  Signal recovery under generative neural network priors has emerged as a
promising direction in statistical inference and computational imaging.
Theoretical analysis of reconstruction algorithms under generative priors is,
however, challenging. For generative priors with fully connected layers and
Gaussian i.i.d. weights, this was achieved by the multi-layer approximate
message (ML-AMP) algorithm via a rigorous state evolution. However, practical
generative priors are typically convolutional, allowing for computational
benefits and inductive biases, and so the Gaussian i.i.d. weight assumption is
very limiting. In this paper, we overcome this limitation and establish the
state evolution of ML-AMP for random convolutional layers. We prove in
particular that random convolutional layers belong to the same universality
class as Gaussian matrices. Our proof technique is of an independent interest
as it establishes a mapping between convolutional matrices and spatially
coupled sensing matrices used in coding theory.
