We can take $A \in {\mathbb{R}}^{m \times n}$ such that every entry is IID Gaussian scaled so that $A_{i,j} \sim \mathcal{N}\left(0, \frac{1}{m}\right)$. Then for any $x \in \mathbb{R}^n$, we get that $\frac{\|Ax\|^2}{\|x\|^2}$ concentrates around $1$, as we increase $m$. In particular it is $subgamma\left(\frac{1}{\sqrt{m}}, \frac{1}{m}\right)$.
Instead if we allow a non-linear function $f(\cdot)$, does that allow us to improve the concentration?
In other words, are there examples of a sequence of family of functions $F_m$, such that if we randomly sample $f_m : \mathbb{R}^n \rightarrow \mathbb{R}^m$ from $F_m$, then $\frac{\|f_m(x_1) - f_m(x_2)\|^2}{\|x_1 - x_2\|^2}$ concentrates around $1$ faster (as a function of $m$)?
This paper proves that it is not possible to improve the concentration (ignoring logarithmic factors).