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

In this paper, we study deep signal representations that are invariant to groups of transformations and stable to the action of diffeomorphisms without losing signal information. This is achieved by generalizing the multilayer kernel construction introduced in the context of convolutional kernel networks and by studying the geometry of the corresponding reproducing kernel Hilbert space. We show that the signal representation is stable, and that models from this functional space, such as a large class of convolutional neural networks with homogeneous activation functions, may enjoy the same stability. In particular, we study the norm of such models, which acts as a measure of complexity, controlling both stability and generalization.