word2vec principle analysis and detailed explanation of code formula

tags: word2vec Neural Networks word embedding Word vector

word2vec is a toolkit for obtaining word vectors that Google launched open source in 2013. The original papers and many analyses on the Internet have been written too rough to understand. Now there are the detailed principle analysis and detailed formula derivation process based on the source code as follows:

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$\sum_{i=1}^{n} \sum_{j=i+1}^{n}<v_{i},v_{j}>x_{i}x_{j}\\ = \frac{1}{2}\sum_{i=1}^{n} \sum_{j=i}^{n}<v_{i},v_{j}>x_{i}x_{j}-\frac{1}{2}\sum_{i=1}^{n} <v_{i},v_{i}>x_{i}x_{i}\\ =\frac{1}{2}(\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{f=1}^{k}v_{if}v_{jf}x_{i}x_{j}-\sum_{i=1}^{n}\sum_{f=1}^{k}v_{if}v_{if}x_{i}x_{i})\\ =\frac{1}{2}(\sum_{f=1}^{k}\sum_{i=1}^{n}v_{if}x_{i}\sum_{j=1}^{n}v_{jf}x_{j}-\sum_{i=1}^{n}\sum_{f=1}^{k}v_{if}v_{if}x_{i}x_{i})\\ =\frac{1}{2}((\sum_{i=1}^{n}v_{if}x_{i})^{2}-\sum_{i=1}^{n}v_{if}^{2}x_{i}^{2})$