Deep Generative Models and Inverse Problems

Linear inverse problems involve the reconstruction of an unknown vector (e.g., a tomography image) from an underdetermined system of noisy linear measurements. Most results in the literature require that the reconstructed signal has some known structure, e.g., it is sparse in some basis (usually Fourier or Wavelet). In this talk, we show how to remove such prior assumptions and rely instead on deep generative models (e.g., Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs)). We show how the problems of image inpainting (completing missing pixels) and super-resolution are special cases of our general framework. We generalize theoretical results on compressive sensing for deep generative models and discuss several open problems.