Abstract
Sparsity has given us MP3, JPEG, MPEG, Faster MRI and many fun mathematical problems. Deep generative models like GANs, VAEs, invertible flows and Score-based models are modern data-driven generalizations of sparse structure. We will start by presenting the CSGM framework by Bora et al. to solve inverse problems like denoising, filling missing data, and recovery from linear projections using an unsupervised method that relies on a pre-trained generator. We generalize compressed sensing theory beyond sparsity, extending Restricted Isometries to sets created by deep generative models. Our recent results include establishing theoretical results for Langevin sampling from full-dimensional generative models, generative models for MRI reconstruction and fairness guarantees for inverse problems.