Paper:
We study the problem of knowledge base (KB) embedding, which is usually addressed through two frameworks-neural KB embedding and tensor decomposition. In this work, we theoretically analyze the neural embedding framework and subsequently connect it with tensor based embedding. Specifically, we show that in neural KB embedding the two commonly adopted optimization solutionsmargin-based and negative sampling losses-are closely related to each other. We also reach the closed-form tensor that is implicitly approximated by popular neural KB approaches, revealing the underlying connection between neural and tensor based KB embedding models. Grounded in the theoretical results, we further present a tensor decomposition based framework KBTD to directly approximate the derived closed form tensor. Under this framework, the neural KB embedding models, such as NTN (Socher et al., 2013), TransE (Bordes et al., 2013), Bilinear (Jenatton et al., 2012), and DISTMULT (Yang et al., 2015), are unified into a general tensor optimization architecture. Finally, we conduct experiments on the link prediction task in WordNet and Freebase, empirically demonstrating the effectiveness of the KBTD framework.

Review:
The paper proposes a new method to train knowledge base embeddings using a least-squares loss. For this purpose, the paper introduces a reweighting scheme of the entries in the original adjacency tensor. The reweighting is derived from an analysis of the cross-entropy loss. In addition, the paper discusses the connections of the margin and cross-entropy loss and evaluates the proposed method on WN18 and FB15k.
 The paper tackles an interesting problem, as learning from knowledge bases via embedding methods has become increasingly important for tasks such as question answering. Providing additional insight into current methods can be an important contribution to advance the state-of-the-art.
However, I'm concerned about several aspects in the current form of the paper. For instance, the derivation in Section 4 is unclear to me, as eq.4 suddenly introduces a weighted sum over expectations using the degrees of nodes. The derivation also seems to rely on a very specific negative sampling assumption (uniform sampling without checking whether the corrupted triple is a true negative). This sampling method isn't used consistently across models and also brings its own problems, e.g., see the LCWA discussion in [4]
In addition, the semantics that are introduced by the weighting scheme are not clear to me either. Using the proposed method, the probability of edges between high-degree nodes are down-weighted, since the ground-truth labels are divided by the node degrees. Since these weighted labels are then fitted using a least-squares loss, this implies that links between high-degree nodes should be less likely, which seems the opposite of what the scores should look like.
With regard to the significance of the contributions: Using a least-squares loss in combination with tensor methods is attractive because it enables ALS algorithms with closed-form updates that can be computed very fast. However, the proposed method still relies on SGD optimization. In this context, it is not clear to me why a tensor framework/least-squares loss would be preferable.
Further comments:
- The paper seems to equate "tensor method" with using a least squares loss. However, this doesn't have to be the case. For instance see [1,2] which propose Logistic and Poisson tensor factorizations, respectively.
- The distinction between tensor factorization and neural methods is unclear. Tensor factorization can be interpreted just as a particular scoring function. For instance, see [5] for a detailed discussion.
- The margin based ranking loss has been proposed earlier than in (Collobert et al, 2011). For instance see [3]
- p1: corrupted triples are not described entirely correct, typically only one of s or o is corrputed. 
- Closed-form tensor in Table 1: This should be least-squares loss of f(s,p,o) and log(...)?
- p6: Adding the constant to the tensor as proposed in (Levy & Goldberg, 2014) can done while gathering the minibatch and is therefore equivalent to the proposed approach.
[1] Nickel et al: Logistic Tensor Factorization for Multi-Relational Data, 2013.
[2] Chi et al: "On tensors, sparsity, and nonnegative factorizations", 2012
[3] Collobert et al: A unified architecture for natural language processing, 2008
[4] Dong et al: Knowledge Vault: A Web-Scale Approach to Probabilistic Knowledge Fusion, 2014
[5] Nickel et al: A Review of Relational Machine Learning for Knowledge Graphs, 2016.