  Deep neural networks as image priors have been recently introduced for
problems such as denoising, super-resolution and inpainting with promising
performance gains over hand-crafted image priors such as sparsity and low-rank.
Unlike learned generative priors they do not require any training over large
datasets. However, few theoretical guarantees exist in the scope of using
untrained neural network priors for inverse imaging problems. We explore new
applications and theory for untrained neural network priors. Specifically, we
consider the problem of solving linear inverse problems, such as compressive
sensing, as well as non-linear problems, such as compressive phase retrieval.
We model images to lie in the range of an untrained deep generative network
with a fixed seed. We further present a projected gradient descent scheme that
can be used for both compressive sensing and phase retrieval and provide
rigorous theoretical guarantees for its convergence. We also show both
theoretically as well as empirically that with deep network priors, one can
achieve better compression rates for the same image quality compared to hand
crafted priors.
