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AAAI2021/tex/theory2.tex

@@ -33,15 +33,20 @@ R\left(c_0,s_0\right) & R\left(c_0,s_0\right)
 \end{equation}
 Each column of $M$ correspond to the semantic information carried by one symbol. In a perfectly compositional language, each symbol represents one specific concept exclusively. Therefore, the similarity between the columns of $M$ and a one-hot vector is align with the compositionality of the emergent language.
 
-Finally, we define \emph{raw mutual information similarity} (denoted as $S_0$) as the average cosine similarity of $M$ columns and one-hot vectors, as Equation~\ref{eq:mis2}. Furthermore, MIS (denoted as $S$) is the normalized raw mutual information similarity into the $[0,1]$ value range.
+Finally, we define \emph{raw mutual information similarity} ($MIS_0$)
+as the average cosine similarity of $M$ columns and one-hot vectors, as
+Equation~\ref{eq:mis2}. Furthermore, $MIS$ is the normalized raw mutual
+information similarity into the $[0,1]$ value range, which can be computed with
+following formula:
 \begin{equation}\label{eq:mis2}\begin{aligned}
-S_0 &= \frac{1}{2}\sum_{j=0}^1\frac{\max_{i=0,1}RI\left(c_i,s_j\right)}{\epsilon + \sqrt{\sum_{i=0}^{1}RI^2\left(c_i,s_j\right)}}, \epsilon > 0\\
-S &= 2S_0 - 1
+MIS_0 &= \frac{1}{2}\sum_{j=0}^1\frac{\max_{i=0,1}RI\left(c_i,s_j\right)}{\epsilon + \sqrt{\sum_{i=0}^{1}RI^2\left(c_i,s_j\right)}}, \epsilon > 0\\
+MIS &= 2MIS_0 - 1
 \end{aligned}\end{equation}
-Generalized to $m$ symbols and $n$ objects, $S$ is as Equation~\ref{eq:mis}
+Generalized to $m$ symbols and $n$ objects, $MIS$ can be computed with
+following formula: 
 \begin{equation}\label{eq:mis2}\begin{aligned}
-S_0 &= \frac{1}{m}\sum_{j=0}^{m-1}\frac{\max_{i\in[0,n-1]}R\left(c_i,s_j\right)}{\epsilon + \sqrt{\sum_{i=0}^{n-1}R^2\left(c_i,s_j\right)}}, \epsilon > 0\\
-S &= \frac{n\cdot S_0 - 1}{n-1}
+MIS_0 &= \frac{1}{m}\sum_{j=0}^{m-1}\frac{\max_{i\in[0,n-1]}R\left(c_i,s_j\right)}{\epsilon + \sqrt{\sum_{i=0}^{n-1}R^2\left(c_i,s_j\right)}}, \epsilon > 0\\
+MIS &= \frac{n\cdot MIS_0 - 1}{n-1}
 \end{aligned}\end{equation}
 
 \begin{figure}[t]