  Trained generative models have shown remarkable performance as priors for
inverse problems in imaging -- for example, Generative Adversarial Network
priors permit recovery of test images from 5-10x fewer measurements than
sparsity priors. Unfortunately, these models may be unable to represent any
particular image because of architectural choices, mode collapse, and bias in
the training dataset. In this paper, we demonstrate that invertible neural
networks, which have zero representation error by design, can be effective
natural signal priors at inverse problems such as denoising, compressive
sensing, and inpainting. Given a trained generative model, we study the
empirical risk formulation of the desired inverse problem under a
regularization that promotes high likelihood images, either directly by
penalization or algorithmically by initialization. For compressive sensing,
invertible priors can yield higher accuracy than sparsity priors across almost
all undersampling ratios, and due to their lack of representation error,
invertible priors can yield better reconstructions than GAN priors for images
that have rare features of variation within the biased training set, including
out-of-distribution natural images. We additionally compare performance for
compressive sensing to unlearned methods, such as the deep decoder, and we
establish theoretical bounds on expected recovery error in the case of a linear
invertible model.
