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.

提出一种基于投影的无监督量化 Wasserstein Procrustes（qWP）CLWE 模型，该模型利用源和目标单语嵌入空间的量化步骤来估计排列矩阵，以改善经验 OT 求解器的逼近质量，从而在计算成本固定的情况下取得了最先进的双语词典归纳（BLI）任务结果。

词嵌入空间的量化瓦瑟斯坦普鲁克斯对齐

Quantized Wasserstein Procrustes Alignment of Word Embedding Spaces

Equilibrium multi-population matching (matching for teams) is a problem from
mathematical economics which is related to multi-marginal optimal transport. A
special but important case is the Wasserstein barycenter problem, which has
applications in image processing and statistics. Two algorithms are presented:
a linear programming algorithm and an efficient nonsmooth optimization
algorithm, which applies in the case of the Wasserstein barycenters. The
measures are approximated by discrete measures: convergence of the
approximation is proved. Numerical results are presented which illustrate the
efficiency of the algorithms.

研究了数学经济学中的平衡多人匹配问题，介绍了与多边最优运输有关的特殊但重要的 Wasserstein barycenter 问题及其在图像处理和统计学中的应用，提出了两种算法：线性规划算法和高效的非平滑优化算法，并利用离散测度逼近来证明了算法的收敛性，最后给出了数值实验结果，展示了算法的高效性。