  Convolutional neural networks (CNN) have led to many state-of-the-art results
spanning through various fields. However, a clear and profound theoretical
understanding of the forward pass, the core algorithm of CNN, is still lacking.
In parallel, within the wide field of sparse approximation, Convolutional
Sparse Coding (CSC) has gained increasing attention in recent years. A
theoretical study of this model was recently conducted, establishing it as a
reliable and stable alternative to the commonly practiced patch-based
processing. Herein, we propose a novel multi-layer model, ML-CSC, in which
signals are assumed to emerge from a cascade of CSC layers. This is shown to be
tightly connected to CNN, so much so that the forward pass of the CNN is in
fact the thresholding pursuit serving the ML-CSC model. This connection brings
a fresh view to CNN, as we are able to attribute to this architecture
theoretical claims such as uniqueness of the representations throughout the
network, and their stable estimation, all guaranteed under simple local
sparsity conditions. Lastly, identifying the weaknesses in the above pursuit
scheme, we propose an alternative to the forward pass, which is connected to
deconvolutional, recurrent and residual networks, and has better theoretical
guarantees.
