# GAN-based Projector for Faster Recovery with Convergence Guarantees in Linear Inverse Problems

## TL;DR

This paper introduces a GAN-based projector for linear inverse problems that accelerates recovery by 60-80 times, guarantees convergence under certain conditions, and reduces measurement requirements by 5-10 times, applicable across various tasks.

## Contribution

It proposes a network-based projector for PGD that speeds up GAN-based recovery, with theoretical convergence guarantees and a method for designing measurement matrices, applicable to multiple inverse problems.

## Key findings

- Achieves 60-80x faster recovery than previous GAN methods.
- Requires 5-10x fewer measurements for similar accuracy.
- Provides convergence guarantees under moderate conditioning.

## Abstract

A Generative Adversarial Network (GAN) with generator $G$ trained to model the prior of images has been shown to perform better than sparsity-based regularizers in ill-posed inverse problems. Here, we propose a new method of deploying a GAN-based prior to solve linear inverse problems using projected gradient descent (PGD). Our method learns a network-based projector for use in the PGD algorithm, eliminating expensive computation of the Jacobian of $G$. Experiments show that our approach provides a speed-up of $60\text{-}80\times$ over earlier GAN-based recovery methods along with better accuracy. Our main theoretical result is that if the measurement matrix is moderately conditioned on the manifold range($G$) and the projector is $\delta$-approximate, then the algorithm is guaranteed to reach $O(\delta)$ reconstruction error in $O(log(1/\delta))$ steps in the low noise regime. Additionally, we propose a fast method to design such measurement matrices for a given $G$. Extensive experiments demonstrate the efficacy of this method by requiring $5\text{-}10\times$ fewer measurements than random Gaussian measurement matrices for comparable recovery performance. Because the learning of the GAN and projector is decoupled from the measurement operator, our GAN-based projector and recovery algorithm are applicable without retraining to all linear inverse problems, as confirmed by experiments on compressed sensing, super-resolution, and inpainting.