I played with this with a colab notebook way back when. I can’t visualize things directly in 4 dimensions, but at the time I came up with the trick of visualizing the pairwise cosine similarity for each pair of features, which gives at least a local sense of what the angles are like.
Trying to squish 9 features into 4 dimensions looks to me like it either ends up with
4 antipodal pairs which are almost orthogonal to one another, and then one “orphan” direction squished into the largest remaining space
OR
3 almost orthogonal antipodal pairs plus a “Y” shape with the narrow angle being 72º and the wide angles being 144º
For reference this is what a square antiprism looks like in this type of diagram:
You maybe got stuck in some of the many local optima that Nurmela 1995 runs into. Genuinely, the best sphere code for 9 points in 4 dimensions is known to have a minimum angular separation of ~1.408 radians, for a worst-case cosine similarity of about 0.162.
You got a lot further than I did with my own initial attempts at random search, but you didn’t quite find it, either.
I played with this with a colab notebook way back when. I can’t visualize things directly in 4 dimensions, but at the time I came up with the trick of visualizing the pairwise cosine similarity for each pair of features, which gives at least a local sense of what the angles are like.
Trying to squish 9 features into 4 dimensions looks to me like it either ends up with
4 antipodal pairs which are almost orthogonal to one another, and then one “orphan” direction squished into the largest remaining space
OR
3 almost orthogonal antipodal pairs plus a “Y” shape with the narrow angle being 72º and the wide angles being 144º
For reference this is what a square antiprism looks like in this type of diagram:
You maybe got stuck in some of the many local optima that Nurmela 1995 runs into. Genuinely, the best sphere code for 9 points in 4 dimensions is known to have a minimum angular separation of ~1.408 radians, for a worst-case cosine similarity of about 0.162.
You got a lot further than I did with my own initial attempts at random search, but you didn’t quite find it, either.