There is a problem with formula 8 in this paper

[KAGAII]

Hello, I recently read your article, and there is a question that has been bothering me. Formula 8 in the article calculates the lower bound of similarity. Both sides of the equation are not equal and cannot be converted into a maximum pooling representation of the paragraph.
Can you answer my confusion? Thank you very much.

[alexlimh]

Thank you for your interest in SLIM. For the lower bound, let's simplify the equation first by setting $N=1$, since the sum operator is a linear operation. we now only need to show that $\max_j e_i^T \phi_j = e_i^T\max_j \phi_j$. As we know that $e_i$ is a one-hot vector and the active dimension $k$ is non-negative, $e_i^T\phi_j = \alpha * \phi_j^{(k)}$, where $\phi_j^{(k)}$ is the $k$-th element of $\phi_j$ and $\alpha = e_j^{(k)}$. Therefore, $\max_j e_i^T \phi_j = \alpha * \max_j \phi_j^{(k)}= \alpha * \max_j$ select $( \phi_j, k)$, where select $( \phi_j, k) = \phi_j^{(k)}$. Now, both the select operation and max operation are element-wise operation, it does not matter if we first select the $k$-th dimension and then take the max, or first take the max for all dimensions and then take the $k$-th dimension. Therefore, $\alpha * \max_j$ select $( \phi_j, k) = \alpha *$ select $( \max_j\phi_j, k) = e_i^T\max_j \phi_j$. I hope this is clear enough.

[KAGAII]

Your answer is excellent, I already understand, let me thank you again！