Projecting on to the multi-layer convolutional sparse coding model

The recently proposed Multi-Layer Convolutional Sparse Coding (ML-CSC) model, consisting of a cascade of convolutional sparse layers, provides a new interpretation of Convolutional Neural Networks (CNNs). Under this framework, the forward pass in a CNN is equivalent to an algorithm that estimates nested sparse representation vectors from a given input signal. Despite having served as a pivotal connection between CNNs and sparse modeling, it is still unclear how to develop pursuit algorithms that serve this model exactly. In this work, we propose a new pursuit formulation by adopting a projection approach. We provide new and improved bounds on the stability of the resulting convolutional sparse representations, and we propose a multi-layer projection algorithm to retrieve them. We demonstrate this algorithm numerically, showing that it is superior to the Layered Basis Pursuit alternative in retrieving the representations of signals belonging to the ML-CSC model.