# Global Guarantees for Enforcing Deep Generative Priors by Empirical Risk

## TL;DR

This paper provides the first theoretical guarantees that enforce deep generative priors via empirical risk minimization have no spurious stationary points, explaining their empirical success in solving inverse problems.

## Contribution

It establishes that under certain conditions, the non-convex optimization landscape is free of spurious stationary points, ensuring reliable solutions for deep generative prior enforcement.

## Key findings

- No spurious stationary points outside neighborhoods of solutions.
- High probability of descent directions away from solutions.
- Global geometry guarantees for non-convex inverse problems.

## Abstract

We examine the theoretical properties of enforcing priors provided by generative deep neural networks via empirical risk minimization. In particular we consider two models, one in which the task is to invert a generative neural network given access to its last layer and another in which the task is to invert a generative neural network given only compressive linear observations of its last layer. We establish that in both cases, in suitable regimes of network layer sizes and a randomness assumption on the network weights, that the non-convex objective function given by empirical risk minimization does not have any spurious stationary points. That is, we establish that with high probability, at any point away from small neighborhoods around two scalar multiples of the desired solution, there is a descent direction. Hence, there are no local minima, saddle points, or other stationary points outside these neighborhoods. These results constitute the first theoretical guarantees which establish the favorable global geometry of these non-convex optimization problems, and they bridge the gap between the empirical success of enforcing deep generative priors and a rigorous understanding of non-linear inverse problems.