  Infinite width limits of deep neural networks often have tractable forms.
They have been used to analyse the behaviour of finite networks, as well as
being useful methods in their own right. When investigating infinitely wide
CNNs it was observed that the correlations arising from spatial weight sharing
disappear in the infinite limit. This is undesirable, as spatial correlation is
the main motivation behind CNNs. We show that the loss of this property is not
a consequence of the infinite limit, but rather of choosing an independent
weight prior. Correlating the weights maintains the correlations in the
activations. Varying the amount of correlation interpolates between
independent-weight limits and mean-pooling. Empirical evaluation of the
infinitely wide network shows that optimal performance is achieved between the
extremes, indicating that correlations can be useful.
