1

It is mentioned in the GloVe: Global Vectors for Word Representation. It says: enter image description here

where $w_i$, $w_j$ and $\tilde{w}_k$ are all word vectors and $F$ is just an unknown function. The author then assumes $F$ as $exp$. What is homomorphism exactly? Why equation (4) can make $F((w_i - w_j)^T\tilde{w}_k)$ homomorphism?

hidemyname
  • 161

1 Answers1

6

copying verbatim from Wikipedia

A homomorphism is a map between two algebraic structures of the same type (that is of the same name), that preserves the operations of the structures. This means a map ${\displaystyle f:A\to B}$ between two sets $A$, $B$ equipped with the same structure such that, if $∗$ is an operation of the structure (supposed here, for simplification, to be a binary operation), then $${\displaystyle f(x*y)=f(x)*f(y)}$$ for every pair $x, y$ of elements of $A$

Xi'an ні війні
  • 766
  • 3
    Specifically, here we have a group homomorphism between the additive group of the reals and the multiplicative group of the nonzero reals. It must satisfy that the neutral element $0$ of the first group must be mapped to the neutral element $1$ of the second one, $F(0)=1$, and that it commutes with the respective group operations, in this case $F(x+y)=F(x)\times F(y)$. – Stephan Kolassa Dec 26 '17 at 09:41