  This paper introduces a new multivariate convolutional sparse coding based on
tensor algebra with a general model enforcing both element-wise sparsity and
low-rankness of the activations tensors. By using the CP decomposition, this
model achieves a significantly more efficient encoding of the multivariate
signal-particularly in the high order/ dimension setting-resulting in better
performance. We prove that our model is closely related to the Kruskal tensor
regression problem, offering interesting theoretical guarantees to our setting.
Furthermore, we provide an efficient optimization algorithm based on
alternating optimization to solve this model. Finally, we evaluate our
algorithm with a large range of experiments, highlighting its advantages and
limitations.
