Paper:
Designing the architecture of deep neural networks (DNNs) is a cumbersome task that requires significant human expertise. There have been attempts to automatize this task, mainly by considering DNN architecture parameters such as the number of layers, the number of neurons and the activation function of each layer as hyper-parameters, and using an external method for optimizing them. This framework is computationally too expensive since each hyper-parameter optimization iteration required a whole model training. Here we propose Farfalle Neural Networks, a novel recurrent structure in which neurons connect each other through their trainable embeddings. Hence, important architecture features such as the number of neurons in each layer and the wiring among the neurons are automatically learned during the training process. We theoretically prove that the proposed model can replace a stack of dense layers, which is used as a part of many DNN architectures. By comparing the performance of dense networks and FNNs we show FNNs achieve comparable or higher accuracy using significantly fewer parameters.

Review:
This paper introduces a new neural network architecture, in which all neurons (called "floating neurons") are essentially endowed with "input" and "output" embedding vectors, the product of which defines the weight of the connection between any two neurons. The authors discuss two network architectures employing floating neurons: (a) multi-layer floating neural networks and (b) farfalle neural network (FNN), in which there is one hidden layer, but additional recurrent connections are introduced between the hidden neurons.
As mentioned by the authors, the proposed architecture is similar to architectures employing low-rank weight matrix factorization. In my opinion, the main novelty lies in: (a) "floating neuron" interpretation, (b) additional weight matrix normalization, and (c) FNN architecture similar to that of a "floating neuron" RNN network with additional restrictions.
I find the proposed idea to be promising and quite intriguing, but I think that the paper has some room for improvement and provided empirical evidence might be insufficient (including for understanding the importance of individual model components), which in turn makes the claims of potential practical attractiveness less justified. I will be happy to update the final score provided with more compelling arguments or empirical analysis of the proposed architecture.
Addressing the following issues might greatly improve the quality of the paper:
1. In Section 4.1, the authors compare FNN and DNN on MNIST and CIFAR10 datasets. My concern is that the authors pick a seemingly arbitrary DNN architecture (just a single one) and restrict comparison to it. One issue is that ~50% accuracy on CIFAR10 can be easily demonstrated by a variety of 5-layer DNN architectures including those much smaller, with just ~600k parameters (!) and possibly even lower. This makes the 90% parameter reduction claim not particularly meaningful. And why were models matched based on the total number of neurons, but not, say, the total number of parameters, or other measures? I believe that these questions require additional discussion and empirical evidence. Just as an example, if it was possible to sample (potentially randomly) different DNN architectures (with a reasonable parameter prior) and compare them with FNNs on a 2D accuracy-parameters plot (or using other important metrics), it would provide much more information to the reader.
2. Another important point that I would like to make is that there is much more that can be done to explore the hyper-parameter space of FNN to isolate which particular factors play a decisive role in its superior accuracy. The authors present us with a specific choice of the normalization function, and values of k and d, but it would be very informative to study how results change when different choices are considered. FNNs differ from DNNs in at least three aspects: usage of low-rank factorization, weight normalization and recurrent structure. How important are these individual aspects? Are some of them redundant, or almost redundant, or do FNNs require all of these components to achieve their peak performance? In other words, I believe that a careful ablation study would greatly improve this publication.
3. As a minor note, I think that the statement that FNNs "are more general" than floating neural networks is only partially correct. If I am not mistaken, FNN can also be "unrolled" and represented as a multi-layer floating neural network with additional parameter sharing. Also, the computational complexity of the constructed FNN (in Theorem 1) appears to be significantly higher than that of the floating neural network (especially for high l). This would imply that FNNs do not necessarily supersede multi-layer floating neural networks, at least when the computational complexity is of importance.
4. There are a few minor misprints throughout the text. For example, in "0<j<=j" in the proof of Theorem 1, or in "R output floating neurons for the final deduction from hidden neuron" (output should be S). Also, I could not find information about the value of d used in the described experiments (which I estimated to be 256; is this correct?).
5. In Section 4.3, the authors propose to use FNNs for the final layers of conventional CNN architectures. The issue is that the VGG16 network chosen for experiments was probably picked because it uses several large fully-connected (FC) layers in its tail whereas all more recent and efficient CNN architectures actually gravitate towards a smaller single FC layer. It is possible that FNNs could still be used in FC layers of these modern networks as well (especially with a large number of classes). But additional empirical results for these architectures would, in my opinion, be much more convincing.
Updated: The authors updated the text and addressed many of my questions. In my opinion, this improved the paper and made some of its claims much better justified. I change the rating to "Weak Accept".