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Deep Convolutional Sparse Coding (D-CSC) is a framework reminiscent of deepconvolutional neural networks (DCNNs), but by omitting the learning of thedictionaries one can more transparently analyse the role of the activationfunction and its ability to recover activation paths through the layers.Papyan, Romano, and Elad conducted an analysis of such an architecture,demonstrated the relationship with DCNNs and proved conditions under which theD-CSC is guaranteed to recover specific activation paths. A technicalinnovation of their work highlights that one can view the efficacy of the ReLUnonlinear activation function of a DCNN through a new variant of the tensor'ssparsity, referred to as stripe-sparsity. Using this they proved thatrepresentations with an activation density proportional to the ambientdimension of the data are recoverable. We extend their uniform guarantees to amodified model and prove that with high probability the true activation istypically possible to recover for a greater density of activations per layer.Our extension follows from incorporating the prior work on one stepthresholding by Schnass and Vandergheynst.

Towards an understanding of CNNs: analysing the recovery of activation pathways via Deep Convolutional Sparse Coding