I have a unit radius sphere with a set $S$ of $n$ points on it. How can I find a map $f:S\to \mathbb{R}^4$ which minimizes $$\sum_{x,y\in S} \bigg( d_{\text{geodesic}} (x,y)^{2} - d(f(x),f(y))^{2}\bigg)^2$$
$d_{\text{geodesic}}(x,y)$ is the length of the shortest path on the sphere from $x$ to $y$, and $d(f(x),f(y))$ is the Euclidean distance in $\mathbb{R}^4$.$\hspace{-0.03 in}\big)$