Interpreting SO(n) multiplication: deep networks generalize by learning an algorithm

We reverse engineer networks trained on SO(n) multiplication (the rotations of the $n$-dimensional sphere) and discover that adding layers results in networks learning clean generalizing algorithms that are not learned by shallow 1- or 2-hidden-layer networks. In deep networks learning SO(3), we discover this task is solved by neurons learning to activate on different axes of the sphere. These `axial'-neurons combine information to compute the answer to the multiplication via Rodrigues' Rotation Formula, which is an efficient algorithm used to compute the rotation of a vector in space, given an axis and angle of rotation. The story is similar in higher dimensions, but neurons learn to rotate about planes instead of an axis. To our knowledge, this is the first Lie Group that has been reverse-engineered and interpreted, whereas prior work has focused on finite groups, e.g., modular addition (cyclic group), alternating groups, etc.