  Measuring sentence similarity is a classic topic in natural language
processing. Light-weighted similarities are still of particular practical
significance even when deep learning models have succeeded in many other tasks.
Some light-weighted similarities with more theoretical insights have been
demonstrated to be even stronger than supervised deep learning approaches.
However, the successful light-weighted models such as Word Mover's Distance
[Kusner et al., 2015] or Smooth Inverse Frequency [Arora et al., 2017] failed
to detect the difference from the structure of sentences, i.e. order of words.
To address this issue, we present Recursive Optimal Transport (ROT) framework
to incorporate the structural information with the classic OT. Moreover, we
further develop Recursive Optimal Similarity (ROTS) for sentences with the
valuable semantic insights from the connections between cosine similarity of
weighted average of word vectors and optimal transport. ROTS is
structural-aware and with low time complexity compared to optimal transport.
Our experiments over 20 sentence textural similarity (STS) datasets show the
clear advantage of ROTS over all weakly supervised approaches. Detailed
ablation study demonstrate the effectiveness of ROT and the semantic insights.
