Coupling Flows

• Consider a continuous random variable $z\in {\mathbb{R}}^D$
• Partition z into two subsets : eg two contiguous sub-vectors using a function g
• Open: Pasted image 20241119165521.png
• \begin{align*} x_{1:d}= z_{1:d} \\ x_{d+1:D}= g(z_{d+1:D}; m(z_{1:d})) \end{align*}
• elements in the first subset are not affected by those in the second
• to invert them is trivial \begin{align*} z_{1:d}= x_{1:d} \\ z_{d+1:D}= g^{-1}(x_{d+1:D}; m(x_{1:d})) \end{align*}
• Additive coupling layer
• Scaling matrix for coupling layers