Item — HN Reader

Slightly off topic: does anyone have good intros to hyperbolic geometry? I was imitating drawn by applications related to hyperbolic embeddings in NLP, but I’m having a hard time understanding why, in what ways and how it could be “better” than euclidean etc

Fun fact: just this week I asked gpt3.5 examples of concrete applications of hyperbolic geometry and it suggested designing transportation networks. When I asked how so, the explanation was that subway lines could make sharper turns in hyperbolic geometry.

2 comments

You’re adding another dimension. Sort of a “half” dimension. If you want a distance function in a space that’s non-Euclidean, you’re likely (always?) going to have something you can represent as hyperbolic geometry. Check out metric spaces and measures on Wikipedia.

For a “vibes” intro, The Hyperbolic Geometry of DMT experiences has a good intro in the beginning. https://youtu.be/loCBvaj4eSg

I’m no expert on hyperbolic geometry. These things just helped me “get it” in the sense of “I get why that’s a thing”. Also, seeing it for yourself doesn’t hurt.

Hyperbolic plane has tree-like structure. If you tried to draw a binary tree of depth 10 on Euclidean paper, you would run out of space. In hyperbolic plane, the straight lines are diverging and new space is created exponentially (very roughly speaking), so such a tree would fit perfectly. This makes hyperbolic embeddings better than Euclidean ones for all kinds of hierarchical data. I would also like to mention that while the uses in machine learning get lots of hype, the ML researchers seem to not be aware of the earlier impressive results on hyperbolic embeddings obtained in other communities (social network analysis/algorithms).

If you want to get intuitions about how this tree-like structure works, it is the best to play HyperRogue. For formal math, I guess it is the best to read the relevant papers.

(I guess the hierarchical structure of transportation networks, from hub airports -> ordinary airports -> major roads -> minor roads, could be interpreted as hyperbolic geometry, in the same way as Internet has hyperbolic geometry. A very liberal and abstract interpretation though. I do not see how making sharper turns makes sense.)