  Objective: This paper investigates how generative models, trained on
ground-truth images, can be used \changes{as} priors for inverse problems,
penalizing reconstructions far from images the generator can produce. The aim
is that learned regularization will provide complex data-driven priors to
inverse problems while still retaining the control and insight of a variational
regularization method. Moreover, unsupervised learning, without paired training
data, allows the learned regularizer to remain flexible to changes in the
forward problem such as noise level, sampling pattern or coil sensitivities in
MRI.
  Approach: We utilize variational autoencoders (VAEs) that generate not only
an image but also a covariance uncertainty matrix for each image. The
covariance can model changing uncertainty dependencies caused by structure in
the image, such as edges or objects, and provides a new distance metric from
the manifold of learned images.
  Main results: We evaluate these novel generative regularizers on
retrospectively sub-sampled real-valued MRI measurements from the fastMRI
dataset. We compare our proposed learned regularization against other unlearned
regularization approaches and unsupervised and supervised deep learning
methods.
  Significance: Our results show that the proposed method is competitive with
other state-of-the-art methods and behaves consistently with changing sampling
patterns and noise levels.
