  To measure the similarity of documents, the Wasserstein distance is a
powerful tool, but it requires a high computational cost. Recently, for fast
computation of the Wasserstein distance, methods for approximating the
Wasserstein distance using a tree metric have been proposed. These tree-based
methods allow fast comparisons of a large number of documents; however, they
are unsupervised and do not learn task-specific distances. In this work, we
propose the Supervised Tree-Wasserstein (STW) distance, a fast, supervised
metric learning method based on the tree metric. Specifically, we rewrite the
Wasserstein distance on the tree metric by the parent-child relationships of a
tree, and formulate it as a continuous optimization problem using a contrastive
loss. Experimentally, we show that the STW distance can be computed fast, and
improves the accuracy of document classification tasks. Furthermore, the STW
distance is formulated by matrix multiplications, runs on a GPU, and is
suitable for batch processing. Therefore, we show that the STW distance is
extremely efficient when comparing a large number of documents.
