I have few points in $S^n$, i.e., the $n$-dimensional unit sphere embedded in $\mathbb{R}^{n+1}$, and I would like to project them down to $S^2$, i.e., the 2-dimensional unit sphere (embedded in $\mathbb{R}^3$) to visualize it with the constraint that neighboring points should be close by. I spent some time playing with t-sne but of course, the points no longer lie on $S^2$. I normalized the projections but that introduces weird distortions, for instance, if the variance of one dataset is very small in $S^n$ as compared to other, I expect the same to hold in their $S^2$ projections; that is not the case upon normalizing t-sne. Any ideas? I would really like something that makes the previous statement hold.
Projecting data from $S^n$ to $S^2$
machine learning;dimensionality reduction;visualization