What is the "Cumulative Distribution of the magnitude of the N-dimensional standard gaussian"

Question

I am confused by this line from a paper:

"Let $F_1(x)$ be the cumulative distribution of the magnitude of an $n$−dimensional standard Gaussian random variable and $F_2(x)$ be the cumulative distribution of the magnitude of a random point in a Euclidean unit ball of radius $r$"

Can someone explain to me what $F_1(x)$ and $F_2(x)$ is? Thanks a lot!

The paper is here: http://www.mit.edu/~har/Dikin.pdf (page 9)

I would also really really appreciate it if someone has the time to explain to me how to sample from this Dikin ellipsoid. I think I've fully understood the definitions, but the part about the pseudo-inverse and vector scaling seems confusing.

Thank you!

Edit: I know how to sample points from the $n$-dimensional Gaussian and the $n$-dimensional Euclidean ball. But it seems to me that $F_1$ and $F_2$ are invertible functions?

Answer 1

Let $X$ be a standard $n$ dimensional Gaussian. Then $|X|$ is a random variable, and thus it has a distribution function. The author is saying let $F_1$ be that distribution function, i.e. $$ F_1(t) = P(|X| \leq t) $$

Similarly, the Euclidean ball of radius $r$ has finite volume and so we can consider the uniform measure $\mu$ on the ball (it will be a rescaling of the usual Lebesgue measure). Let $Y$ be a random vector sampled from $\mu$. Then $|Y|$ is a random variable and so it has a distribution function. The author is saying let $F_2$ be that distribution function, i.e. $$ F_2(t) = P(|Y| \leq t) $$

Edit: As for invertibilitity, you are correct that they are invertible. This is because $F_1,F_2$ are strictly increasing (a fact that you need to prove).

I suggest you make a separate, more specific, question about any technicalities that stump you in the paper.