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Do we have cross-language vector space for word embedding?

When measure similarity for apple/Pomme/mela/Lacus/苹果/りんご, they should be the same

If would be great if there's available internet service of neuron network which already be trained by multiple language

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You can try to read about MUSE (Multilingual Unsupervised and Supervised Embeddings) by Facebook. You can read it from its Github or this article. They also provide the FastText dictionary format (.vec file) for some languages.

Their original paper shows how it aligns the vector of words from two different languages:

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  • $\begingroup$ Thank you very much, it helps me a lot! $\endgroup$ – gloomyson Apr 17 at 3:28
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For cross-language word representation the trend now is:

Remember that you can also do the task in 2 steps: Translate the words to a reference language (e.g english), then represent each one of them using any word representation model (in the reference language).

The 2-steps option is also good as specific-language word representation models are more accurate, and there are a bench of easy-to-use libraries for single-language translation (i.e py-translator) and representation (i.e Universal sentence encoder by Google).

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