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Beyond Fine-Tuning: Procrustes Alignment of Multilingual Embeddings for Low-Resource Cross-Lingual Retrieval

Multilingual sentence-embedding models are widely used for cross-lingual retrieval; however, their performance drops significantly in low-resource languages. The Urdu language, which is considered a low-resource language by the NL community, poses this challenge, despite being spoken by over 246 million people worldwide. Its distribution in training corpora results in poor alignment with English within shared embedding spaces. To resolve this misalignment without model fine-tuning, we apply Procrustes transformation, which is an orthogonal post-hoc alignment method with a closed-form solution. We utilize SQuAD and UQA datasets to learn a rotation matrix from a small set of sentence pairs and evaluate its effect across five multilingual embedding models (MiniLM, DistilUSE, E5-Base, LaBSE, and E5-Large) and perform geometric alignment, cross-lingual retrieval, and question-answering tasks on these models. We find that cosine distances between parallel pairs decrease by up to 38.67%, and retrieval accuracy improves by 12.49% points in Recall@1. We also analyze that models with better pre-trained cross-lingual representations exhibit a saturation effect, showing minimal retrieval change even as geometric tightening increases. Our error analysis reveals that morphologically complex queries and colloquial expressions remain challenging, indicating representational limitations beyond the scope of a linear transformation. These findings demonstrate that a computationally inexpensive alignment step can meaningfully improve cross-lingual retrieval for low-resource languages, with implications for retrieval-augmented generation (RAG) in resource-constrained settings.