  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
convolutional neural networks (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.
