  The ability to measure similarity between documents enables intelligent
summarization and analysis of large corpora. Past distances between documents
suffer from either an inability to incorporate semantic similarities between
words or from scalability issues. As an alternative, we introduce hierarchical
optimal transport as a meta-distance between documents, where documents are
modeled as distributions over topics, which themselves are modeled as
distributions over words. We then solve an optimal transport problem on the
smaller topic space to compute a similarity score. We give conditions on the
topics under which this construction defines a distance, and we relate it to
the word mover's distance. We evaluate our technique for k-NN classification
and show better interpretability and scalability with comparable performance to
current methods at a fraction of the cost.
