We provide a novel thinking of regularization neural networks. We smooth the objective of neural networks w.r.t small adversarial perturbations of the inputs. Different from previous works, we assume the adversarial perturbations are caused by the movement field. When the magnitude of movement field approaches 0, we call it virtual movement field. By introducing the movement field, we cast the problem of finding adversarial perturbations into the problem of finding adversarial movement field. By adding proper geometrical constraints to the movement field, such smoothness can be approximated in closed-form by solving a min-max problem and its geometric meaning is clear. We define the approximated smoothness as the regularization term. We derive three regularization terms as running examples which measure the smoothness w.r.t shift, rotation and scale respectively by adding different constraints. We evaluate our methods on synthetic data, MNIST and CIFAR-10. Experimental results show that our proposed method can significantly improve the baseline neural networks. Compared with the state of the art regularization methods, proposed method achieves a tradeoff between accuracy and geometrical interpretability as well as computational cost.
This paper tackles the overfitting problem when training neural networks based on regularization technique. More precisely, the authors propose new regularization terms that are related to the underlying virtual geometrical transformations (shift, rotation and scale) of the input data (signal, image and video). By formalizing the geometrical transformation process of a given image, the authors deduce constraints on the objective function which depend on the magnitude of the applied transformation. The proposed method is compared to three methods: one baseline and two methods of the literature (AT and VAT). The comparison is done on three datasets (synthetic data, MNIST and CIFAR10) in terms of test errors (for classification problems) and running time.
The paper is well formalized and the idea is interesting. The regularization approach is novel compared to the methods of the literature. 
Main concerns: 
1)	The experimental validation of the proposed approach is not consistent:
The description of the baseline method is not detailed in the paper. 
A priori, the baseline should naturally be the method without your regularization terms.
But, this seems to be contrary with what you displayed in Figure 3. 
Indeed, in Figure 3, there is three different graphs for the baseline method (i.e., one for each regularization term). It seems that the baseline method depends on the different kinds of regularization term, why? Same question for AT and VAT methods. 
In practice, what is the magnitude of the perturbations? 
Please, explain the axis of all the figures. 
Please, explain how do you mix your different regularization terms in your method that you call VMT-all? 
All the following points are related to the experiment for which you presented the results in Table 2: 
Please, provide the results of all your methods on the synthetic dataset (only VMT-shift is provided). What is VMF? Do you mean VMT? 
For the evaluations, it would be more rigorous to re-implement also the state-of-the-art methods for which you only give the results that they report in their paper. Especially, because you re-implemented AT with L-2 constraint, so, it seems straightforward to re-implement also AT with L-infinite constraint. Same remark for the dropout regularization technique, which is easy to re-implement on the dense layers of your neural networks, within the Tensorflow framework. 
As you mentioned, your main contribution is related to running time, thus, you should give the running time in all experiments. 
2)	The method seems to be a tradeoff between accuracy and running time:
The VAT method performs better than all your methods in all the datasets. 
The baseline method is faster than all the methods (Table 3). 
This being said, the proposed method should be clearly presented in the paper as a tradeoff between accuracy and running time. 
3)	The positioning of the proposed approach is not so clear:  
As mentioned above, your method is a tradeoff between accuracy and running time. But you also mentioned (top of page 2) that the contribution of your paper is also related to the interpretability in terms of ‘’Human perception’’. Indeed, you clearly mentioned that the methods of the literature lacks interpretability. You also mentioned that your method is more ‘’geometrically’’ interpretable than methods of the literature. The link between interpretability in terms of “human perception” and “geometry” is not obvious. Anyway, the interpretability point is not sufficiently demonstrated, or at least, discussed in the paper. 
4)	Many typos in the paper : 
Section 1: “farward-backward”
Section 2.1: “we define the movement field V of as a n+1…”
Section 2.2: “lable” - “the another” - “of how it are generated” – Sentence “Since V is normalized.” seems incomplete… - \mathcal{L} not defined - Please, precise the simplifications like \mathcal{L}_{\theta} to \mathcal{L} 
Section 3: “DISCUSSTION”
Section 4.1: “negtive”
Figure 2: “negetive”
Table 2: “VMF”
Section 4.2: “Tab 2.3” does not exist 
Section 4.3: “consists 9 convolutional” – “nerual networks”…
Please, always use the \eqref latex command to refer to equations.
There is many others typos in the paper, so, please proofread the paper…