Using a dictionary to map independently trained word embeddings to a shared space has shown to be an effective approach to learn bilingual word embeddings. In this work, we propose a multi-step framework of linear transformations that generalizes a substantial body of previous work. The core step of the framework is an orthogonal transformation, and existing methods can be explained in terms of the additional normalization, whitening, re-weighting, de-whitening and dimensionality reduction steps. This allows us to gain new insights into the behavior of existing methods, including the effectiveness of inverse regression, and design a novel variant that obtains the best published results in zero-shot bilingual lexicon extraction. The corresponding software is released as an open source project.
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