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First steps towards a mathematical theory of deep convolutional neuralnetworks for feature extraction were made---for the continuous-time case---inMallat, 2012, and Wiatowski and B\"olcskei, 2015. This paper considers thediscrete case, introduces new convolutional neural network architectures, andproposes a mathematical framework for their analysis. Specifically, weestablish deformation and translation sensitivity results of local and globalnature, and we investigate how certain structural properties of the inputsignal are reflected in the corresponding feature vectors. Our theory appliesto general filters and general Lipschitz-continuous non-linearities and poolingoperators. Experiments on handwritten digit classification and facial landmarkdetection---including feature importance evaluation---complement thetheoretical findings.

Discrete Deep Feature Extraction: A Theory and New Architectures