Generative models, such as GANs, learn an explicit low-dimensional representation of a particular class of images, and so they may be used as natural image priors for solving inverse problems such as image restoration and compressive sensing. GAN priors have demonstrated impressive performance on these tasks, but they can exhibit substantial representation error for both in-distribution and out-of-distribution images, because of the mismatch between the learned, approximate image distribution and the data generating distribution. In this paper, we demonstrate a method for reducing the representation error of GAN priors by modeling images as the linear combination of a GAN prior with a Deep Decoder. The deep decoder is an underparameterized and most importantly unlearned natural signal model similar to the Deep Image Prior. No knowledge of the specific inverse problem is needed in the training of the GAN underlying our method. For compressive sensing and image superresolution, our hybrid model exhibits consistently higher PSNRs than both the GAN priors and Deep Decoder separately, both on in-distribution and out-of-distribution images. This model provides a method for extensibly and cheaply leveraging both the benefits of learned and unlearned image recovery priors in inverse problems.
Summary: This paper proposes to use a combination of a pretrained GAN and an untrained deep decoder as the image prior for image restoration problem. The combined model jointly infers the latent code for the trained GAN and the parameters in the untrained deep decoder. It also jointly infers the mixing coefficient alpha and beta during test time for each image, thus learning how much we should rely on GAN. The proposed hybrid model is helpful on compressed sensing experiments on the CelebA dataset; however, it is only marginally better than deep decoder on image super resolution and out-of-distribution compressed sensing.
Detailed comments:
-	The writing is clear and I was able to understand the model part of the paper. The algorithm box is helpful. However, I would still appreciate if the authors can provide an overall model figure in the model section to help understanding. 
-	Jointly learning the mixing coefficient is an interesting part of the model.
-	The motivation in the abstract and intro could be strengthened. A smaller version of Figure 1 can be probably moved to the beginning of the paper to illustrate the problem of GAN. But even with the help of Figure 1, it is still unclear what is the fundamental problem for GAN. Simply combining a GAN with an untrained decoder model doesn’t help elucidate the source of the problem. 
-	The proposed Hybrid model seems to help on compressed sensing experiments on CelebA. However, it doesn’t help much on out-of-distribution experiments. Moreover, in the super-resolution task, as shown in Figure 5, the improvement over deep decoder is also not significant.
-	The out-of-distribution experiments seems lack of thorough study. In particular, the paper only studies the transfer between CelebA -> Caltech-UCSD Bird dataset. It would be better if the paper can study a variety of other image datasets as well. Also some visualization on the Bird dataset should also be included.
-	Effect of n_pre needs to be further investigated. Why not directly train both models together? It would be good if the authors could comment on how sensitive the n_pre is and what is the intuition.
-	For figures, I would recommend rename “Hybrid” to “Hybrid (Ours)” to highlight the paper’s contribution, and use a brighter color.
-	Figure 5 should be renamed as a Table. 
-	Hyperparameter details should be moved to the Experiment section.
Conclusion:
The paper proposes a simple combination of a trained GAN and an untrained decoder model for the task of image restoration. Although the method is clear and straightforward, in the experiments, the influence of the new model component seems marginal. Moreover, the motivation is not strong enough. Therefore, I recommend weak reject.