# Asymptotics of MAP Inference in Deep Networks

## TL;DR

This paper rigorously analyzes the performance of MAP inference in deep networks using ML-VAMP, providing exact characterization of mean squared error in high-dimensional limits.

## Contribution

It introduces a tractable method, ML-VAMP, for analyzing MAP inference in deep networks with exact performance guarantees.

## Key findings

- ML-VAMP accurately characterizes MAP inference performance.
- Mean squared error can be exactly predicted in high-dimensional limits.
- Provides rigorous theoretical analysis for deep network inference.

## Abstract

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.