Experiments by<cite> Bolukbasi et al. (2016)</cite> found that debiasing word embeddings using the subspace projection method precludes gender-biased analogies from holding. However, as we noted earlier, despite this method being intuitive, there is no theoretical guarantee that the debiased vectors are perfectly unbiased or that the debiasing method works for embedding models other than SGNS. In this section, we prove that for any embedding model that does implicit matrix factorization (e.g., GloVe, SGNS), debiasing embeddings post hoc using the subspace projection method is, under certain conditions, equivalent to training on a perfectly unbiased corpus without reconstruction error.