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Multilingual representations pre-trained with monolingual data exhibitconsiderably unequal task performances across languages. Previous studiesaddress this challenge with resource-intensive contextualized alignment, whichassumes the availability of large parallel data, thereby leavingunder-represented language communities behind. In this work, we attribute thedata hungriness of previous alignment techniques to two limitations: (i) theinability to sufficiently leverage data and (ii) these techniques are nottrained properly. To address these issues, we introduce supervised andunsupervised density-based approaches named Real-NVP and GAN-Real-NVP, drivenby Normalizing Flow, to perform alignment, both dissecting the alignment ofmultilingual subspaces into density matching and density modeling. Wecomplement these approaches with our validation criteria in order to guide thetraining process. Our experiments encompass 16 alignments, including ourapproaches, evaluated across 6 language pairs, synthetic data and 5 NLP tasks.We demonstrate the effectiveness of our approaches in the scenarios of limitedand no parallel data. First, our supervised approach trained on 20k paralleldata (sentences) mostly surpasses Joint-Align and InfoXLM trained on over 100kparallel sentences. Second, parallel data can be removed without sacrificingperformance when integrating our unsupervised approach in our bootstrappingprocedure, which is theoretically motivated to enforce equality of multilingualsubspaces. Moreover, we demonstrate the advantages of validation criteria overvalidation data for guiding supervised training.

Constrained Density Matching and Modeling for Cross-lingual Alignment of Contextualized Representations