I am trying to understand this paper, but in this paper the following "$\|$" symbol is frequently used.
$$p_i(j) \propto \exp \{ - \| A x_i - A x_j \|_2^2\}$$
I wonder to know what does it mean, what are these two "$2$"s doing here?
And also wonder to know does the above "$\|$" symbol has different meaning than the below "$\|$" symbol (considering there is only one "$2$" here)?
$$p_i(j) \propto \exp \left\{ -\dfrac{\|x_i - x_j\|^2}{2 \sigma_i^2} \right\}$$
Thanks in advance.