Paper:
An Inner-loop Free Solution to Inverse Problems using Deep Neural Networks
We propose a new method that uses deep learning techniques to accelerate the popular alternating direction method of multipliers (ADMM) solution for inverse problems. The ADMM updates consist of a proximity operator, a least squares regression that includes a big matrix inversion, and an explicit solution for updating the dual variables. Typically, inner loops are required to solve the first two subminimization problems due to the intractability of the prior and the matrix inversion. To avoid such drawbacks or limitations, we propose an inner-loop free update rule with two pre-trained deep convolutional architectures. More specifically, we learn a conditional denoising auto-encoder which imposes an implicit data-dependent prior/regularization on ground-truth in the first sub-minimization problem. This design follows an empirical Bayesian strategy, leading to so-called amortized inference. For matrix inversion in the second sub-problem, we learn a convolutional neural network to approximate the matrix inversion, i.e., the inverse mapping is learned by feeding the input through the learned forward network. Note that training this neural network does not require ground-truth or measurements, i.e., data-independent. Extensive experiments on both synthetic data and real datasets demonstrate the efficiency and accuracy of the proposed method compared with the conventional ADMM solution using inner loops for solving inverse problems.

Review:
An interesting paper that solves linear inverse problems using a combination of two networks: one that learns a proximal operator to the signal class of interest, and the other that serves as a proxy for a large scale matrix inversion. The proximal operator is reusable whenever the signal domain is unchanged. One would need to retrain only the matrix inversion network when the underlying problem is changed. This is a significant advantage towards reusability of the training procedure.
Strengths
+ Novelty 
+ A very interesting problem
Weaknesses
- An important reference is missing
- Other less important references are missing
- Bare-bones evaluation
The paper provides an approach to solve linear inverse problems by reducing training requirements. While there is some prior work in this area (notably the reference below and reference [4] of the paper), the paper has some very interesting improvements over them. In particular, the paper combines the best parts of [4] (decoupling of signal prior from the specific inverse problem being solved) and Lista (fast implementation). That said, evaluation is rather skim — almost anecdotal at times — and this needs fixing. There are other concerns as well on the specific choices made for the matrix inversion that needs clarification and justifications.
1) One of the main parts of the paper is a network learnt to invert (I + A^T A). The paper used the Woodbury identity to change it to a different form and learns the inverse of (I+AA^T) since this is a smaller matrix to invert. At test time, we need to apply not just this network but also "A" and "A^T" operators.
A competitor to this is to  learn a deep network that inverts (I+A^T A). A key advantage of this is that we do not need apply A and A^T during test time. It is true that that this network learns to inverts a larger matrix ... But at test time we have increased rewards. Could we see a comparison against this ? both interms of accuracy as well as runtime (at training and testing) ?
2) Important reference missing. The paper is closely related to the idea of unrolling, first proposed in, “Lista”
http://yann.lecun.com/exdb/publis/pdf/gregor-icml-10.pdf
While there are important similarities and differences between the proposed work and Lista, it is important that the paper talks about them and places itself in appropriate context. 
3) Evaluation is rather limited to a visual comparison to very basic competitors (bicubic and wiener filter). It would be good to have comparisons to 
- Specialized DNN: this would provide the loss in performance due to avoiding the specialized network.
- Speed-ups over [4] given the similarity (not having this is ok given [4] is an arXiv paper)
- Quantitative numbers that capture actual improvements over vanilla priors like TV and wavelets and gap to specialized DNNs.
Typos
- Figure 2: Syntehtic