# Quantized Wasserstein Procrustes Alignment of Word Embedding Spaces

arXiv: https://arxiv.org/abs/2212.02468 · HTML: https://mlsanity.com/paper/2212.02468/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 37 references that are in the map.
- First posted on arXiv 5 Dec 2022.

## Builds on (most cited first, 19 of 19)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Enriching Word Vectors with Subword Information](https://mlsanity.com/paper/1607.04606/) (2016)
- [Word Translation Without Parallel Data](https://mlsanity.com/paper/1710.04087/) (2017)
- [Exploiting Similarities among Languages for Machine Translation](https://mlsanity.com/paper/1309.4168/) (2013)
- [Unsupervised Neural Machine Translation](https://mlsanity.com/paper/1710.11041/) (2017)
- [A robust self-learning method for fully unsupervised cross-lingual mappings of word embeddings](https://mlsanity.com/paper/1805.06297/) (2018)
- [A Survey Of Cross-lingual Word Embedding Models](https://mlsanity.com/paper/1706.04902/) (2017)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Unsupervised Alignment of Embeddings with Wasserstein Procrustes](https://mlsanity.com/paper/1805.11222/) (2018)
- [Offline bilingual word vectors, orthogonal transformations and the inverted softmax](https://mlsanity.com/paper/1702.03859/) (2017)
- [Loss in Translation: Learning Bilingual Word Mapping with a Retrieval Criterion](https://mlsanity.com/paper/1804.07745/) (2018)
- [How to (Properly) Evaluate Cross-Lingual Word Embeddings: On Strong Baselines, Comparative Analyses, and Some Misconceptions](https://mlsanity.com/paper/1902.00508/) (2019)
- [Do We Really Need Fully Unsupervised Cross-Lingual Embeddings?](https://mlsanity.com/paper/1909.01638/) (2019)
- [Unsupervised Hyperalignment for Multilingual Word Embeddings](https://mlsanity.com/paper/1811.01124/) (2018)
- [Towards cross-lingual distributed representations without parallel text trained with adversarial autoencoders](https://mlsanity.com/paper/1608.02996/) (2016)
- [Revisiting Adversarial Autoencoder for Unsupervised Word Translation with Cycle Consistency and Improved Training](https://mlsanity.com/paper/1904.04116/) (2019)
- [On a Novel Application of Wasserstein-Procrustes for Unsupervised Cross-Lingual Learning](https://mlsanity.com/paper/2007.09456/) (2020)
- [Improving Approximate Optimal Transport Distances using Quantization](https://mlsanity.com/paper/2102.12731/) (2021)
- [Wasserstein-Fisher-Rao Document Distance](https://mlsanity.com/paper/1904.10294/) (2019)

## Built on by (most cited first, 6 of 6)

- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Discovering Universal Geometry in Embeddings with ICA](https://mlsanity.com/paper/2305.13175/) (2023)
- [Uncovering Challenges of Solving the Continuous Gromov-Wasserstein Problem](https://mlsanity.com/paper/2303.05978/) (2023)
- [An in depth look at the Procrustes-Wasserstein distance: properties and barycenters](https://mlsanity.com/paper/2507.00894/) (2025)
- [Sinhala-English Word Embedding Alignment: Introducing Datasets and Benchmark for a Low Resource Language](https://mlsanity.com/paper/2311.10436/) (2023)
- [How Good is BLI as an Alignment Measure: A Study in Word Embedding Paradigm](https://mlsanity.com/paper/2511.13040/) (2025)

## Abstract (arXiv)

Optimal Transport (OT) provides a useful geometric framework to estimate the permutation matrix under unsupervised cross-lingual word embedding (CLWE) models that pose the alignment task as a Wasserstein-Procrustes problem. However, linear programming algorithms and approximate OT solvers via Sinkhorn for computing the permutation matrix come with a significant computational burden since they scale cubically and quadratically, respectively, in the input size. This makes it slow and infeasible to compute OT distances exactly for a larger input size, resulting in a poor approximation quality of the permutation matrix and subsequently a less robust learned transfer function or mapper. This paper proposes an unsupervised projection-based CLWE model called quantized Wasserstein Procrustes (qWP). qWP relies on a quantization step of both the source and target monolingual embedding space to estimate the permutation matrix given a cheap sampling procedure. This approach substantially improves the approximation quality of empirical OT solvers given fixed computational cost. We demonstrate that qWP achieves state-of-the-art results on the Bilingual lexicon Induction (BLI) task.