Yes. So the idea was that an orthogonal rotation will 'encrypt' the embeddings without affecting performance, since orthogonality preserves cosine similarity. It's a good idea, but we can un-rotate the embeddings using our GAN.
EDIT: Perfect symmetries, for example, feature-less spheres, or the analogues of platonic solids would break this. If the embedded space has no geometric symmetries you would be in business.
Re-aligning, essentially would be akin to solving a graph-isomorphism problem.
Lie algebraic formulation would make it less generic than an arbitrary graph-isomorphism problem. Essentially reduce it to a high dimensional procrustes problem. Generic graph isomorphism can be quite a challenge.
https://en.m.wikipedia.org/wiki/Procrustes_analysis
EDIT: Sinkhorn balancing over a set of points (say a d-dimensional tetrahedron, essentially a simplex) furthest from each other might be a good first cut to try. You might have already done so, I haven't read your paper yet.
The GAN works way better than traditional OT methods though. I really don't know why, this is the part that feels like magic to me.
All the best.
I can totally understand the professors point, little bit of alignment data ought significantly increase the chance of success. Otherwise it will have to rely on these small deviations from symmetry to anchor the orientation.