# Wasserstein-Fisher-Rao Document Distance

arXiv: https://arxiv.org/abs/1904.10294 · HTML: https://mlsanity.com/paper/1904.10294/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Apr 2019.

## Built on by (most cited first, 3 of 3)

- [Variational Regularized Unbalanced Optimal Transport: Single Network, Least Action](https://mlsanity.com/paper/2505.11823/) (2025)
- [Quantized Wasserstein Procrustes Alignment of Word Embedding Spaces](https://mlsanity.com/paper/2212.02468/) (2022)
- [Unsupervised Sentence-embeddings by Manifold Approximation and Projection](https://mlsanity.com/paper/2102.03795/) (2021)

## Abstract (arXiv)

As a fundamental problem of natural language processing, it is important to measure the distance between different documents. Among the existing methods, the Word Mover's Distance (WMD) has shown remarkable success in document semantic matching for its clear physical insight as a parameter-free model. However, WMD is essentially based on the classical Wasserstein metric, thus it often fails to robustly represent the semantic similarity between texts of different lengths. In this paper, we apply the newly developed Wasserstein-Fisher-Rao (WFR) metric from unbalanced optimal transport theory to measure the distance between different documents. The proposed WFR document distance maintains the great interpretability and simplicity as WMD. We demonstrate that the WFR document distance has significant advantages when comparing the texts of different lengths. In addition, an accelerated Sinkhorn based algorithm with GPU implementation has been developed for the fast computation of WFR distances. The KNN classification results on eight datasets have shown its clear improvement over WMD.