Multilingual representations pre-trained with monolingual data exhibit considerably unequal task performances across languages. Previous studies address this challenge with resource-intensive contextualized alignment, which assumes the availability of large parallel data, thereby leaving under-represented language communities behind. In this work, we attribute the data hungriness of previous alignment techniques to two limitations: (i) the inability to sufficiently leverage data and (ii) these techniques are not trained properly. To address these issues, we introduce supervised and unsupervised density-based approaches named Real-NVP and GAN-Real-NVP, driven by Normalizing Flow, to perform alignment, both dissecting the alignment of multilingual subspaces into density matching and density modeling. We complement these approaches with our validation criteria in order to guide the training process. Our experiments encompass 16 alignments, including our approaches, evaluated across 6 language pairs, synthetic data and 4 NLP tasks. We demonstrate the effectiveness of our approaches in the scenarios of limited and no parallel data. First, our supervised approach trained on 20k parallel data mostly surpasses Joint-Align and InfoXLM trained on much larger parallel data. Second, parallel data can be removed without sacrificing performance when integrating our unsupervised approach in our bootstrapping procedure, which is theoretically motivated to enforce equality of multilingual subspaces. Moreover, we demonstrate the advantages of validation criteria over validation data for guiding supervised training. Our code is available at \url{https://github.com/AIPHES/Real-NVP}.

本研究采用密度匹配和密度建模的方法进行多语种词向量的对齐，提出基于Normalizing Flow的Real-NVP和GAN-Real-NVP两种监督学习和无监督学习的密度对齐方法，并通过验证标准来指导训练过程。实验结果表明，我们的方法在有限或没有平行数据的情况下均能显著提高多项NLP任务的表现。

约束密度匹配与建模：用于跨语言上下文表示对齐