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I would like to obtain word embeddings (vector representation of words).

Do you think it is better to try to directly obtain a word embedding using some neural network approach or is it better to learn character embeddings then, use them to build word embeddings?

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    Use the words directly, not the characters. Yann LeCun did a text prediction study years ago where he found much higher accuracy using characters rather than words. That was the only evidence; other studies showed words were better.
    – Mitch
    Commented May 26, 2017 at 13:05

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I recommend that you read this article by Ling et al:

@Conference{Lingetal2015,
Author = {Wang Ling and Chris Dyer and Alan Black and Isabel Trancoso},
Title = {Two/too simple adaptations of word2vec for syntax problems},
Booktitle = {Proceedings of NAACL},
Year = {2015},
}

It contains a guidance on doing word embeddings properly.

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