the orthogonal group is very simple

Soledad Villar (JHU) and I have been kicking around ideas for machine learning methods that are tailored to classical (mechanical and electromagnetic) physical systems. The question is: What is the simplest representation of objects in this theory that permits highly expressive machine-learning methods but constrained to obey fundamental symmetries, like translation, rotation, reflection, and boost. Since almost all (maybe exactly all) of classical physics obeys rotation and reflection, one of the relevant groups is the orthogonal group O(3) (or O(d) in general). This group turns out to be extremely simple (and extremely constrained). We might be able to make extremely expressive machines with very simple internals, if we have this group deliver the main symmetry or equivariance. We played around with possible abstracts or scopes for a paper. Yes, a purely theoretical paper for machine learning. That puts me out of my comfort zone! We also read some group theory, which I (hate to admit that I) find very confusing.