  Word embeddings are ubiquitous in NLP and information retrieval, but it is
unclear what they represent when the word is polysemous. Here it is shown that
multiple word senses reside in linear superposition within the word embedding
and simple sparse coding can recover vectors that approximately capture the
senses. The success of our approach, which applies to several embedding
methods, is mathematically explained using a variant of the random walk on
discourses model (Arora et al., 2016). A novel aspect of our technique is that
each extracted word sense is accompanied by one of about 2000 "discourse atoms"
that gives a succinct description of which other words co-occur with that word
sense. Discourse atoms can be of independent interest, and make the method
potentially more useful. Empirical tests are used to verify and support the
theory.
