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I'm reading the paper titled "man to computer programmer is woman to homemaker. Debiasing word embeddings". I'm right now trying to figure out the math and logic behind it and was doing OK till I got to the neutralizing and equalizing part. Can anyone explain the equation in Step3?

I understand that it's some how taking the average of the bias directions out of the set's average and then for each word adds this debiased average to the debiased word vector, but I can't seem to get the math between them where it says $\sqrt{1-||v||^2}.$

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Thanks in advance!

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It looks like the term $\sqrt{1-||\nu||^2}$ was added in order to ensure that the norm of $\vec{w}$ is $1$.

Since $\nu$ and $\vec{w}_B-\mu_B$ are orthogonal, in the red-underlined formula you get $||\vec{w}||^2 = ||\nu||^2 + (1-||\nu||^2)\frac{||\vec{w}_B-\mu_B||^2}{||\vec{w}_B-\mu_B||^2} = 1$.

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