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I have a set of predefined categories(like: cars, girls, etc.) and a set short texts and I need to assign one or several categories to each text. What is the best way to do this? I don't have trainings texts to make supervised learning classification. Is Word2Vec the best choice or there is better approach?

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If you don't have labeled data, Word2vec can be one solution.

Perhaps have a look at FastText unsupervised (recently released by Facebook) which is similar but uses character n-grams. Character n-grams helps to take into account the morphology, is more robust to typos and make it possible to compute vectors for unknown words.

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