  Recent work in the literature has shown experimentally that one can use the
lower layers of a trained convolutional neural network (CNN) to model natural
textures. More interestingly, it has also been experimentally shown that only
one layer with random filters can also model textures although with less
variability. In this paper we ask the question as to why one layer CNNs with
random filters are so effective in generating textures? We theoretically show
that one layer convolutional architectures (without a non-linearity) paired
with the an energy function used in previous literature, can in fact preserve
and modulate frequency coefficients in a manner so that random weights and
pretrained weights will generate the same type of images. Based on the results
of this analysis we question whether similar properties hold in the case where
one uses one convolution layer with a non-linearity. We show that in the case
of ReLu non-linearity there are situations where only one input will give the
minimum possible energy whereas in the case of no nonlinearity, there are
always infinite solutions that will give the minimum possible energy. Thus we
can show that in certain situations adding a ReLu non-linearity generates less
variable images.
