  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
$\textit{relational inner product association}$ (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.
