Harmonic Decompositions of Convolutional Networks

The renewed interest in convolutional neural networks [12, 15] in computer vision and signal processing has lead to a major leap in generalization performance on common task benchmarks, supported by the recent advances in graphical processing hardware and the collection of huge labelled datasets for training and evaluation. Convolutional neural networks pose major a challenge to statistical learning theory. First and foremost a convolutional network learns from data, jointly, both a feature representation through its hidden layers and a prediction function through its ultimate layer. A convolutional neural network implements a function unfolding as a composition of basic functions (respectively nonlinearity, convolution, and pooling), which appear to model well visual information in images. Yet the relevant function spaces to analyze their statistical performance remain unclear. The analysis of convolutional neural networks (CNNs) has been an active research topic. Different viewpoints have been developed. A straightforward viewpoint is to dismiss completely the grid-or latticestructure of images and analyze a multi-layer perceptron (MLP) instead acting on vectorized images, which has the downside the set aside the most interesting property CNNs which is to model well images that is data with a 2D lattice structure.