Word embeddings are often criticized for capturing undesirable word associations such as gender stereotypes. However, methods for measuring and removing such biases remain poorly understood. We show that for any embedding model that implicitly does matrix factorization, debiasing vectors post hoc using subspace projection (Bolukbasi et al., 2016) is, under certain conditions, equivalent to training on an unbiased corpus. We also prove that WEAT, the most common association test for word embeddings, systematically overestimates bias. Given that the subspace projection method is provably effective, we use it to derive a new measure of association called the relational inner product association (RIPA). Experiments with RIPA reveal that, on average, skipgram with negative sampling (SGNS) does not make most words any more gendered than they are in the training corpus. However, for gender-stereotyped words, SGNS actually amplifies the gender association in the corpus.

A surprising property of word vectors is that word analogies can often be solved with vector arithmetic. However, it is unclear why arithmetic operators correspond to non-linear embedding models such as skip-gram with negative sampling (SGNS). We provide a formal explanation of this phenomenon without making the strong assumptions that past theories have made about the vector space and word distribution. Our theory has several implications. Past work has conjectured that linear substructures exist in vector spaces because relations can be represented as ratios; we prove that this holds for SGNS. We provide novel justification for the addition of SGNS word vectors by showing that it automatically down-weights the more frequent word, as weighting schemes do ad hoc. Lastly, we offer an information theoretic interpretation of Euclidean distance in vector spaces, justifying its use in capturing word dissimilarity.