Solving Inverse Problems Using a Deep Generative Prior

Abstract

In an inverse problem, the objective is to recover a signal from its measurements, given the knowledge of the measurement operator. In this thesis, we address the problems of compressive sensing (CS) and compressive phase retrieval (CPR) using a generative prior model with sparse latent sampling. These problems are ill-posed and have infinite solutions. Structural assumptions such as smoothness, sparsity and non-negativity are imposed on the solution to obtain a unique and meaningful solution. The standard CS and CPR formulations impose a sparsity prior on the signal. Recently, generative modeling approaches have removed the sparsity constraint and shown superior performance over traditional CS and CPR techniques in recovering signals from fewer measurements. Generative model uses a pre-trained network, the generator of a Generative Adversarial Network (GAN) or the decoder of a Variational Autoencoder (VAE) to model the distribution of the signal and impose a Set-Restricted Eigenvalue Condition (S - REC) on the measurement operator. The S - REC property places a condition on the L2-norm of the difference in signal and measurement domain for signals coming from the set S. Solving CS and CPR using generative models have some limitations. The reconstructed signal is constrained to lie in the range-space of the generator. The reconstruction process is slow because the latent space is optimized through gradient-descent (GD) and requires several restarts. It has been argued that the distribution of natural images is not confined to a single manifold, but a union of submanifolds. To take advantage of this property, we propose a sparsity-driven latent space sampling (SDLSS) framework, where sparsity is imposed in the latent space. The effect is to divide the latent space into subspaces such that the generator models maps each subspace into a submanifold. We propose a proximal meta-learning (PML) algorithm to optimize the parameters of the generative model along with the latent code. The PML algorithm reduces the number of gradient steps required during testing and imposes sparsity in the latent space. We derive the sample complexity bounds within the SDLSS framework for the linear CS model, which is a generalization of the result available in the literature. The results demonstrate that, for a higher degree of compression, the SDLSS method is more efficient than the state-of-the-art deep compressive sensing (DCS) method. We consider both linear and learned nonlinear sensing mechanisms, where the nonlinear operator is a learned fully connected neural network or a convolutional neural network, and show that the learned nonlinear version is superior to the linear one. As an application of the nonlinear sensing operator, we consider compressive phase retrieval, wherein the problem is to reconstruct a signal from the magnitude of its compressed linear measurements. We adapt the S-REC imposed on the measurement operator and propose a novel cost function. The SDLSS framework along with PML algorithm is applied to optimize the sparse latent space such that the adapted S-REC loss and data-fitting error are minimized. The reconstruction process is fast and requires few gradient steps during testing compared with the state-of-art deep phase retrieval technique. Experiments are conducted on standard datasets such as MNIST, Fashion-MNIST, CIFAR-10, and CelebA to validate the efficiency of SDLSS framework for CS and CPR. The results show that, for a given dataset, there exists an effective input latent dimension for the generative model. Performance quantification is carried out by employing three objective metrics: peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and reconstruction error (RE) per pixel, which are averaged across the test dataset.