  We introduce general scattering transforms as mathematical models of deep
neural networks with l2 pooling. Scattering networks iteratively apply complex
valued unitary operators, and the pooling is performed by a complex modulus. An
expected scattering defines a contractive representation of a high-dimensional
probability distribution, which preserves its mean-square norm. We show that
unsupervised learning can be casted as an optimization of the space contraction
to preserve the volume occupied by unlabeled examples, at each layer of the
network. Supervised learning and classification are performed with an averaged
scattering, which provides scattering estimations for multiple classes.
