  Product embeddings have been heavily investigated in the past few years,
serving as the cornerstone for a broad range of machine learning applications
in e-commerce. Despite the empirical success of product embeddings, little is
known on how and why they work from the theoretical standpoint. Analogous
results from the natural language processing (NLP) often rely on
domain-specific properties that are not transferable to the e-commerce setting,
and the downstream tasks often focus on different aspects of the embeddings. We
take an e-commerce-oriented view of the product embeddings and reveal a
complete theoretical view from both the representation learning and the
learning theory perspective. We prove that product embeddings trained by the
widely-adopted skip-gram negative sampling algorithm and its variants are
sufficient dimension reduction regarding a critical product relatedness
measure. The generalization performance in the downstream machine learning task
is controlled by the alignment between the embeddings and the product
relatedness measure. Following the theoretical discoveries, we conduct
exploratory experiments that supports our theoretical insights for the product
embeddings.
