One way to pose/(think about) the problem is that there are two finite metric spaces linked by an unknown odometry (damn you autocorrect). The problem is to recover that unknown isometry.

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.

Dupe: https://news.ycombinator.com/item?id=44054425

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

I am not familiar with the standards of publishing in machine learning, but as someone trained in a mathematics background, this paper seems relatively light on details and heavy on exposition. Is that typical? Is this a really novel idea? Not trying to be snarky, just trying to understand how meaningful this is.

Let's assume that monotonocity of pair-wise distances are preserved.

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.

I’ve never liked that this was called “the platonic representation hypothesis”. Lots of weird baggage attached and seems like a waste of a good name.

Cyberphrenology. In any two random graphs, you'll find an isomorphic graph which is can be up to log of the size of the graphs.

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.

What is the (co)homology of this space?

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