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Harnessing the Universal Geometry of Embeddings

arxiv.org

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Re: Harnessing the Universal Geometry of Embeddings

#11

Post by srean »

jxmorris12 wrote: post #3

Hi HN, I'm Jack, the last author of this paper. It feels good to release this, the fruit of a two-year quest to "align" two vector spaces without any paired data. It's fun to look back a bit and note that at least two people told me this wasn't possible: 1. An MIT professor who works on similar geometry alignment problems didn't want to work on this with me because he was certain we would need at least a little bit o…

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.

Re: Harnessing the Universal Geometry of Embeddings

#12

Post by jxmorris12 »

srean wrote: post #11
jxmorris12 wrote: post #3

Hi HN, I'm Jack, the last author of this paper. It feels good to release this, the fruit of a two-year quest to "align" two vector spaces without any paired data. It's fun to look back a bit and note that at least two people told me this wasn't possible: 1. An MIT professor who works on similar geometry alignment problems didn't want to work on this with me because he was certain we would need at least a little bit o…

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.

Re: Harnessing the Universal Geometry of Embeddings

#13

Post by jxmorris12 »

Lerc wrote: post #10
jxmorris12 wrote: post #3

Hi HN, I'm Jack, the last author of this paper. It feels good to release this, the fruit of a two-year quest to "align" two vector spaces without any paired data. It's fun to look back a bit and note that at least two people told me this wasn't possible: 1. An MIT professor who works on similar geometry alignment problems didn't want to work on this with me because he was certain we would need at least a little bit o…

I must admit reading the abstract made me think to myself that I should read the paper in skeptical mode. Does this extend to being able to analytically determine which concepts are encodable in one embedding but not another? An embedding from a deft tiny stories LLM presumably cannot encode concepts about RNA replication. Assuming that is true. If you can detect when you are trying to put a square peg into a round h…

Very fair!

> Does this extend to being able to analytically determine which concepts are encodable in one embedding but not another? An embedding from a deft tiny stories LLM presumably cannot encode concepts about RNA replication.

Yeah, this is a great point. We're mostly building off of this prior work on the Platonic Representation Hypothesis (https://arxiv.org/abs/2405.07987). I think our findings go-so-far as to apply to large-enough models that are well-enough trained on The Internet. So, text and images. Maybe audio, too, if the audio is scraped from the Internet.

So I don't think your tinystories example qualifies for the PRH, since it's not enough data and it's not representative of the whole Internet. And RNA data is (I would guess) something very different altogether.

> Assuming that is true. If you can detect when you are trying to put a square peg into a round hole, does this mean you have the ability to remove square holes from a system?

Not sure I follow this part.

Re: Harnessing the Universal Geometry of Embeddings

#14

Post by srean »

jxmorris12 wrote: post #12
srean wrote: post #11

Earlier quoted context omitted.

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.

Re: Harnessing the Universal Geometry of Embeddings

#15

Post by jxmorris12 »

srean wrote: post #14
jxmorris12 wrote: post #12

Earlier quoted context omitted.

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…

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.

Re: Harnessing the Universal Geometry of Embeddings

#16

Post by srean »

jxmorris12 wrote: post #15
srean wrote: post #14

Earlier quoted context omitted.

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…

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.

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.

Re: Harnessing the Universal Geometry of Embeddings

#17

Post by jxmorris12 »

srean wrote: post #16
jxmorris12 wrote: post #15

Earlier quoted context omitted.

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.

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.

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.

Re: Harnessing the Universal Geometry of Embeddings

#19

Post by nimish »

jxmorris12 wrote: post #3

Hi HN, I'm Jack, the last author of this paper. It feels good to release this, the fruit of a two-year quest to "align" two vector spaces without any paired data. It's fun to look back a bit and note that at least two people told me this wasn't possible: 1. An MIT professor who works on similar geometry alignment problems didn't want to work on this with me because he was certain we would need at least a little bit o…

Hooray, finally we are getting the geometric analysis of embedding spaces we need. Information geometry and differential geometry is finally getting its moment in the sun!

Re: Harnessing the Universal Geometry of Embeddings

#20

Post by Lerc »

jxmorris12 wrote: post #13
Lerc wrote: post #10

Earlier quoted context omitted.

I must admit reading the abstract made me think to myself that I should read the paper in skeptical mode. Does this extend to being able to analytically determine which concepts are encodable in one embedding but not another? An embedding from a deft tiny stories LLM presumably cannot encode concepts about RNA replication. Assuming that is true. If you can detect when you are trying to put a square peg into a round h…

Very fair! > Does this extend to being able to analytically determine which concepts are encodable in one embedding but not another? An embedding from a deft tiny stories LLM presumably cannot encode concepts about RNA replication. Yeah, this is a great point. We're mostly building off of this prior work on the Platonic Representation Hypothesis ( https://arxiv.org/abs/2405.07987 ). I think our findings go-so-far as…

>So I don't think your tinystories example qualifies for the PRH, since it's not enough data and it's not representative of the whole Internet. And RNA data is (I would guess) something very different altogether.

My thought there was that you'd be comparing tinystories to a model that trained on the entire internet. The RNA related information would be a subset of the second representation that has no comparable encoding in the tinystories space. Can you detect that? If both models have to be of sufficient scale to work the question becomes "what is the scale, is it sliding or a threshold? "

>> Assuming that is true. If you can detect when you are trying to put a square peg into a round hole, does this mean you have the ability to remove square holes from a system?

>Not sure I follow this part.

Perhaps the metaphor doesn't work so well. If you can detect if something is encodable in one embedding model but not another. Can you then leverage that detection ability in order to modify an embedding model so that it cannot represent an idea.

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