  Embeddings are a basic initial feature extraction step in many machine
learning models, particularly in natural language processing. An embedding
attempts to map data tokens to a low-dimensional space where similar tokens are
mapped to vectors that are close to one another by some metric in the embedding
space. A basic question is how well can such embedding be learned? To study
this problem, we consider a simple probability model for discrete data where
there is some "true" but unknown embedding where the correlation of random
variables is related to the similarity of the embeddings. Under this model, it
is shown that the embeddings can be learned by a variant of low-rank
approximate message passing (AMP) method. The AMP approach enables precise
predictions of the accuracy of the estimation in certain high-dimensional
limits. In particular, the methodology provides insight on the relations of key
parameters such as the number of samples per value, the frequency of the terms,
and the strength of the embedding correlation on the probability distribution.
Our theoretical findings are validated by simulations on both synthetic data
and real text data.
