  Despite their impressive performance, deep convolutional neural networks
(CNNs) have been shown to be sensitive to small adversarial perturbations.
These nuisances, which one can barely notice, are powerful enough to fool
sophisticated and well performing classifiers, leading to ridiculous
misclassification results. In this paper we analyze the stability of
state-of-the-art deep-learning classification machines to adversarial
perturbations, where we assume that the signals belong to the (possibly
multi-layer) sparse representation model. We start with convolutional sparsity
and then proceed to its multi-layered version, which is tightly connected to
CNNs. Our analysis links between the stability of the classification to noise
and the underlying structure of the signal, quantified by the sparsity of its
representation under a fixed dictionary. In addition, we offer similar
stability theorems for two practical pursuit algorithms, which are posed as two
different deep-learning architectures - the layered Thresholding and the
layered Basis Pursuit. Our analysis establishes the better robustness of the
later to adversarial attacks. We corroborate these theoretical results by
numerical experiments on three datasets: MNIST, CIFAR-10 and CIFAR-100.
