I want to do an integral ${x}$ over multidimensional Gaussian $e^{-|y|^2}$ where $y = LeakyRelu(Mx + b)$. Eventually, I want to do this integral after multiple transformations (i.e. $y_{i+1}= LeakyRelu(M_{i} y_{i} + b_i)$ and $y_0 = x$) but for now I want to make sure it's doable after a single transformation.
The element-wise non-linearity of LeakyRelu is causing a lot of trouble for me. Since $LeakyRelu(x)$ multiplies $x$ by a constant for $x < 0$, I'm thinking I have to do two integrals per element of $y$ ($y \geq 0$, $y < 0$), which makes this exponential in dimension of $y$. If there is any shortcut here that would be nice, but I don't expect it to exist. Also, since $y$ is generally a linear combination of $x$ plus a shift, the bounds of integral are not independent.
Any idea on how I could approach this problem? I don't want to go the Monte Carlo route.