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The success of deep convolutional architectures is often attributed in partto their ability to learn multiscale and invariant representations of naturalsignals. However, a precise study of these properties and how they affectlearning guarantees is still missing. In this paper, we consider deepconvolutional representations of signals; we study their invariance totranslations and to more general groups of transformations, their stability tothe action of diffeomorphisms, and their ability to preserve signalinformation. This analysis is carried by introducing a multilayer kernel basedon convolutional kernel networks and by studying the geometry induced by thekernel mapping. We then characterize the corresponding reproducing kernelHilbert space (RKHS), showing that it contains a large class of convolutionalneural networks with homogeneous activation functions. This analysis allows usto separate data representation from learning, and to provide a canonicalmeasure of model complexity, the RKHS norm, which controls both stability andgeneralization of any learned model. In addition to models in the constructedRKHS, our stability analysis also applies to convolutional networks withgeneric activations such as rectified linear units, and we discuss itsrelationship with recent generalization bounds based on spectral norms.

Group Invariance, Stability to Deformations, and Complexity of Deep Convolutional Representations