Comment by srean
2 months ago
Doesn't the space of embeddings have some symmetries, that when applied does not change the output sequence ?
For example, global rotation that does not change embedded vector x embedded vector dot-product and changes query vector x embedded dot-product in an equivariant way.
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.
I can understand that two relatively rotated embeddings from the same or similar dataset can be realigned as long as they don't have internal geometric symmetries. The same way we can re-align two globes -- look for matching shapes, continents.
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.
Right, that's why the baselines here come from the land of Optimal Transport, which looks at the world through isomorphisms, exactly as you've suggested.
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.
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