  Many practical machine learning tasks employ very deep convolutional neural
networks. Such large depths pose formidable computational challenges in
training and operating the network. It is therefore important to understand how
fast the energy contained in the propagated signals (a.k.a. feature maps)
decays across layers. In addition, it is desirable that the feature extractor
generated by the network be informative in the sense of the only signal mapping
to the all-zeros feature vector being the zero input signal. This "trivial
null-set" property can be accomplished by asking for "energy conservation" in
the sense of the energy in the feature vector being proportional to that of the
corresponding input signal. This paper establishes conditions for energy
conservation (and thus for a trivial null-set) for a wide class of deep
convolutional neural network-based feature extractors and characterizes
corresponding feature map energy decay rates. Specifically, we consider general
scattering networks employing the modulus non-linearity and we find that under
mild analyticity and high-pass conditions on the filters (which encompass,
inter alia, various constructions of Weyl-Heisenberg filters, wavelets,
ridgelets, ($\alpha$)-curvelets, and shearlets) the feature map energy decays
at least polynomially fast. For broad families of wavelets and Weyl-Heisenberg
filters, the guaranteed decay rate is shown to be exponential. Moreover, we
provide handy estimates of the number of layers needed to have at least
$((1-\varepsilon)\cdot 100)\%$ of the input signal energy be contained in the
feature vector.
