our dictionary learning model that involves efficient optimal updates. We also provide a convergence analysis of the proposed method for a highly nonconvex problem. Our numerical experiments show the usefulness of our schemes in inverse problem settings, such as in dynamic MRI and inpainting.