(Statistics) Trying to understand an equation

Question

I'm trying to understand the equation below from this paper.

But there is 2 things I don't quite understand (I'm not a math expert):

• $E_{z,y}\sim_{P_{data(z, y)}}$, means $E_{z,y}$ follows a data distribution of $_{P_{data(z, y)}}$ ? What does that mean exactly?
• $||\phi_i(y) - \phi_i(G(z))||$, the double $||$ means: "take the absolute values of $\phi_i(y) - \phi_i(G(z))$" right?

Thanks you for your help.

Answer 1

1. $E_{z,y}$ means the expected value of $\|\phi_i(y)-\phi_i(G(z))\|$, where the expectation is taken over both $y$ and $z$. It is $y$ and $z$ that are distributed as $P_{data(y,z)}$.
2. $\|\phi_i(y)-\phi_i(G(z))\|$ is the $L^2$ norm of $\phi_i(y)-\phi_i(G(z))$. That is the quare root of the sum of squares.