Paper:
Multi-relational graph embedding which aims at achieving effective representations with reduced low-dimensional parameters, has been widely used in knowledge base completion. Although knowledge base data usually contains tree-like or cyclic structure, none of existing approaches can embed these data into a compatible space that in line with the structure. To overcome this problem, a novel framework, called Riemannian TransE, is proposed in this paper to embed the entities in a Riemannian manifold. Riemannian TransE models each relation as a move to a point and defines specific novel distance dissimilarity for each relation, so that all the relations are naturally embedded in correspondence to the structure of data. Experiments on several knowledge base completion tasks have shown that, based on an appropriate choice of manifold, Riemannian TransE achieves good performance even with a significantly reduced parameters.

Review:
The paper proposes a new approach to compute embeddings of multi-relational data such as knowledge graphs. For this purpose, the paper introduces a variant of TransE that operates on Riemannian manifolds, in particular, Euclidean, Spherical, and Hyperbolic space. This approach is motivated by the results of Nickel & Kiela (2017), who showed tha  Hyperbolic space can provide important advantages for embedding graphs with hierarchical structure.
Hyperbolic and Riemannian embeddings are a promising research area that fits well into ICLR. Extending hyperbolic, and more general, Riemannian embeddings to multi-relational data is an important aspect in this context, as it allows to extend such methods to new applications such as Knowledge Graph Completion. Overall, the paper is written well and mostly good to understand. However, I am concerned about multiple aspects in the current version:
- What is the motivation for using this particular form of translation? In Riemannian manifolds, the analogue of vector addition and subtraction is typically taken as the exponential or logarithmic map (as expm and logm in Euclidean space are exactly vector addition and subtraction). For the spherical and hyperbolic manifold both maps have closed form expressions and are differentiable. It is therefore not clear to me what the advantage of the proposed approach is compared to these standard methods. In any case, it would be important to include them in the experimental results.
- It is also not clear to me whether the benefits of hyperbolic embeddings translate into the setting that is proposed here. The advantage of hyperbolic embeddings is that they impose a hierarchical structure in the latent space. The method proposed in this paper uses then a single (hyperbolic) embedding of entities across all relation types. This implies that there should be a single consistent hierarchy that explains all the links in all relations. This seems unlikely and might explain some of the hyperbolic results. A more detailed discussion and motivation would be important here.
- Regarding the experimental results: Why are the results for HolE and ComplEx so different? As [1,2] showed, both models are identical, and for that reason should get close to identical results. The large differences seem inconsistent with these results. Furthermore it seems that the results reported in this paper do not match previously reported results. What is the reason for these discrepancies?
[1] Hayashi, K., and Shimbo, M. "On the equivalence of holographic and complex embeddings for link prediction", 2017.
[2] Trouillon, T, and Nickel, M. "Complex and Holographic Embeddings of Knowledge Graphs: A Comparison", 2017.