About Hermitian product
EJHyun opened this issue · 2 comments
Dear author
Hi im student from South Korea
I have a question about hermitian product which is used in ComplEx's score function.
I would be glad if you explain how the equation (11) was made..
In detail, lets assume we have entity A's embedding vector of dimension 100. (Tensor shape (1, 100))
To treat this embedding vector as a complex vector,
ComplEx divides this 100-dimension embedding into 2 pieces, Re and Im.
(Real, Imaginary each would have 50-dimension in this case)
Let's assume we have another entity B's embedding too.
Then the similarity(= inner product) of these two entities can be computed with hermitian product.
(Let's say <> is a sum of element-wise multiplication>
Similarity(A,B) = <Re(A) , Re(B)> + <-Im(A) , Im(B)>
This is what I understood(Am I right?) and here is the question.
ComplEx's score function calculates score with 3 vectors (2 entities and 1 relation)...
How can I do this?? I mean, How did you made equation (11)??
Uhh..lets say relation between A, B is R.
The paper says we can calculate 3 vector's hermitian product just like the equation (11) does.
<Re(R), Re(A), Re(B)> + <Re(R), Im(A), Im(B)> + <Im(R), Re(A), Im(B)> - <Im(R), Im(A), Re(B)>
How did you made this equation??
I'll be really glad if you tell me how..
Thank you!
Hi !
So as mentionned in the foot note 2 in page 4 of the ICML paper, there is no "proper" extension of the Hermitian product to three dimension, we simply extended it as the sum of the element wise products of three vectors instead of two. It's nothing more than the distribution of the products:
<R,A,B> = <Re(R) + i Im(R), Re(A) + i Im(A), Re(B) + i Im(B) >
= SUM_j ( ( Re(r_j) + i Im(r_j) ) * ( Re(a_j) + i Im(a_j) ) * ( Re(b_j) + i Im(b_j) ) )
= SUM_j ( ( Re(r_j) * Re(a_j) * Re(b_j) ) + ( Re(r_j) * Im(a_j) * Im (b_j) ) + ( Im(r_j) * Re(a_j) * Im(b_j) ) - ( Im(r_j) * Im(a_j) * Re(b_j) ) )
= <Re(R), Re(A), Re(B)> + <Re(R), Im(A), Im(B)> + <Im(R), Re(A), Im(B)> - <Im(R), Im(A), Re(B)>
Write it down it'll be clearer, hope it helps !
Also, a 100-dimension complex embedding has a 100-dimension real part and a 100-dimension imaginary part, not 50 !
Thank you so much!
Your answer was truly helpful
Thanks!!