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Yes. So the idea was that an orthogonal rotation will 'encrypt' the embeddings without affecting performance, since orthogonality preserves cosine similarity. I

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.

srean 1y ago
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 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.
- jxmorris12 1y ago
  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.
  - srean 1y ago
    Got you. I can understand that this has a chance of working if the embeddings have converted to their global optimum. Otherwise all bets ought to be off. 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.
    - jxmorris12 1y ago
      Yeah, we didn't get around to testing what the impact would be of having a small amount of aligned data. I've seen other papers asserting that as few as five pairs is enough to go a long way.

srean 1y ago

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 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.

jxmorris12 1y ago
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.
- srean 1y ago
  Got you. I can understand that this has a chance of working if the embeddings have converted to their global optimum. Otherwise all bets ought to be off. 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.
  - jxmorris12 1y ago
    Yeah, we didn't get around to testing what the impact would be of having a small amount of aligned data. I've seen other papers asserting that as few as five pairs is enough to go a long way.