In the following equation, $$\large R(Y) = \lambda \left\| \frac {x^\top Y} {\|x\|} \right\|^2_2$$ What do the double bars with range $2$ to $2$ on the right hand side mean? I need to work with this equation, but I don't understand what it is doing. Note, $x$ and $Y$ are both matrices, and $\lambda$ is just a constant.
What does do double vertical bars with numerical parameters on the right hand side mean
notation
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I do not know the details of the adversarial networks however I can offer a general answer for probability theory which might be close to the answer.
In a measure-theoretic setting $P(A||\mathscr{G})$ is sometimes written to denote the conditional probability of the event $A$ with respect to the $\sigma$-field $\mathscr{G}$ where $P$ is a probability measure on the measurable space $(\Omega,\mathscr{F})$ where $\mathscr{F}$ is a larger $\sigma$-field satisfying $\mathscr{G}\subseteq\mathscr{F}$. Random variables $Y$ and $X$ can generate such a $\sigma$-fields, say $X$ generates $\mathscr{G}$ and $Y$ generates $\mathscr{F}$, then $P(A||\mathscr{G})=P(Y\in A||X)$. The specific relationship satisfied is
$$\int_{G}P(Y\in A||X)dP=P(\{Y\in A\}\cap\{X\in G\}) \hspace{10pt}\text{for all}\hspace{10pt} G\in\mathscr{G}\hspace{10pt}(1)$$
The $\hat{p}(y)$ in your equation probably (I am guessing here) denotes an estimate using a sample of random data $Y$ observed at $Y=y$. This estimate $\hat{p}(y)$ will be a random variable so perhaps all of the above will apply and the $||$ notation simply hints at the measure-theoretic machinery I allude to.
In the special case where
$$\int_{G}P(Y\in A||X)dP=P(Y\in A||X)\int_{G}dP\\=P(Y\in A||X)P(X\in G)$$
then the above equation reduces to
$$P(Y\in A||X)dP=P(\{Y\in A\}\cap\{X\in G\})/P(X\in G)\\=P([\{Y\in A\}\cap\{X\in G\}]|X\in G)$$
using the traditional $|$ notation signifying the $P(A,B)/P(B)=P(A|B)$ definition. In general the two definitions do not coincide - I believe $\mathscr{G}$ being generated by a countable class $\mathcal{A}$ might be a sufficient condition, that is $\mathscr{G}=\sigma(\mathcal{A})$ where $|\mathcal{A}|=\aleph_{0}$.
The general formatting of " $X_{y=n_1}^{n_2}\;Z$ " pretty strongly implies "Make a bunch of expressions by replacing $y$ with each of the values from $n_1$ to $n_2$ in $Z$, and then combing those statements using $X$". This is the general pattern of $\Sigma$, $\Pi$, $\bigcap$, etc. I guess the word for it would be "n-ary $X$". There are variations and details that aren't in question here.
In this case, the "$Z$" is an $\exists$ statement; depending on the context you might call that a boolean, or a predicate, or a "statement", but in any case it's consistent with $⋀$ being logical conjunction, as suggested by Tyma Gaidash and Antares in the comments.
I can't definitively rule out an "exterior product" or "wedge product", but neither of them sound like they'd apply to existence expressions.
Best Answer
In the notation $$\left\|x\right\|_\color{blue}{2}^\color{red}{2}$$
See here for the more general $p$-norm, of which this is a special case: $$\left\| x \right\| _p = \left( |x_1|^p + |x_2|^p + \dotsb + |x_n|^p \right) ^{1/p}$$