Paper:
Deep ADMM-Net for Compressive Sensing MRI
Compressive Sensing (CS) is an effective approach for fast Magnetic Resonance Imaging (MRI). It aims at reconstructing MR image from a small number of undersampled data in k-space, and accelerating the data acquisition in MRI. To improve the current MRI system in reconstruction accuracy and computational speed, in this paper, we propose a novel deep architecture, dubbed ADMM-Net. ADMMNet is defined over a data flow graph, which is derived from the iterative procedures in Alternating Direction Method of Multipliers (ADMM) algorithm for optimizing a CS-based MRI model. In the training phase, all parameters of the net, e.g., image transforms, shrinkage functions, etc., are discriminatively trained end-to-end using L-BFGS algorithm. In the testing phase, it has computational overhead similar to ADMM but uses optimized parameters learned from the training data for CS-based reconstruction task. Experiments on MRI image reconstruction under different sampling ratios in k-space demonstrate that it significantly improves the baseline ADMM algorithm and achieves high reconstruction accuracies with fast computational speed.

Review:
The authors present a deep architecture that implements an ADMM for compressed sensing applied to MRI recovery. The ADMM algorithm has proven to be useful for solving problems with differentiable and non-differentiable terms, and therefore has a clear link with compressed sensing. Experiments prove some gain in performance with respect to the state of the art, specially in terms of computational cost at test time.
The idea of reimplementing an iterative algorithm in a deep architecture is not new, but solving a CS problem for MRI recovery with a Deep-ADMM architecture is novel as far as I can say.  I am a bit worried about the correspondence between the "classical" and the "deep" learning problems. Indeed, the classical problem is very well stated as minimizing the augmented Lagrangian. However, when discussing the implementation of the convolutional layer, the authors say that they increase the network capacity by decoupling the transform matrix of the convolutional from the one of the reconstruction layer. Therefore, we are not solving for the augmented Lagrangian anymore, but for something else. What is the interpretation in terms of "classical" learning? Can you rewrite the optimization problem you are actually solving for and discuss similarities and differenceswith the original augmented Lagrangian?  The authors claim something before equation (8) that is not clear to me and would require explanation or proof. When Implementing the non-linear transform layer (I.e. the shrinkage), how do you know that using a piecewise linear function is more general than trying to implement the shrinkage. I mean, I understand that fitting a piecewise linear function can approximate many other functions and therefore has some flexibility: but can it implement any shrinkage operator with piecewise linear functions (recall that the control points are predefined). My understanding is that even if more flexible, this layer is perhaps not more general than the original shrinkage.  When writing the gradients used in the back-propagation, I'd rather see in the main text the actual gradients and not general chain rules that do not provide any insight on the effects of each back-propagation step.  The experimental results should be sharpened-up. It is not really convincing to me that ADMM-NET-14 does not reach state-of-the-art performance, but ADMM-NET-15 does. It is very intuitive, since you get one iteration more of the ADMM. But how can I choose the number of iterations a priori? In regular ADMM different convergence criteria can be evaluated, but with your approach the number of iterations has to be fixed in advance. Importantly, this is critical, because the number of iterations defines the number of layers, and therefore the amount of parameters of the network. Therefore, while for classical ADMM, the number of parameters is fixed with the number of iterations, here it grows. Therefore, the number of data must grow with the number of iterations. Can you please comment on that and provide some insights? In this topic, I would also appreciate to see that the parameters also converge (as the MMSE does in Fig. 6b).   Typos: - Generally speaking, check the usage of the definite article "the" very carefully. -P1L19 techniques are -P1L22 CS-MRI (without the) -P1L29 uses groupS -P1L36 proven (not proved) -P2L56 non-local techniqueS -P1L58 propose (without d)