Paper:
As observed in previous works, many word embedding methods exhibit two interesting properties: (1) words having similar semantic meanings are embedded closely; (2) analogy structure exists in the embedding space, such that "Paris is to France as Berlin is to Germany". We theoretically analyze the inner mechanism leading to these nice properties. Specifically, the embedding can be viewed as a low rank transformation from the word-context co-occurrence space to the embedding space. Such embedding transformation will preserve the relative distances among words. Furthermore, previous studies empirically observed that the parameter α has a strong influence on the performance of word embedding but did not provide theoretical explanation. We provide a theoretical explanation for this behavior, and derive a method to automatically find its optimal value. The experiments on real datasets verify our analysis.

Review:
This paper explores the role of the implicit alpha parameter when learning word embeddings. More concretely, word embeddings work by either implicitly or explicitly factorizing a co-occurrence matrix, and the underlying parameter alpha controls how the singular values are weighted between the word and the context vectors. The authors provide theoretical insights on the role of alpha in relation with the original co-occurrence matrix, and propose a new method to find its optimal value.
I think that this is overall a solid work. The paper provides a new perspective in the workings of word embeddings that I find interesting, it is theoretically well-founded (although I did not check all derivations in detail), and the presentation is clear.
However, I think that the paper does a poor job in putting its contributions into context in relation to previous work. In particular, the role of alpha in word embeddings was already studied empirically by Artetxe et al. (CoNLL'18, https://www.aclweb.org/anthology/K18-1028.pdf) for both word analogy and word similarity tasks, to the extent that Figure 2 in both papers is showing the exact same curves. However, the authors do not even cite it. As acknowledged in the paper, other authors like Levy et al. (TACL'15) also observed that the value of alpha was important in their experiments.
I think that the right narrative for the paper should more in the line of "previous work showed that alpha behaves this and that way; we provide a theoretical explanation for this behavior, and derive a method to automatically find its optimal value". However, starting from the title ("word embedding re-examined: is the symmetrical factorization optimal?", when it was already known that it wasn't) and the abstract (where only the statement that "we propose a method to find the optimal alpha" corresponds to a novel contribution), the paper does a poor job in identifying and properly contextualizing its real contributions. More importantly, the paper does not try to establish any connection between the authors own theory and the empirical findings from previous work.
In terms of the actual content, the authors constantly claim that word2vec is performing a symmetric factorization (e.g. "the original word2vec is implicitly performing a symmetric factorization, thus implying the alpha equal to 0.5) as if it was something obvious or well-known. I might be missing something here, but I do not see why this is the case. Following your notation, let's say that word2vec is implicitly factorizing M = E*C^T, where E are the word embeddings and C are the context embeddings. One could multiply E with any arbitrary invertible matrix W, and C by the transpose of its inverse W^-T, which could be chosen to completely break any symmetry, yet the objective value of word2vec would not change at all, as (E*W)*(C*W^-T)^T = E*C^T. In other words, there is nothing in the training objective of word2vec that forces a symmetric factorization, and there is always an optimal solution with respect to this training objective that is arbitrarily asymmetric.
Another point that raises concerns to me is that the optimal value of alpha is determined by the vocabulary of the evaluation task. It would make sense if the optimal alpha depended on the nature of the task (e.g. syntactic vs semantic), but I do not have any intuition (nor do the authors provide) as of why the vocabulary would be anyhow relevant. More importantly, this does not seem generalizable beyond a few intrinsic tasks as, in the general case, one wants good embeddings for the full vocabulary. In either case, I think that this point deserves more attention in the paper.
I also find the experimental evaluation to be somewhat weak. In particular, the proposed theory focuses in two phenomena (word similarity and word analogy) as stated in the abstract itself, but the empirical evaluation is limited to the word similarity task.
Also, this is a minor detail and it did not influence my score, but I dislike that the authors use "word2vec" to refer to skip-gram with negative sampling throughout the paper. I would suggest to either use SGNS (which is quite standard) or simply skip-gram.