  Deep generative priors are a powerful tool for reconstruction problems with
complex data such as images and text. Inverse problems using such models
require solving an inference problem of estimating the input and hidden units
of the multi-layer network from its output. Maximum a priori (MAP) estimation
is a widely-used inference method as it is straightforward to implement, and
has been successful in practice. However, rigorous analysis of MAP inference in
multi-layer networks is difficult. This work considers a recently-developed
method, multi-layer vector approximate message passing (ML-VAMP), to study MAP
inference in deep networks. It is shown that the mean squared error of the
ML-VAMP estimate can be exactly and rigorously characterized in a certain
high-dimensional random limit. The proposed method thus provides a tractable
method for MAP inference with exact performance guarantees.
