Tensor Convolutional Dictionary Learning with CP Low-Rank activations

In this paper, we propose an extension of the standard CDL problem with tensor representation, where each activation is constrained to be "low-rank" through a Canonical Polyadic decomposition. We show that this important additional constraint increases the robustness of the CDL with respect to strong noise and improve the interpretability of the results. Additionally, we discuss in details the benefits of this representation. Then, we propose two new algorithms, based on respectively ADMM or FISTA, that efficiently solve this problem, by leveraging the low-rank property and achieve a lower complexity than the leading CDL algorithms. Finally, we evaluate our approach on a wide range of experiments, highlighting the modularity and the important advantages of this tensorial lowrank formulation.