This paper proposes a knowledge graph embedding model where entities and relations are represented in the quaternion space - this generalises some recent trends in knowledge graph embeddings where the complex space was being used to overcome the limitations of some existing models [1, 2]. The dependencies between latent features are modeled via the Hamilton product, which seems to encourage more compact interactions between latent features. The model design is sound and, with little added complexity in comparison with other methods (a slightly more convoluted scoring function, and an uncommon initialisation scheme) the model achieves robust results in comparison with other methods in the literature (see Tab. 4 - although a bit far from SOTA [2]) and the resulting models are fairly parameter efficient (see Tab. 6). Small issue: "euclidean" should be "Euclidean". [1] proceedings.mlr.press/v48/trouillon16.pdf [2] https://arxiv.org/abs/1806.07297

This paper introduces quaternion embeddings for entities and relations in a knowledge graph. Quaternions are vectors in hypercomplex space with 3 imaginary components (2 more than usual complex numbers). It also proposes to use the Hamilton product to score the existence of a triple. Hamilton product are non-commutative which is highly desirable property. Also Hamilton product, represents scaling and smooth rotation and are easier to compose and avoid the problem of gimbal lock. They are also numerically stable when compared to rotational matrices. The score function is s(e1, r, e2) is defined first by a Hamilton product between e1 and normalized r (to only model rotation), followed by a inner product with e2 to output a scalar score. The paper also shows that both complex embeddings and distmult are special cases of quaternion embeddings. The paper also gets convincing results on benchmark datasets such as FB15k-237 and WN18RR. The ablation tests are convincing and it has comparable number of parameters wrt the baseline. Originality: The paper presents a novel way of representing entities and relations Clarity: The paper is well-written and was easy to follow. Significance: the results are significance, and the paper will be impactful.

After rebuttal: Thanks for providing more results. That addressed my 3rd concern to some extent (although it seems like your model requires many more epochs compared to ComplEx and I wonder how ComplEx will perform given the same number of epochs and using uniform negative sampling, but this is not a major concern). I'm not yet convinced about the issue I raised regarding relation normalization though. "We also found that the relation normalization can improve the ComplEx model as well. But it is till worse than QuatE." Why not provide some actual numbers similar to the other cases so we can see how much better QuatE is compared to ComplEx when they both use relation normalization? And in case QuatE actually outperforms ComplEx significantly in presence of relation normalization, what is special about QuatE compared to ComplEx that makes it benefit from relation normalization to the extent that QuatE + relation normalization outperforms ComplEx + relation normalization, while ComplEx without relation normalization outperforms QuatE without relation normalization? =============================== Knowledge graph completion has been vastly studied during the past few years with more than 30 embedding-based approaches proposed for the task. Inspired by the recent success of ComplEx and RotatE in using embeddings with complex numbers, this paper proposes to use quaternion embeddings instead of complex embeddings. 1- Novelty: My major reservation for this work concerns the novelty. While going from real-valued embeddings to complex-valued embeddings was novel and addressed some of the issues with simple approaches based on real-valued embeddings (e.g., they addressed the symmetry issue of DistMult), going beyond complex-valued embeddings may not be super novel. This is especially concerning as I think with the same number of parameters, ComplEx may work quite competitive with QuatE. 2- My second concern is regarding one of the experiments in the ablation study. According to Table 7, when the authors remove the relation normalization, then the performance of QuatE drops significantly showing (in three cases) sub-par results even compared to ComplEx. This raises the question that maybe the performance boost is only coming from relation normalization and not from using quaternions. 3- The results may be a bit unfair towards previous approaches. The authors report the results of several of the previous models directly from their papers while they use a somewhat different experimental setup. In particular, the embedding dimensions and the number of negative examples the authors generate per positive example are (for some datasets) larger than those used in previous papers. Also, the authors use a different way of generating negative examples (uniform sampling) rather than the corruption method introduced by Bordes et al. 2013 and used by the reported baselines. 4- How many epochs does it take your model to train? I'm assuming with a uniform negative sampling, it should take many epochs. Is this correct? 5- I may be missing something but the results in Table 4 and 5 do not seem to match. According to table 5, for WN18PR, QuatE outperforms RotatE on most relations with a large margin. However, the overall MRR of QuatE on this dataset is only slightly better than RotatE. Why is this happening? Minor comments: 6- The correct reference for the proof of isomorphism between ComplEx and HolE is https://arxiv.org/pdf/1702.05563.pdf 7- "Compared to Euler angles, quaternion can avoid the problem of gimbal lock" requires much more explanation. 8- In line 172, I think "antisymmetric" should be replaced with "asymmetric" 9- I liked the criticism on fixing the composition patterns. 10- The paragraph on "connection to DistMult and ComplEx" is kind of obvious and not necessary,

Using quaternions seems novel. The experimental results look good too. However, the necessity of quaternions is not very clear, i.e., why more degrees of freedom is needed? And, do quaternions solve the problem in the example of "elder brother"? In the appendix, octonions are considered, but similarly, there is almost no explanation why this is needed. In the conclusion, "good generalization" is mentioned as an advantage, but I did not see any corresponding rigorous argument in this paper.