We propose a novel node embedding of directed graphs to statistical manifolds, which is based on a global minimization of pairwise relative entropy and graph geodesics in a non-linear way. Each node is encoded with a probability density function over a measurable space. Furthermore, we analyze the connection between the geometrical properties of such embedding and their efficient learning procedure. Extensive experiments show that our proposed embedding is better in preserving the global geodesic information of graphs, as well as outperforming existing embedding models on directed graphs in a variety of evaluation metrics, in an unsupervised setting.
This paper proposed another graph embedding method. It focuses on directed graphs, and it embedded the graph nodes into exponential power distributions, which include the Gaussian distribution as a special case. The method is implemented by optimizing with respect to the free distributions on a statistical manifold so as to achieve the minimum distortion between the input/output distances.  The method is tested on several directed graph datasets and showed superior performance based on several metrics.
Overall, the submission forms a complete and novel contribution in the area of graph embeddings. A key novelty is that the authors used the asymmetry of KL divergences to model the asymmetry of the distances in directed graphs, and they use the fact that KL is unbounded to model the infinite distances in undirected graphs. The proposed method has three main hyperparameters, \lambda in eq.(1), \beta in eq.(2), and the dimensionality of the target embedding. The author showed that \lambda and \beta are not sensitive and can be set to the default values, and 2-dimensional distributions already give much better results as compared to alternative embeddings. Moreover, the author proposed a scalable implementation based on sampling. Furthermore, the authored justified their choice of the target embedding space through some minor theoretical analysis given in section 3.
The writing quality and clarity are good (well above average).
To further improve this paper (e.g., in the final version), the authors are suggested to incorporate the following comments:
In the experimental evaluation, it should include some cases when the dimensionality of the target embedding has a large value (e.g., 50). This will make the evaluation more complete.
There are some typos and unusual expressions.  For example, page 3, what is "a good proposal function"?
After eq.(1), mention \lambda is a hyperparameter (that is not to be learned).
Theorem 1 (2), mention the Fisher information matrix is wrt the coordinate system (\sigma^1, \cdots,\sigma^k, \mu^1, \cdots, \mu^k)
Ideally, the experiments can include an undirected graph and show for example that the advantages of the proposed method become smaller in this case.