Word embeddings are meant to encode meaning without regard to the frequencies of the words. The underlying assumption is that word meaning is independent of word frequency — the meaning of 'dog' does not depend on how often it is used. (Of course, word frequency can be relevant to other aspects of meaning, such as semantic change.)
However, what we have seen in these posts is that frequency effects are pervasive in word embeddings, across methods and corpus size. In other words, word embeddings do not encode meaning without regard to the frequencies of words. Here are some of the frequency effects from the previous posts:
Another set of results has to do with the embedding methods:
Since particular frequency effects vary by embedding method, a given frequency effect is not due to language, or the corpus, alone, but is due in large part to the embedding method. Sgns and ft are similar algorithmically in trying to opimize predictions, and glove and ppmi are similar in working more directly with counts. These facts make it tempting to attribute the difference in frequency effects to the differences in the two general approaches. On the other hand,  have shown the mathematical similarity of sgns and ppmi, which makes the differences harder to understand.
We have also seen that some frequency effects vary by the corpus (shifted similarities and hubs). Since I've only used two corpora, we can't say for sure whether this variation is due to size (though that seems likely) or something else. Similarly the "power law" of nearest neighbors did not hold for the methods and corpora here, even though  showed it for a large corpus using a different embdding method, but we cannot say whether the power law is due to the larger size of the corpus in  or the method.
Given that frequency effects seem to be pervasive in word embeddings, one question is whether they affect uses of word embeddings. In order to determine that, we would need ways to mitigate the frequency effects. In upcoming posts, I will propose a framework for classifying the types of mitigation effects, both those that have been proposed in the literature and some new ones. I will also show that the mitigation effects may have positive effects, improving some evaluation scores.
A second question is whether the frequency effects discussed here occur with other methods and with other corpora. Furthermore, all the frequency effects have been with corpora that are large in English. (Vanity Fair does have some French, and the large corpora used for testing in the literature are also not completely monolingual.) Given these I plan on looking at methods, more corpora, and more languages in the future.
 Omar Levy and Yoav Goldberg. 2014. Neural Word Embedding as Implicit Matrix Factorization. In Advances in neural information processing systems 3:2177-2185.
 Tobias Schnabel, Igor Labutov, David Mimno, and Thorsten Joachims. 2015. Evaluation methods for unsupervised word embeddings. Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing.