更改

*当<math>\det(\partial_{x'}f(x))\neq0</math>:

*当<math>\det(\partial_{x'}f(x))\neq0</math>:

<math>\begin{gathered}EI(f)=I(do(x\sim U([-L,L]^n));y)\approx-\frac{n+n\ln(2\pi)+\sum_{i=1}^n\ln\sigma_i^2}2+n\ln(2L)+\operatorname{E}_{x\sim U([-L,L]^n)}(\ln|\det(\partial_{x^{\prime}}f(x)))|)\end{gathered} </math>

<math>\begin{gathered}EI(f)=I(do(x\sim U([-\frac{L}{2},\frac{L}{2}]^n));y)\approx-\frac{n+n\ln(2\pi)+\sum_{i=1}^n\ln\sigma_i^2}2+n\ln(2L)+\mathbb{E}_{x\sim U([-\frac{L}{2},\frac{L}{2}]^n)}(\ln|\det(\partial_{x^{\prime}}f(x)))|)\end{gathered} </math>

其中<math>U\left(\left[-L, L\right]^n\right) </math>表示范围在<math>\left[-L ,L\right] </math>上的<math>n </math>维均匀分布，<math>\det </math>表示函数<math>f </math>的雅可比行列式。维度平均EI为：

其中<math>U\left(\left[-L, L\right]^n\right) </math>表示范围在<math>\left[-L ,L\right] </math>上的<math>n </math>维均匀分布，<math>\det </math>表示函数<math>f </math>的雅可比行列式。维度平均EI为：

<math>

<math>

\begin{gathered}

\begin{gathered}

\mathcal{J}\equiv \frac{EI(f)}{n}\approx -\frac{1+\ln(2\pi)+\frac{1}{n}\cdot\sum_{i=1}^n\ln\sigma_i^2}{2}+\ln(2L)+\frac{1}{n}\cdot\mathbb{E}_{x\sim U([-L,L]^n)}(\ln|\det(\partial_{x^{\prime}}f(x)))|)

\mathcal{J}\equiv \frac{EI(f)}{n}\approx -\frac{\ln(2\pi e)}{2}-\frac{\sum_{i=1}^n\ln\sigma_i}{n}+\ln(2L)+\frac{1}{n}\cdot\mathbb{E}_{x\sim U([-\frac{L}{2},\frac{L}{2}]^n)}(\ln|\det(\partial_{x^{\prime}}f(x)))|)

\end{gathered}  

\end{gathered}  

</math>

</math>