GAN-Based Projector for Faster Recovery With Convergence Guarantees in Linear Inverse Problems

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 illposed 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 60-80× 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 δ-approximate, then the algorithm is guaranteed to reach O(δ) reconstruction error in O(log(1/δ)) 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 5-10× 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.

Fixed-Point and Objective Convergence of Plug-and-Play Algorithms

“…It was shown in [11] that PnP convergence is guaranteed for ISTA and ADMM for a specially trained CNN denoiser, provided the data-fidelity is strongly convex. Apart from [11], PnP convergence has been established for CNN denoisers [47], [48], generative denoisers [5], and GAN-based projectors [49]. Moreover, it was shown in [50] that the DnCNN denoiser can be approximately expressed as the proximal operator of a nonconvex function.…”

Principal Component Wavelet Networks for Solving Linear Inverse Problems

“…Differential unrolled ADMM (DU-ADMM) has been proposed [15], which improves on the original OneNet, showing faster convergence and a reduction in overfitting during training. In a similar vein, Reference [16] used projected gradient descent (PGD) to allow a significant speed up in the convergence, by removing the need to compute the Jacobian of the gradient. Nevertheless, the training still requires generating artificial perturbed images, which the system learns to project back into the original so could be prone to bias due to the choice of perturbations.…”

Compressed Sensing via Measurement-Conditional Generative Models

“…[9] first employed pre-trained generators to reconstruct signals, providing a recovery guarantee. Thereafter, a number of studies [13], [17], [19], [20], [22], [23] have been conducted to enhance the performance of CSPG. In contrast to CSPG, CSUG [12], [30] includes methods based on the deep image prior [31] so that the weights of an untrained generator can be trained using only one measurement vector y te .…”

“…As neural networks (NNs) have achieved enormous success in both supervised learning including regression and classification tasks and unsupervised learning such as clustering and density estimation tasks, many researchers have recently devoted considerable effort to leveraging NNs as a structural assumption for CS [4], [9]- [20]. VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.…”

“…Unfortunately, prior works [9], [13], [17], [19], [20], [22], [23] in CSPG solely concentrate on training generators without conditioning on any measurement vector. As y can be viewed as a (linearly) compressed signal of the original one x, it includes the information of x, thus helping generators trained to estimate x by additionally using y.…”