A ubiquitous and fundamental task across the natural sciences is an
imaging inverse problem, where the goal is to reconstruct a desired
image from a small number of noisy measurements. Due to the ill-posed
nature of such problems, it is desirable to enforce that the
reconstructed image obeys particular structural properties believed to
be obeyed by the image of interest. To minimize the amount of
measurements required, the desired properties often have a
low-dimensional structure. Such properties are known as priors and the
dominant paradigm over the last two decades or so has been to exploit
the sparsity of natural images in a hand-crafted basis. In recent
years, however, the field of machine learning, and deep learning in
particular, has demonstrated the effectiveness of data-driven priors
in the form of generative models. These models represent signals as
lying on an explicitly parameterized low-dimensional manifold, and
have shown to generate highly realistic, yet synthetic images from a
number of complex image classes, ranging from human faces to proteins.
This dissertation proposes a novel framework for image recovery by
exploiting these data-driven priors, and offers three main
contributions. First, we rigorously prove that these learned models
can help recover images from fewer nonlinear measurements than
traditional hand-crafted techniques in the challenging inverse
problem, phase retrieval. We additionally discuss how our developed
theory has a broader applicability to more general settings without
structural information on the image. Finally, we present a method
using invertible generative models to overcome dataset biases and
representational issues in previous generative prior-based approaches,
and theoretically analyze the method’s recovery performance in
compressive sensing. This thesis, more broadly, offers a new paradigm
for image recovery under deep generative priors and gives concrete
empirical and theoretical evidence towards the benefits of utilizing
such learned priors in a variety of inverse problems.