Paper:
In recent years Convolutional Neural Networks (CNN) have been used extensively for Superresolution (SR). In this paper, we use inverse problem and sparse representation solutions to form a mathematical basis for CNN operations. We show how a single neuron is able to provide the optimum solution for inverse problem, given a low resolution image dictionary as an operator. Introducing a new concept called Representation Dictionary Duality, we show that CNN elements (filters) are trained to be representation vectors and then, during reconstruction, used as dictionaries. In the light of theoretical work, we propose a new algorithm which uses two networks with different structures that are separately trained with low and high coherency image patches and show that it performs faster compared to the state-of-the-art algorithms while not sacrificing from performance.

Review:
The method proposes a new architecture for solving image super-resolution task. They provide an analysis that connects aims to establish a connection between how CNNs for solving super resolution and solving sparse regularized inverse problems.
The writing of the paper needs improvement. I was not able to understand the proposed connection, as notation is inconsistent and it is difficult to figure out what the authors are stating. I am willing to reconsider my evaluation if the authors provide clarifications.
The paper does not refer to recent advances in the problem, which are (as far as I know), the state of the art in the problem in terms of quality of the solutions. This references should be added and the authors should put their work into context.
1) Arguably, the state of the art in super resolution are techniques that go beyond L2 fitting. Specifically, methods using perceptual losses such as:
Johnson, J. et al "Perceptual losses for real-time style transfer and super-resolution." European Conference on Computer Vision. Springer International Publishing, 2016.
Ledig, Christian, et al. "Photo-realistic single image super-resolution using a generative adversarial network." arXiv preprint arXiv:1609.04802 (2016).
PSNR is known to not be directly related to image quality, as it favors blurred solutions. This should be discussed.
2) The overall notation of the paper should be improved. For instance, in (1), g represents the observation (the LR image), whereas later in the text, g is the HR image. 
3) The description of Section 2.1 is quite confusing in my view. In equation (1), y is the signal to be recovered and K is just the downsampling plus blurring. So assuming an L1 regularization in this equation assumes that the signal itself is sparse. Equation (2) changes notation referring y as f. 
4) Equation (2) seems wrong. The term multiplying K^T is not the norm (should be parenthesis).
5) The first statement of Section 2.2. seems wrong. DL methods do state the super resolution problem as an inverse problem. Instead of using a pre-defined basis function they learn an over-complete dictionary from the data, assuming that natural images can be sparsely represented. Also, this section does not explain how DL is used for super resolution. The cited work by Yang et al learns a two coupled dictionaries (one for LR and HL), such that for a given patch, the same sparse coefficients can reconstruct both HR and LR patches. The authors just state the sparse coding problem.
6) Equation (10) should not contain the \leq \epsilon.
7) In the second paragraph of Section 3, the authors mention that the LR image has to be larger than the HR image to prevent border effects. This makes sense. However, with the size of the network (20 layers), the change in size seems to be quite large. Could you please provide the sizes? When measuring PSNR, is this taken into account? 
8) It would be very helpful to include an image explaining the procedure described in the second paragraph of Section 3.
9) I find the description in Section 3 quite confusing. The authors relate the training of a single filter (or neuron) to equation (7), but they define D, that is not used in all of Section 2.1. And K does not show in any of the analysis given in the last paragraph of page 4. However, D and K seem two different things (it is not just one for the other), see bellow.
10) I cannot understand the derivation that the authors do in the last paragraph of page 4 (and beginning of page 5). What is phi_l here? K in equation (7) seems to match to D here, but D here is a collection of patches and in (7) is a blurring and downsampling operator. I cannot review this section. I will wait for the author's response clarifications.
11) The authors describe a change in roles between the representations and atoms in the training and testing phase respectively. I do not understand this. If I understand correctly, the final algorithm, the authors train a CNN mapping LR to HR images. The network is used in the same way at training and testing.
12) It would be useful to provide more details about the training of the network. Please describe the training set used by Kim et al. Are the two networks trained independently? One could think of fine-tuning them jointly (including the aggregation).
13) The authors show the advantage of separating networks on a single image, Barbara. It would be good to quantify this better (maybe in terms of PSNR?). This observation might be true only because the training loss, say than the works cited above. Please comment on this.
14) In figures 3 and 4, the learned filters are those on the top (above the yellow arrow). It is not obvious to me that the reflect the predominant structure in the data. (maybe due to the low resolution).
15) This work is related to (though clearly different)  that of LISTA (Learned ISTA) type of networks, proposed in:
Gregor, K., & LeCun, Y. (2010). Learning fast approximations of sparse coding. In Proceedings of the 27th International Conference on Machine Learning (ICML) 
Which connect the network architecture with the optimization algorithm used for solving the sparse coding problem. Follow up works have used these ideas for solving inverse problems as well.