Paper:
A DATA-DRIVEN AND DISTRIBUTED APPROACH TO SPARSE SIGNAL REPRESENTATION AND RECOVERY
In this paper, we focus on two challenges which offset the promise of sparse signal representation, sensing, and recovery. First, real-world signals can seldom be described as perfectly sparse vectors in a known basis, and traditionally used random measurement schemes are seldom optimal for sensing them. Second, existing signal recovery algorithms are usually not fast enough to make them applicable to real-time problems. In this paper, we address these two challenges by presenting a novel framework based on deep learning. For the first challenge, we cast the problem of finding informative measurements by using a maximum likelihood (ML) formulation and show how we can build a data-driven dimensionality reduction protocol for sensing signals using convolutional architectures. For the second challenge, we discuss and analyze a novel parallelization scheme and show it significantly speeds-up the signal recovery process. We demonstrate the significant improvement our method obtains over competing methods through a series of experiments.

Review:
This paper proposes a (CNNs) architecture for encoding and decoding images for compressed sensing. 
In standard compressed sensing (CS), encoding usually is linear and corresponds to multiplying by a fat matrix that is iid gaussian. The decoding is performed with a recovery algorithm that tries to explain the linear measurements but also promotes sparsity. Standard decoding algorithms include Lasso (i.e. l1 regularization and a MSE constraint) 
or iterative algorithms that promote sparsity by construction. 
This paper instead proposes a joint framework to learn a measurement matrix Phi and a decoder which is another CNN in a data-driven way. The proposed architecture is novel and interesting.  
I particularly liked the theoretical motivation of the used MSE loss by maximizing mutual information. 
The use of parallel convolutions is also neat and can significantly accelerate inference, which can be useful for some applications. 
The empirical performance is very good and matches or outperforms previous state of the art reconstruction algorithms D-AMP and Learned D-Amp. 
On comparisons with prior/concurrent work: The paper is essentially a CNN autoencoder architecture but specifically designed for compressed sensing problems. 
There is vast literature on CNN autoencoders including (Jiang 2017 and Shi 2017) paper cited by the authors. I think it is fine to not compare against those since they divide the images into small blocks and hence have are a fundamentally different approach. This is fine even if block-reconstruction methods outperform this paper, in my opinion: new ideas should be allowed to be published even if they do not beat SOTA, as long as they have clearly novel ideas. It is important however to discuss these differences as the authors have done in page 2.  
Specific comments: 
1. It would be interesting to see a comparison to D-Amp and LDAmp for different number of measurements or for different SNRs (i.e. when y = Phi x+ noise ). I suspect each method will be better for a different regime?
2. The paper: `The Sparse Recovery Autoencoder' (SRA) by Wu et al. https://arxiv.org/abs/1806.10175
is related in that it learns both the sensing matrix and a decoder and is also focused on compressed sensing, but for non-image data. The authors should discuss the differences in architecture and training. 
3. Building on the SRA paper, it is possible that the learned Phi matrix is used but then reconstruction is done with l1-minimization. How does that perform for the matrices learned with DeepSSRR?
4. Why is Figure 1 going from right to left?