We consider the problem of representation learning for graph data. Convolutional neural networks can naturally operate on images, but have significant challenges in dealing with graph data. Given images are special cases of graphs with nodes lie on 2D lattices, graph embedding tasks have a natural correspondence with image pixel-wise prediction tasks such as segmentation. While encoder-decoder architectures like U-Net have been successfully applied on many image pixel-wise prediction tasks, similar methods are lacking for graph data. This is due to the fact that pooling and up-sampling operations are not natural on graph data. To address these challenges, we propose novel graph pooling (gPool) and unpooling (gUnpool) operations in this work. The gPool layer adaptively selects some nodes to form a smaller graph based on their scalar projection values on a trainable projection vector. We further propose the gUnpool layer as the inverse operation of the gPool layer. The gUnpool layer restores the graph into its original structure using the position information of nodes selected in the corresponding gPool layer. Based on our proposed gPool and gUnpool layers, we develop an encoder-decoder model on graph, known as the graph U-Net. Our experimental results on node classification tasks demonstrate that our methods achieve consistently better performance than previous models.
This paper proposes pooling and upsampling operations for graph structured data, to be interleaved with graph convolutions, following the spirit of fully convolutional networks for image pixel-wise prediction. Experiments are performed on node classification benchmarks, showing an improvement w.r.t. architectures that do not perform any downsampling/upsampling operations.
Given that the main contribution of the paper is the introduction of a pooling operation for graph structured data, it might be a good idea to evaluate the operation in a task that does require some kind of downsampling, such as graph classification / regression. Moreover, authors should compare to other graph pooling methods.
Authors claim that one of the motivations to perform their pooling operation is to increase the receptive field. It would be worth comparing pooling/upsamping to dilated convolutions to see if they have the same effect on the performance when dealing with graphs. 
Some choices in the method seem rather arbitrary, such as the tanh non-linearity in \tilde y. Could the authors elaborate on that? How important is the gating?
It would be interesting to analyze which nodes where selected by the pooling operators. Are those nodes close together or spread out in the previous graph?
The proposed unpooling operation seems to be the same as unpooling performed to upsample images, that is using skip connections to track indices, by recovering the position where the max value comes from and setting the rest to 0. Have the authors tried other upsampling strategies analogous to the ones typically used for images (e.g. upsampling with nearest neighbors)?
When skipping information from the downsampling path to the upsampling path, is there a concatenation or a summation? How do both operations compare? (note that concatenation introduces many more parameters) How about only skipping only the indices (no summation nor concatenation)? This kind of analysis, as it has been done in the computer vision literature, would be interesting.
What is the influence of the first embedding layer to reduce the dimensionality of the features?
How do the models in Table 2 compare in terms of number of parameters?
 What's the influence of imposing larger weights on self loop in the graph?
What about experiments in inductive settings?
Please add references for the following claim "U-Net models with depth 3 or 4 are commonly used..."
Please double check your references, e.g. in the introduction, citations used for CNNs do not always correspond to CNN architectures.
The literature review could be significantly improved, missing relevant papers to discuss include:
- Gori et al. A new model for learning in graph domains, 2005.
- Scarselli et al. The graph neural network model, 2009.
- Bruna et al. Spectral networks and locally connected networks on graphs, 2014.
- Henaff et al. Deep convolutional networks on graph-structured data, 2015.
- Niepert et al. Learning convolutional neural networks for graphs, 2016.
- Atwood and Towsley. Diffusion-convolutional neural networks, 2016.
- Bronstein et al. Geometric deep learning: going beyond Euclidean data, 2016.
- Monti et al. Geometric deep learning on graphs and manifolds using mixture model cnns, 2017.
- Fey et al. SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernels, 2017.
- Gama et al. Convolutional Neural Networks Architectures for Signals Supported on Graphs, 2018.
As well as other pixel-wise architecture for image-based tasks such as:
- Long et al. Fully Convolutional Networks for Semantic Segmentation, 2015.
- Jegou et al. The one hundred layers tiramisu: fully convolutional densenets for semantic segmentation, 2016.
- Isola et al. Image-to-image translation with conditional adversarial networks, 2016.
- Zhao et al. Stacked What-Where auto-encoders, 2015.