Harnessing the Universal Geometry of Embeddings

While you can make embeddings across different LLMs similar (i.e. universally looking) the few percentage of R2 that you are missing in translating the universal representation into a native representation are precisely those that make the hidden states executable in the LLM (which makes them useful).

They do this with simple embedding models because there the purpose of the embeddings is to measure similarity, but if you would like to use this principle to turn latent representations of one LLM into latent representations that are understandable/executably by a different LLM, you will fail.

Note this is version 4 of the paper and the original post was version 1 (I think?)

OpenReview (for NeurIPS) for the curious: https://openreview.net/forum?id=jiCLUPq5xv

This, like graph isometry, can be very computationally intensive in the worst case. However, heuristics to aid matching one vertex on one graph to another vertex on another graph using local, semilocal structural signatures can be very effective on particular cases.

One can of course argue that the spaces are not designed as metric spaces. Even if true, these might be metrizable topological spaces.

More generally, if these are indeed non-metric spaces one can still pose it as finding the unknown isomorphism between two poset spaces.

In my other comment I was using the property of maximal chains -- Identify the longest chains in both posets. The isomorphism must map the longest chain in Poset 1 directly to a longest chain in Poset 2, preserving the exact linear order.

Without knowing the details of how the paper solved the problem, my first attempt would be to find the diametrically distant pair of points in the two different embeddings and assume that the pair is the same pair. Then find the next distant pairs and so on.

After sufficiently many such pairs have been found, or better still, the largest d-simplex is found, find that scaled rigid body transformation that makes the corresponding pairs coincide. Proceeding this way ought to be less work than solving a generic graph isomorphism problem.

But the idea makes sense, of course there is still recoverable data in embeddings, that's the point. Though as I constantly find the more you try to squeeze into an n bit vector the more watered down everything gets.

I suppose a latent space could be encrypted/mapped in some way to resolve that, but how many people are exposing their vectors in the first place?

And if the LLM has been trained up to the limit of what data it can hold, it is going to be random. Proof below if it isn't obvious.

The entire effort of all people who are trying to understand how LLMs work, how they represent their data, its all bound to fail.

Proof: a LLM is a very good approximation of the Solomonov/Levin/Kolmogorov universal probability function on tokens. As such, it will be random--pure white noise--because if you found any patterns in there, you could exploit the regularity and come up with a smaller set of weights for the same LLM.

There are no patterns there to be found. They have all been factored out by training the neural net until it couldn't learn any more.