Question about Gaussian operations in different regions

[Xiaohan-Z]

Thank you for your outstanding work!
I have a question about global Gaussian calculation. The number and positions of Gaussians in each local region are different. How does this formula $z^{t+1}:=\frac{1}{K}\sum_{k=1}^{K}(x_i^k)^{t+1}$ sum up the local Gaussians?