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--- 신경망:공동_방법 [2022/12/05 15:49] – [공동 방법(cavity method)] jiwon
+++ 신경망:공동_방법 [2023/09/05 15:46] (current) – external edit 127.0.0.1
@@ Line 48: / Line 48: @@
 가 된다.
+마찬가지로 아래 그림의 회색 영역처럼 스핀 변수 하나와 인접한 기능 노드들을 한꺼번에 추가하는 경우도 생각해볼 수 있을 것이다.
+{{ :신경망:fig2.3.png?500 |}}
+이 때 새로운 해밀토니안은
+$$H^{\text{new}} = H^{\text{old}} -\sum_{b\in\partial i}J_b\prod_{j\in\partial b}\sigma_j$$
+이고, 분배함수는
+\begin{align*}
+Z^{\text{new}}=&\sum_{\sigma^{\text{old}}}\sum_{\sigma_i}\exp\left(-\beta H^{\text{old}} + \beta\sum_{b\in\partial i}J_b\prod_{j\in\partial b}\sigma_j\right)\\
+=&\sum_{\sigma^{\text{old}}}\sum_{\sigma_i}\exp\left(-\beta H^{\text{old}} + \beta\sum_{b\in\partial i}J_b\sigma_i\prod_{j\in\partial b\backslash i}\sigma_j\right)\\
+=&Z^{\text{old}}\sum_{\sigma^{\text{old}}}\sum_{\sigma_i}\frac{\exp\left(-\beta H^{\text{old}}\right)}{Z^{\text{old}}}\exp\left(\beta\sum_{b\in\partial i}J_b\sigma_i\prod_{j\in\partial b\backslash i}\sigma_j\right)
+\end{align*}
+가 된다. 위에서와 같이
+$$P_{\text{cavity}}(\{\sigma_j\vert j\in\partial b\backslash i;b\in\partial i\}) = \sum_{\{\sigma_j\vert j\not\in\partial b\backslash i;b\not\in\partial i\}}\frac{\exp\left(-\beta H^{\text{old}}\right)}{Z^{\text{old}}}\approx \left(\prod_{j\in\partial b\backslash i}\prod_{b\in\partial i}q_{j\rightarrow b}(\sigma_j)\right)$$
+로 근사하면
+\begin{align*}
+Z^{\text{new}}=&Z^{\text{old}}\sum_{\{\sigma_j\vert j\in\partial b\backslash i;b\in\partial i\}}\sum_{\sigma_i}P_{\text{cavity}}(\{\sigma_j\vert j\in\partial b\backslash i;b\in\partial i\})\exp\left(\beta\sum_{b\in\partial i}J_b\sigma_i\prod_{j\in\partial b\backslash i}\sigma_j\right)\\
+\approx&Z^{\text{old}}\sum_{\{\sigma_j\vert j\in\partial b\backslash i;b\in\partial i\}}\sum_{\sigma_i}\left(\prod_{j\in\partial b\backslash i}\prod_{b\in\partial i}q_{j\rightarrow b}(\sigma_j)\right)\exp\left(\beta\sum_{b\in\partial i}J_b\sigma_i\prod_{j\in\partial b\backslash i}\sigma_j\right)\\
+=&Z^{\text{old}}\sum_{\sigma_i}\prod_{b\in\partial i}\left[\sum_{\{\sigma_j\vert j\in\partial b\backslash i\}}\prod_{j\in\partial b\backslash i}q_{j\rightarrow b}(\sigma_j)\exp\left(\beta J_b\sigma_i\prod_{j\in\partial b\backslash i}\sigma_j\right)\right]\\
+\end{align*}
+로 쓸 수 있다. 괄호 안의 항을
+\begin{align*}
+\Lambda_{b\rightarrow i}^{\sigma_i} \equiv&\sum_{\{\sigma_j\vert j\in\partial b\backslash i\}}\prod_{j\in\partial b\backslash i}q_{j\rightarrow b}(\sigma_j)\exp\left(\beta J_b\sigma_i\prod_{j\in\partial b\backslash i}\sigma_j\right)\\
+=&\sum_{\{\sigma_j\vert j\in\partial b\backslash i\}}\prod_{j\in\partial b\backslash i}\frac{1+\sigma_jm_{j\rightarrow b}}2\cosh(\beta J_b)\left(1+\sigma_i\prod_{j\in\partial b\backslash i}\sigma_j\tanh(\beta J_b)\right)
+\end{align*}
+로 정의하고, 각 항을 나누어 계산하면
+$$\sum_{\{\sigma_j\vert j\in\partial b\backslash i\}}\prod_{j\in\partial b\backslash i}\frac{1+\sigma_jm_{j\rightarrow b}}2 = 1$$
+\begin{align*}
+&\sum_{\{\sigma_j\vert j\in\partial b\backslash i\}}\prod_{j\in\partial b\backslash i}\frac{1+\sigma_jm_{j\rightarrow b}}2\sigma_i\sigma_j\tanh(\beta J_b)\\
+=&\sigma_i\tanh(\beta J_b)\sum_{\{\sigma_j\vert j\in\partial b\backslash i\}}\prod_{j\in\partial b\backslash i}\frac{\sigma_j+m_{j\rightarrow b}}2\\
+=&\sigma_i\tanh(\beta J_b)\prod_{j\in\partial b\backslash i}m_{j\rightarrow b}
+\end{align*}
+$$\Lambda_{b\rightarrow i}^{\sigma_i} = \cosh(\beta J_b)\left(1+\sigma_i\tanh(\beta J_b)\prod_{j\in\partial b\backslash i}m_{j\rightarrow b}\right)$$
+가 되므로 새로운 분배함수는
+$$Z^{\text{new}} = Z^{\text{old}}\sum_{\sigma_i}\prod_{b\in\partial i}\Lambda_{b\rightarrow i}^{\sigma_i} =  Z^{\text{old}}\left(\prod_{b\in\partial i}\Lambda_{b\rightarrow i}^++\prod_{b\in\partial i}\Lambda_{b\rightarrow i}^-\right)$$
+와 같이 쓸 수 있다.
+====참고문헌====
+Haiping Huang, Statistical Physics of Neural Networks, Springer, 2021